Stochastic Mechanics
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and Image Synthesis
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Stochastic Optimization
Stochastic Control
Stochastic Models in Life Sciences
Edited by
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Applications of
Mathematics
Stochastic Modelling
and Applied Probability
21
B. Rozovskii
M.Yor
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Applications of Mathematics
1 Fleming/Rishel, Deterministic and Stochastic Optimal Control A975)
2 Marchuk, Methods of Numerical Mathematics 1975,2nd. ed. 1982)
3 Balakrishnan, Applied Functional Analysis A976,2nd. ed. 1981)
4 Borovkov, Stochastic Processes in Queueing Theory A976)
5 Liptser/Shiryaev, Statistics of Random Processes I: General Theory A977,2nd. ed. 2001)
6 Liptser/Shiryaev, Statistics of Random Processes II: Applications A978,2nd. ed. 2001)
7 Vorob'ev, Game Theory: Lectures for Economists and Systems Scientists A977)
8 Shiryaev, Optimal Stopping Rules A978)
9 Ibragimov/Rozanov, Gaussian Random Processes A978)
10 Wonham, Linear Multivariable Control: A Geometric Approach A979, 2nd. ed. 1985)
11 Hida, Brownian Motion A980)
12 Hestenes, Conjugate Direction Methods in Optimization A980)
13 Kallianpur, Stochastic Filtering Theory A980)
14 Krylov, Controlled Diffusion Processes A980)
15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams A980)
16 Ibragimov/Has'minskii, Statistical Estimation: Asymptotic Theory A981)
17 Cesari, Optimization: Theory and Applications A982)
18 Elliott, Stochastic Calculus and Applications A982)
19 Marchuk/Shaidourov, Difference Methods and Their Extrapolations A983)
20 Hijab, Stabilization of Control Systems A986)
21 Protter, Stochastic Integration and Differential Equations A990,2nd. ed. 2003)
22 Benveniste/Metivier/Priouret, Adaptive Algorithms and Stochastic Approximations A990)
23 Kloeden/Platen, Numerical Solution of Stochastic Differential Equations
A992, corr. 3rd printing 1999)
24 Kushner/Dupuis, Numerical Methods for Stochastic Control Problems in Continuous
Time A992)
25 Fleming/Soner, Controlled Markov Processes and Viscosity Solutions A993)
26 Baccelli/Bremaud, Elements of Queueing Theory A994,2nd ed. 2003)
27 Winkler, Image Analysis, Random Fields and Dynamic Monte Carlo Methods
A995,2nd. ed. 2003)
28 Kalpazidou, Cycle Representations of Markov Processes A995)
29 Elliott/Aggoun/Moore, Hidden Markov Models: Estimation and Control A995)
30 Hernandez-Lerma/Lasserre, Discrete-Time Markov Control Processes A995)
31 Devroye/Gyorfi/Lugosi, A Probabilistic Theory of Pattern Recognition A996)
32 Maitra/Sudderth, Discrete Gambling and Stochastic Games A996)
33 Embrechts/Kluppelberg/Mikosch, Modelling Extremal Events for Insurance and Finance
A997. corr. 4th printing 2003)
34 Duflo, Random Iterative Models A997)
35 Kushner/Yin, Stochastic Approximation Algorithms and Applications A997)
36 Musiela/Rutkowski, Martingale Methods in Financial Modelling A997)
37 Yin, Continuous-Time Markov Chains and Applications A998)
38 Oembo/Zeitouni, Large Deviations Techniques and Applications A998)
39 Karatzas, Methods of Mathematical Finance A998)
40 Fayolle/Iasnogorodski/Malyshev, Random Walks in the Quarter-Plane A999)
41 Aven/Jensen, Stochastic Models in Reliability A999)
42 Hernandez-Lerma/Lasserre, Further Topics on Discrete-Time Markov Control Processes
A999)
43 Yong/Zhou, Stochastic Controls. Hamiltonian Systems and HJB Equations A999)
44 Serfozo, Introduction to Stochastic Networks A999)
45 Steele, Stochastic Calculus and Financial Applications B001)
46 Chen/Yao, Fundamentals of Queuing Networks: Performance, Asymptotics,
and Optimization B001)
47 Kushner, Heavy Traffic Analysis of Controlled Queueing and Communications Networks
B001)
48 Fernholz, Stochastic Portfolio Theory B002)
49 Kabanov/Pergamenshchikov, Two-Scale Stochastic Systems B003)
50 Han, Information-Spectrum Methods in Information Theory B003)
(continued after index)

Philip E. Protter
Stochastic
Integration
and Differential
Equations
Second Edition
Springer

Author
Philip E. Protter
Cornell University
School of Operations Res. and
Industrial Engineering
Rhodes Hall
14853 Ithaca, NY
USA
e-mail: pep4@c0rnell.edu
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B. Rozovskii M. Yor
Center for Applied Mathematical Universite de Paris VI
Sciences Laboratoire de Probabilites
University of Southern California et Modeles Aleatoires
1042 West 36th Place, 175, rue du Chevaleret
Denney Research Building 308 75013 Paris, France
Los Angeles, CA 90089, USA
Mathematics Subject Classification B000): Primary: 60H05,60H10,60H20
Secondary: 60G07,60G17,60G44,60G51
Cover pattern by courtesy of Rick Durrett (Cornell University, Ithaca)
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Printed on acid-free paper 41/3142db-543210

To Diane and Rachel

Preface to the Second Edition
It has been thirteen years since the first edition was published, with its subtitle
"a new approach." While the book has had some success, there are still almost
no other books that use the same approach. (See however the recent book by
K. Bichteler [15].) There are nevertheless of course other extant books, many
of them quite good, although the majority still are devoted primarily to the
case of continuous sample paths, and others treat stochastic integration as
one of many topics. Examples of alternative texts which have appeared since
the first edition of this book are: [32], [44], [87], [110], [186], [180], [208], [216],
and [226]. While the subject has not changed much, there have been new
developments, and subjects we thought unimportant in 1990 and did not
include, we now think important enough either to include or to expand in this
book.
The most obvious changes in this edition are that we have added exercises
at the end of each chapter, and we have also added Chap. VI which intro-
introduces the expansion of nitrations. However we have also completely rewritten
Chap. III. In the first edition we followed an elementary approach which was
P. A. Meyer's original approach before the methods of Doleans-Dade. In or-
order to remain friends with Freddy Delbaen, and also because we now agree
with him, we have instead used the modern approach of predictability rather
than naturality. However we benefited from the new proof of the Doob-Meyer
Theorem due to R. Bass, which ultimately uses only Doob's quadratic martin-
martingale inequality, and in passing reveals the role played by totally inaccessible
stopping times. The treatment of Girsanov's theorem now includes the case
where the two probability measures are not necessarily equivalent, and we
include the Kazamaki-Novikov theorems. We have also added a section on
compensators, with examples. In Chap. IV we have expanded our treatment
of martingale representation to include the Jacod-Yor Theorem, and this has
allowed us to use the Emery-Azema martingales as a class of examples of mar-
martingales with the martingale representation property. Also, largely because of
the Delbaen-Schachermayer theory of the fundamental theorems of mathe-
mathematical finance, we have included the topic of sigma martingales. In Chap. V

VIII Preface to the Second Edition
we added a section which includes some useful results about the solutions of
stochastic differential equations, inspired by the review of the first edition by
E. Pardoux [191]. We have also made small changes throughout the book;
for instance we have included specific examples of Levy processes and their
corresponding Levy measures, in Sect. 4 of Chap. I.
The exercises are gathered at the end of the chapters, in no particular
order. Some of the (presumed) harder problems we have designated with a
star (*), and occasionally we have used two stars (**). While of course many
of the problems are of our own creation, a significant number are theorems
or lemmas taken from research papers, or taken from other books. We do not
attempt to ascribe credit, other than listing the sources in the bibliography,
primarily because they have been gathered over the past decade and often
we don't remember from where they came. We have tried systematically to
refrain from relegating a needed lemma as an exercise; thus in that sense the
exercises are independent from the text, and (we hope) serve primarily to
illustrate the concepts and possible applications of the theorems.
Last, we have the pleasant task of thanking the numerous people who
helped with this book, either by suggesting improvements, finding typos and
mistakes, alerting me to references, or by reading chapters and making com-
comments. We wish to thank patient students both at Purdue University and
Cornell University who have been subjected to preliminary versions over the
years, and the following individuals: C. Benes, R. Cont, F. Diener, M. Di-
ener, R. Durrett, T. Fujiwara, K. Giesecke, L. Goldberg, R. Haboush, J. Ja-
cod, H. Kraft, K. Lee, J. Ma, J. Mitro, J. Rodriguez, K. Schurger, D. Sezer,
J. A. Trujillo Ferreras, R. Williams, M. Yor, and Yong Zeng. Th. Jeulin,
K. Shimbo, and Yan Zeng gave extraordinary help, and my editor C. Byrne
gives advice and has patience that is impressive. Over the last decade I have
learned much from many discussions with Darrell Duffle, Jean Jacod, Tom
Kurtz, and Denis Talay, and this no doubt is reflected in this new edition.
Finally, I wish to give a special thanks to M. Kozdron who hastened the ap-
appearance of this book through his superb help with ВД^Х, as well as his own
advice on all aspects of the book.
Ithaca, NY Philip Protter
August 2003

Preface to the First Edition
The idea of this book began with an invitation to give a course at the Third
Chilean Winter School in Probability and Statistics, at Santiago de Chile, in
July, 1984. Faced with the problem of teaching stochastic integration in only
a few weeks, I realized that the work of C. Dellacherie [42] provided an outline
for just such a pedagogic approach. I developed this into a series of lectures
(Protter [201]), using the work of K. Bichteler [14], E. Lenglart [145] and
P. Protter [202], as well as that of Dellacherie. I then taught from these lecture
notes, expanding and improving them, in courses at Purdue University, the
University of Wisconsin at Madison, and the University of Rouen in France.
I take this opportunity to thank these institutions and Professor Rolando
Rebolledo for my initial invitation to Chile.
This book assumes the reader has some knowledge of the theory of stochas-
stochastic processes, including elementary martingale theory. While we have recalled
the few necessary martingale theorems in Chap. I, we have not provided
proofs, as there are already many excellent treatments of martingale the-
theory readily available (e.g., Breiman [23], Dellacherie-Meyer [45, 46], or Ethier-
Kurtz [71]). There are several other texts on stochastic integration, all of which
adopt to some extent the usual approach and thus require the general theory.
The books of Elliott [63], Kopp [130], Metivier [158], Rogers-Williams [210]
and to a much lesser extent Letta [148] are examples. The books of McK-
ean [153], Chung-Williams [32], and Karatzas-Shreve [121] avoid the general
theory by limiting their scope to Brownian motion (McKean) and to contin-
continuous semimartingales.
Our hope is that this book will allow a rapid introduction to some of the
deepest theorems of the subject, without first having to be burdened with the
beautiful but highly technical "general theory of processes."
Many people have aided in the writing of this book, either through dis-
discussions or by reading one of the versions of the manuscript. I would like to
thank J. Azema, M. Barlow, A. Bose, M. Brown, C. Constantini, C. Dellache-
Dellacherie, D. Duffie, M. Emery, N. Falkner, E. Goggin, D. Gottlieb, A. Gut, S. He,
J. Jacod, T. Kurtz, J. de Sam Lazaro, R. Leandre, E. Lenglart, G. Letta,

X Preface to the First Edition
S. Levantal, P. A. Meyer, E. Pardoux, H. Rubin, T. Sellke, R. Stockbridge,
C. Strieker, P. Sundar, and M. Yor. I would especially like to thank J. San Mar-
Martin for his careful reading of the manuscript in several of its versions.
Svante Janson read the entire manuscript in several versions, giving me
support, encouragement, and wonderful suggestions, all of which improved
the book. He also found, and helped to correct, several errors. I am extremely
grateful to him, especially for his enthusiasm and generosity.
The National Science Foundation provided partial support throughout the
writing of this book.
I wish to thank Judy Snider for her cheerful and excellent typing of several
versions of this book.
Philip Protter

Contents
Introduction 1
I Preliminaries 3
1 Basic Definitions and Notation 3
2 Martingales 7
3 The Poisson Process and Brownian Motion 12
4 Levy Processes 19
5 Why the Usual Hypotheses? 34
6 Local Martingales 37
7 Stieltjes Integration and Change of Variables 39
8 Naive Stochastic Integration Is Impossible 43
Bibliographic Notes 44
Exercises for Chapter I 45
II Semimartingales and Stochastic Integrals 51
1 Introduction to Semimartingales 51
2 Stability Properties of Semimartingales 52
3 Elementary Examples of Semimartingales 54
4 Stochastic Integrals 56
5 Properties of Stochastic Integrals 60
6 The Quadratic Variation of a Semimartingale 66
7 Ito's Formula (Change of Variables) 78
8 Applications of Ito's Formula 84
Bibliographic Notes 92
Exercises for Chapter II 94
III Semimartingales and Decomposable Processes 101
1 Introduction 101
2 The Classification of Stopping Times 103
3 The Doob-Meyer Decompositions 105
4 Quasimartingales 116

XII Contents
5 Compensators 118
6 The Fundamental Theorem of Local Martingales 124
7 Classical Semimartingales 127
8 Girsanov's Theorem 131
9 The Bichteler-Dellacherie Theorem 143
Bibliographic Notes 147
Exercises for Chapter III 147
IV General Stochastic Integration and Local Times 153
1 Introduction 153
2 Stochastic Integration for Predictable Integrands 153
3 Martingale Representation 178
4 Martingale Duality and the Jacod-Yor Theorem on
Martingale Representation 193
5 Examples of Martingale Representation 200
6 Stochastic Integration Depending on a Parameter 205
7 Local Times 210
8 Azema's Martingale 227
9 Sigma Martingales 233
Bibliographic Notes 235
Exercises for Chapter IV 236
V Stochastic Differential Equations 243
1 Introduction 243
2 The И? Norms for Semimartingales 244
3 Existence and Uniqueness of Solutions 249
4 Stability of Stochastic Differential Equations 257
5 Fisk-Stratonovich Integrals and Differential Equations 270
6 The Markov Nature of Solutions 291
7 Flows of Stochastic Differential Equations: Continuity and
Differentiability 301
8 Flows as Diffeomorphisms: The Continuous Case 310
9 General Stochastic Exponentials and Linear Equations 321
10 Flows as Diffeomorphisms: The General Case 328
11 Eclectic Useful Results on Stochastic Differential Equations . .. 338
Bibliographic Notes 347
Exercises for Chapter V 349
VI Expansion of Filtrations 355
1 Introduction 355
2 Initial Expansions 356
3 Progressive Expansions 369
4 Time Reversal 377
Bibliographic Notes 383
Exercises for Chapter VI 384

Contents XIII
References 389
Symbol Index 403
Subject Index 407

Introduction
In this book we present a new approach to the theory of modern stochastic
integration. The novelty is that we define a semimartingale as a stochastic pro-
process which is a "good integrator" on an elementary class of processes, rather
than as a process that can be written as the sum of a local martingale and an
adapted process with paths of finite variation on compacts: This approach has
the advantage over the customary approach of not requiring a close analysis of
the structure of martingales as a prerequisite. This is a significant advantage
because such an analysis of martingales itself requires a highly technical body
of knowledge known as "the general theory of processes." Our approach has a
further advantage of giving traditionally difficult and non-intuitive theorems
(such as Strieker's Theorem) transparently simple proofs. We have tried to
capitalize on the natural advantage of our approach by systematically choos-
choosing the simplest, least technical proofs and presentations. As an example we
have used К. М. Rao's proofs of the Doob-Meyer decomposition theorems
in Chap. Ill, rather than the more abstract but less intuitive Doleans-Dade
measure approach.
In Chap. I we present preliminaries, including the Poisson process, Brown-
ian motion, and Levy processes. Naturally our treatment presents those prop-
properties of these processes that are germane to stochastic integration.
In Chap. II we define a semimartingale as a good integrator and establish
many of its properties and give examples. By restricting the class of integrands
to adapted processes having left continuous paths with right limits, we are
able to give an intuitive Riemann-type definition of the stochastic integral as
the limit of sums. This is sufficient to prove many theorems (and treat many
applications) including a change of variables formula ("Ito's formula").
Chapter III is devoted to developing a minimal amount of "general the-
theory" in order to prove the Bichteler-Dellacherie Theorem, which shows that
our "good integrator" definition of a semimartingale is equivalent to the usual
one as a process X having a decomposition X — M + A, into the sum of a
local martingale M and an adapted process A having paths of finite variation
on compacts. Nevertheless most of the theorems covered en route (Doob-

2 Introduction
Meyer, Meyer-Girsanov) are themselves key results in the theory. The core
of the whole treatment is the Doob-Meyer decomposition theorem. We have
followed the relatively recent proof due to R. Bass, which is especially simple
for the case where the martingale jumps only at totally inaccessible stopping
times, and in all cases uses no mathematical tool deeper than Doob's quadratic
martingale inequality. This allows us to avoid the detailed treatment of nat-
natural processes which was ubiquitous in the first edition, although we still use
natural processes from time to time, as they do simplify some proofs.
Using the results of Chap. Ill we extend the stochastic integral by continu-
continuity to predictable integrands in Chap. IV, thus making the stochastic integral
a Lebesgue-type integral. We use predictable integrands to develop a theory of
martingale representation. The theory we develop is an I? theory, but we also
prove that the dual of the martingale space Ц} is В MO and then prove the
Jacod-Yor Theorem on martingale representation, which in turn allows us to
present a class of examples having both jumps and martingale representation.
We also use predictable integrands to give a presentation of semimartingale
local times.
Chapter V serves as an introduction to the enormous subject of stochastic
differential equations. We present theorems on the existence and uniqueness
of solutions as well as stability results. Fisk-Stratonovich equations are pre-
presented, as well as the Markov nature of the solutions when the differentials
have Markov-type properties. The last part of the chapter is an introduction
to the theory of flows, followed by moment estimates on the solutions, and
other minor but useful results. Throughout Chap. V we have tried to achieve
a balance between maximum generality and the simplicity of the proofs.
Chapter VI provides an introduction to the theory of the expansion of ni-
nitrations (known as "grossissements de nitrations" in the French literature). We
present first a theory of initial expansions, which includes Jacod's Theorem.
Jacod's Theorem gives a sufficient condition for semimartingales to remain
semimartingales in the expanded filtration. We next present the more diffi-
difficult theory of progressive expansion, which involves expanding nitrations to
turn a random time into a stopping time, and then analyzing what happens
to the semimartingales of the first filtration when considered in the expanded
filtration. Last, we give an application of these ideas to time reversal.

Preliminaries
1 Basic Definitions and Notation
We assume as given a complete probability space (Q, T', P). In addition we are
given a filtration (Tt)o<t<oo- By a filtration we mean a family of cr-algebras
(•^t)o<t<oo that is increasing, i.e., Ts С Tt if s < t. For convenience, we will
usually write F for the filtration (Tt)o<t<oo-
Definition. A filtered complete probability space (fi,T,F, P) is said to sat-
satisfy the usual hypotheses if
(i) To contains all the P-null sets of T;
(ii) Tt = f]u>t Tu, all i, 0 < t < oo; that is, the filtration F is right continuous.
We always assume that the usual hypotheses hold.
Definition. A random variable T : п —> [0, oo] is a stopping time if the
event {T < t} € Tt, every t,0 <t <oo.
One important consequence of the right continuity of the filtration is the
following theorem.
Theorem 1. The event {T < t} € Tt, 0 < t < oo, if and only if T is a
stopping time.
Proof. Since {T < t} = f)t+e>u>t{T < UK апУ ? > 0, we have {T <
t} € f)u>tTu = Tt, so T is a stopping time. For the converse, {T < t} =
{Jt>e>olT < ^ - ?}. and {T<t-e}€ Tts, hence also in Tt. D
A stochastic process X on (fl, T, P) is a collection of R-valued or Re-
Revalued random variables (^Q)o<t<oo- The process X is said to be adapted if
Xt ? Tt (that is, is Tt measurable) for each t. We must take care to be precise
about the concept of equality of two stochastic processes.
Definition. Two stochastic processes X and Y are modifications if Xt =Yt
a.s., each t. Two processes X and Y are indistinguishable if a.s., for all t,
Xt = Yt.

4 I Preliminaries
If X and Y are modifications there exists a null set, Nt, such that if ш $. Nt,
then Xt(u) = Yt{u). The null set Nt depends on t. Since the interval [0, oo)
is uncountable the set TV = Uo<t<oo Nt could have any probability between 0
and 1, and it could even be non-measurable. If X and Y are indistinguishable,
however, then there exists one null set TV such that if w $. N, then Xt{u)) =
Yt(uj), for all t. In other words, the functions t н-> Xt(uj) and t н-» Yt{u>) are
the same for all w $ N, where P(N) = 0. The set TV is in Tt, all t, since J-q
contains all the P-null sets of T. The functions t i-> Xt(w) mapping [0, oo)
into R are called the sample paths of the stochastic process X.
Definition. A stochastic process X is said to be cadlag if it a.s. has sam-
sample paths which are right continuous, with left limits. Similarly, a stochastic
process X is said to be caglad if it a.s. has sample paths which are left
continuous, with right limits. (The nonsensical words cadlag and caglad are
acronyms from the French for continu a droite, limites a gauche and continu
a gauche, limites a droite, respectively.)
Theorem 2. Let X and Y be two stochastic processes, with X a modifica-
modification of Y. If X and Y have right continuous paths a.s., then X and Y are
indistinguishable.
Proof. Let A be the null set where the paths of X are not right continuous,
and let В be the analogous set for Y. Let Nt = {w : Xt(w) ф Yt(w)}, and
let TV = \JteqNt, where Q denotes the rationale in [0, oo). Then P{N) = 0.
Let M = A U В U N, and P(M) = 0. We have Xt{u) = Yt{u) for all t e Q,
ш $ M. If t is not rational, let tn decrease to t through Q. For w $ M,
Xtn(cu) = Ytn(u), each n, and Xt(w) = lim^ooXtn(ш) — limn-.oo^^w) =
Yt(w). Since P{M) = 0, X and Y are indistinguishable. ?
Corollary. Let X and Y be two stochastic processes which are cadlag. If X
is a modification of Y, then X and Y are indistinguishable.
Cadlag processes provide natural examples of stopping times.
Definition. Let X be a stochastic process and let Л be a Borel set in R.
Define
T(w) = M{t > 0 : Xt € A}.
Then T is called a hitting time of Л for X.
Theorem 3. Let X be an adapted cadlag stochastic process, and let Л be an
open set. Then the hitting time of Л is a stopping time.
Proof. By Theorem 1 it suffices to show that {Г < t} € Tt, 0 < t < oo. But
{T<t}= U {XseA},
s6Qn[0,t)
since Л is open and X has right continuous paths. Since {Xs € Л} = Х8(Л) е
J-s, the result follows. ?

1 Basic Definitions and Notation 5
Theorem 4. Let X be an adapted cadlag stochastic process, and let Л be a
closed set. Then the random variable
T(w) = ini{t >0:I((wN/l or Xt-(w) € A}
is a stopping time.
Proof. By Xt_(w) we mean lims_tiS<t Xs(w). Let An = {x : d(x, A) < 1/n},
where d(x, A) denotes the distance from a point x to A. Then An is an open
set and
{T<t} = {xt e Л or Xt_ e Л} и {f| (J {xs e AJ}. a
» s€Qn[0,t)
It is a very deep result that the hitting time of a Borel set is a stopping
time. We do not have need of this result.
The next theorem collects elementary facts about stopping times; we leave
the proof to the reader.
Theorem 5. Let S, T be stopping times. Then the following are stopping
times:
(i) S AT = min(S,T);
(ii) SvT = ma,x(S,T);
(in) S + T;
(iv) aS, where a > 1.
The (T-algebra Tt can be thought of as representing all (theoretically) ob-
observable events up to and including time t. We would like to have an analogous
notion of events that are observable before a random time.
Definition. Let Г be a stopping time. The stopping time <r-algebra Тт
is defined to be
{Ле7:Лп{Т<«}€7(, alH > 0}.
The previous definition is not especially intuitive. However it does well
represent "knowledge" up to time T, as the next theorem illustrates.
Theorem 6. Let T be a finite stopping time. Then Тт is the smallest a-
algebra containing all cadlag processes sampled at T. That is,
J~t = <t{Xt;X all adapted cadlag processes}.
Proof. Let G — a{XT;X all adapted cadlag processes}. Let A € Тт- Then
Xt = lyil{t>T}1 is a cadlag process, and Хт = 1л- Hence A € Q, and Тт С Q.
1 1л is the indicator function of A : 1л(ш) = I ' '
10, w f A.

6 I Preliminaries
Next let X be an adapted cadlag process. We need to show XT is Тт
measurable. Consider X(s,u) as a function from [0, oo) x fi into R. Construct
4> ¦ {T < t} ->• [0, oo) x П by ip(u) = (Т(ш),ш). Then since X is adapted and
cadlag, we have Хт = Xoip is a measurable mapping from ({T < t}, TtD{T <
t}) into (R, B), where В are the Borel sets of R. Therefore
{co:X(T(Lj),u;)eB}n{T<t}
is in Tt, and this implies Хт € Тт- Therefore Q С Тт- ?
We leave it to the reader to check that if S < T a.s., then Ts с Тт, and
the less obvious (and less important) fact that Ts ОТт = Tsat-
If X and Y are cadlag, then Xt = Yt a.s. each t implies that X and Y are
indistinguishable, as we have already noted. Since fixed times are stopping
times, obviously if Хт = Yr a.s. for each finite stopping time T, then X and
Y are indistinguishable. If X is cadlag, let AX denote the process AXt =
Xt — Xt-. Then AX is not cadlag, though it is adapted and for a.a. w,
t i—> AXt = 0 except for at most countably many t. We record here a useful
result.
Theorem 7. Let X be adapted and cadlag. If АХт1{т<оо} = 0 a-s- for each
stopping time T, then AX is indistinguishable from the zero process.
Proof. It suffices to prove the result on [0, to] for 0 < to < oo. The set {t :
|AXt| > 0} is countable a.s. since X is cadlag. Moreover
oo 1
= \J{f.\AXt\>-}
n=l
and the set {t : |AXj| > 1/n} must be finite for each n, since ?o < oo. Using
Theorem 4 we define stopping times for each n inductively as follows:
грП,1
rpn,k
Then Tn'k > Tn'k~l a.s.
{|AXt| ;
= inf{t
= mi{t
on {T1'
>0} = L
n
>0:
>Tn,
JilAJ
,fc
АХ4|
k-i.
oo}.
AXt\ > -}.
n
Moreover,
L{T".fe<oo}l >
where the right side of the equality is a countable union. The result follows. D
Corollary. Let X and Y be adapted and cadlag. If for each stopping time
T, АХт1{т<оо} = AYt1{t<oo} a-s-5 then AX and AY are indistinguishable.

2 Martingales 7
A much more general version of Theorem 7 is true, but it is a very deep
result which uses Meyer's "section theorems," and we will not have need of
it. See, for example, Dellacherie [41] or Dellacherie-Meyer [45].
A fundamental theorem of measure theory that we will need from time
to time is known as the Moiiotone Class Theorem. Actually there are several
such theorems, but the one given here is sufficient for our needs.
Definition. A monotone vector space Hona space Q is defined to be
the collection of bounded, real-valued functions /onfi satisfying the three
conditions:
(i) ti is a vector space over R;
(ii) 1q e H (i.e., constant functions are in H); and
(iii) if (/„)„>! С П, and 0 < ^ < /2 <•¦•</„<••-, and limn-.*, fn = f,
and / is bounded, then / G H.
Definition. A collection M. of real functions defined on a space fl is said to
be multiplicative if /, g e M implies that fg € M.
For a collection of real-valued functions ЛЛ defined on 0, we let cr{.M}
denote the space of functions defined on fl which are measurable with respect
to the ст-algebra on Q generated by {f~1(A}; Л ? В (Ж), f G M}.
Theorem 8 (Monotone Class Theorem). Let Ai be a multiplicative class
of bounded real-valued functions defined on a space fl, and let Л = а{ЛЛ}. If
ti is a monotone vector space containing ЛЛ, then ti contains all bounded, Л
measurable functions.
Theorem 8 is proved in Dellacherie-Meyer [45, page 14] with the additional
hypothesis that H is closed under uniform convergence. This extra hypothesis
is unnecessary, however, since every monotone vector space is closed under
uniform convergence. (See Sharpe [215, page 365].)
2 Martingales
In this section we give, mostly without proofs, only the essential results from
the theory of continuous time martingales. The reader can consult any of
a large number of texts to find excellent proofs; for example Dellacherie-
Meyer [46], or Ethier-Kurtz [71]. Also, recall that we will always assume as
given a filtered, complete probability space (fi,.F, F, P), where the filtration
F = (^t)o<t<oo is assumed to be right continuous.
Definition. A real-valued, adapted process X — (Xt)o<t<00 is called a mar-
tingale(resp. supermartingale, submartingale) with respect to the filtra-
filtration F if
(i) Xt e L^dP); that is, E{\Xt\} < oo;
(ii) if s < t, then E{Xt\Ts} = Xs, a.s. (resp. E{Xt\Ts} < Xs, resp. > Xs).

8 I Preliminaries
Note that martingales are only defined on [0, oo); that is, for finite t and not
t = oo. It is often possible to extend the definition to t = oo.
Definition. A martingale X is said to be closed by a random variable Y if
E{\Y\} < oo and Xt = E{Y\ft}, 0<t<oo.
A random variable Y closing a martingale is not necessarily unique. We
give a sufficient condition for a martingale to be closed (as well as a construc-
construction for closing it) in Theorem 12.
Theorem 9. Let X be a supermartingale. The function t i—> E{Xt) is right
continuous if and only if there exists a modification Y of X which is cadlag.
Such a modification is unique.
By uniqueness we mean up to indistinguishability. Our standing assump-
assumption that the "usual hypotheses" are satisfied is used implicitly in the state-
statement of Theorem 9. Also, note that the process Y is, of course, also a super-
supermartingale. Theorem 9 is proved using Doob's upcrossing inequalities. If X is
a martingale then 11—> E{Xt} is constant, and hence it has a right continuous
modification.
Corollary. If X = (Xt)o<t<<x> is a martingale then there exists a unique
modification Y of X which is cadlag.
Since all martingales have right continuous modifications, we will always
assume that we are taking the right continuous version, without any special
mention. Note that it follows from this corollary and Theorem 2 that a right
continuous martingale is cadlag.
Theorem 10 (Martingale Convergence Theorem). Let X be a right
continuous supermartingale, sup0<t<oo-E{|Xt|} < oo. Then the random vari-
variable Y = limt^oo^t a.s. exists, and E{\Y\} < oo. Moreover if X is
a martingale closed by a random variable Z, then Y also closes X and
A condition known as uniform integrability is sufficient for a martingale
to be closed.
Definition. A family of random variables {Ua)a^A is uniformly integrable
if r
lim sup / \Ua\dP = 0.
™^°° a J{\Ua\>n}
Theorem 11. Let {Ua)aeA be a subset of L1. The following are equivalent:
(i) (Ua)a?A is uniformly integrable.
(ii) supa€j4 .E{|[/a|} < oo, and for every e > 0 there exists 5 > 0 such that
ЛеТ, Р(Л) < 5, imply E{\UalA\} < e.
Vo<t<oo ^* denotes the smallest cr-algebra generated by (JFt), alii, 0 < t < oo.

2 Martingales 9
(in) There exists a positive, increasing, convex function G(x) defined on
[0, oo) such that lim^oo ^jp = +oo and supa E{G о \Ua\} < oo.
The assumption that G is convex is not needed for the implications (Hi) =Ф>
(ii) and (in) =Ф> (i).
Theorem 12. Let X be a right continuous martingale which is uniformly
integrable. Then Y = Нгщ-юо Xt a.s. exists, E{\Y\} < oo, and Y closes X as
a martingale.
Theorem 13. Let X be a (right continuous) martingale. Then (Xt)t>o is
uniformly integrable if and only ifY = limt^ooXt exists a.s., E{\Y\} < oo,
and (Xt)o<t<oo is a martingale, where Xoo = Y.
If X is a uniformly integrable martingale, then Xt converges to Xoo = Y in
L1 as well as almost surely. The next theorem we use only once (in the proof
of Theorem 28), but we give it here for completeness. The notation (Xn)n<0
refers to a process indexed by the non-positive integers: ... ,X-2, X~i, Xo-
Theorem 14 (Backwards Convergence Theorem). Let (Xn)n<o be a
martingale. Then linin^-oo Xn = E{X0\ Dn=-oo ^n} a-$- and in L1.
A less probabilistic interpretation of martingales uses Hilbert space theory.
Let Y e L2{U,T,P). Since Tt С Т, the spaces L2{U,Tt,P) form a family of
Hilbert subspaces of Ь2(П,Т, Р). Let WtY denote the Hilbert space projection
of rontoL2@,ft,P).
Theorem 15. Let Y € I2(Q,T, P). The process Xt - VtY is a uniformly
integrable martingale.
Proof. It suffices to show E{Y\Jrt) — VtY. The random variable E{Y\Tt}
is the unique Tt measurable r.v. such that JAYdP = fA E{Y\!Ft)dP, for
any event A e Tt. We have JA YdP = JA ^YdP + JA{Y -"•* Y)dP. But
JA(Y-^Y)dP = J lA{Y-^Y)dP. Since 1A e I2(Q,Ft,P), and (Y-**Y)
is in the orthocomplement of L2{U,TUP), we have / 1A(Y - VtY)dP = 0,
and thus by uniqueness E{Y\Tt) = VtY. Since |Г'У||ь2 < ЦУЦ^г, by part (iii)
of Theorem 11 we have that X is uniformly integrable (take G{x) = x2). ?
The next theorem is one of the most useful martingale theorems for our
purposes.
Theorem 16 (Doob's Optional Sampling Theorem). Let X be a right
continuous martingale, which is closed by a random variable X^. Let S and
T be two stopping times such that S <T a.s. Then Xs and Xt are integrable
and
Xs = E{XT\Ts] a.s.
Theorem 16 has a similar version for supermartingales.

10
I Preliminaries
Theorem 17. Let X be a right continuous supermartingale (resp. martin-
martingale), and let S and T be two bounded stopping times such that S < T a.s.
Then Xs and Хт are integrable and
Xs > E{XT\?s} a.s. {resp. =).
If Г is a stopping time, then so is t A T = mm(t, T), for each t > 0.
Definition. Let X be a stochastic process and let T be a random time. XT
is said to be the process stopped at T if Xj = XtAT-
Note that if X is adapted and cadlag and if T is a stopping time, then
is also adapted. A martingale stopped at a stopping time is still a martingale,
as the next theorem shows.
Theorem 18. Let X be a uniformly integrable right continuous martingale,
and let T be a stopping time. Then XT — (XtAT)o<t<oo is also a uniformly
integrable right continuous martingale.
Proof. XT is clearly right continuous. By Theorem 16
e Тт. Thus,
However for H e Tt we have
Therefore
= E{XT\Tt},
since Хт1{т<г} is Ft measurable. Thus XT is a uniformly integrable Tt mar-
martingale by Theorem 13. ?
Observe that the difficulty in Theorem 18 is to show that XT is a martin-
martingale for the filtration {Tt)o<t<oo- It is a trivial consequence of Theorem 16 that
XT = XtAT is a martingale for the filtration (Qt)o<t<oo given by Qt = FtAT-
Corollary. Let Y be an integrable random variable and let S, T be stopping
times. Then
= E{E{Y\TT}\Ts]
= E{Y\Fsat).

2 Martingales 11
Proof. Let Yt = E{Y\!Ft}. Then YT is a uniformly integrable martingale and
= Y$ = E{YT\FS}
= E{E{Y\TtWs}.
Interchanging the roles of T and 5 yields
Ysat = Yj? = E{YS\TT}
= E{E{Y\FsWt}-
Finally, E{Y\FSat} = YSat- ?
The next inequality is elementary, but indispensable.
Theorem 19 (Jensen's Inequality). Let <p : E —> R 6e convex, and let X
and <p(X) be integrable random variables. For any a-algebra Q,
<P°E{X\G}<E{<p(X)\G}.
Corollary 1. Let X be a martingale, and let <p be convex such that <p(Xt)
is integrable, 0 < t < oo. Then </?(X) is a submartingale. In particular, if M
is a martingale, then \M\ is a submartingale.
Corollary 2. Let X be a submartingale and let <p be convex, non-decreasing,
and such that <p(Xt)o<t<oo is integrable. Then <p(X) is also a submartingale.
We end our review of martingale theory with Doob's inequalities; the most
important is when p = 2.
Theorem 20. Let X be a positive submartingale. For all p > 1, with q con-
conjugate to p (i.e., ^ + ^ = 1), we have
| sup \Xt\\\LP<q sup \\Xt\\
t t
We let X* denote sups \XS\. Note that if M is a martingale with M^ e L2,
then \M\ is a positive submartingale, and taking p = 2 we have
This last inequality is called Doob's maximal quadratic inequality.
An elementary but useful result concerning martingales is the following.
Theorem 21. Let X = (Xt)o<t<oo be an adapted process with cadlag paths.
Suppose E{\Xt\} < oo and E{X?} = 0 for any stopping time T, finite or
not. Then X is a uniformly integrable martingale.
Proof. Let 0 < s < t < oo, and let Л € Ts. Let
\ u, if ш е A,
00, if LJ f Л.

12 I Preliminaries
Then ил are stopping times for all и > s. Moreover
f XudP = I XUAdP - f X^dP
Ja J Jq\a
= _/
Jn\
)п\л
since Е{Хил} = 0 by hypothesis, for и > s. Thus for Л е !FS and s < t,
E{XtlA} = E{XS1A} = -E{Xooln\A}, which implies E{Xt\Fs} = X., and
X is a martingale, 0 < t < oo. П
Definition. A martingale X with Xq = 0 and E{Xf} < oo for each t > 0 is
called a square integrable martingale. If E{X^a\ < oo as well, then X is
called an L2 martingale.
Clearly, any I? martingale is also a square integrable martingale. See also
Sect. 3 of Chap. IV.
3 The Poisson Process and Brownian Motion
The Poisson process and Brownian motion are the two fundamental examples
in the theory of continuous time stochastic processes. The Poisson process is
the simpler of the two, and we begin with it. We recall that we assume given
a filtered probability space (Q,^", F, P) satisfying the usual hypotheses.
Let (Tn)n>o be a strictly increasing sequence of positive random variables.
We always take To = 0 a.s. Recall that the indicator function l{t>Tn} is defined
as
_/l, if t>Tn(cj),
1{|
Definition. The process N = (NtH<t<oo denned by
with values in NU{oo} where N = {0,1,2,... } is called the counting process
associated to the sequence (Tn)n>\.
If we set T — supn Tn, then
[Г„,оо) = {N>n} = {(t,w) : Щи) > n}
as well as
[Tn,Tn+l) = {N = n}, and [r,oo) = {7V = oo}.

3 The Poisson Process and Brownian Motion 13
The random variable T is the explosion time of N. If T = oo a.s., then N is a
counting process without explosions. For T = oo, note that for 0 < s < t < oo
we have
The increment Nt — Ns counts the number of random times Tn that occur
between the fixed times s and t.
As we have defined a counting process it is not necessarily adapted to the
filtration F. Indeed, we have the following.
Theorem 22. A counting process N is adapted if and only if the associated
random variables (Tn)n>i are stopping times.
Proof. If the (Tn)n>o are stopping times (with To = 0 a.s.), then the event
{Nt = n} = {uj : Tn{uj) < t < Tn+iH} € Tu
for each n. Thus Nt e Tt and N is adapted. If N is adapted, then {Tn < t} =
{Nt > n\ € Tt, each t, and therefore Tn is a stopping time. D
Note that a counting process without explosions has right continuous paths
with left limits; hence a counting process without explosions is cadlag.
Definition. An adapted counting process TV is a Poisson process if
(i) for any s, t, 0 < s < t < oo, Nt — Ns is independent of Т3\
(ii) for any s, t, u, v, 0 < s <t < oo, 0 < и < v < oo, t — s — v — u, then the
distribution of Nt — Ns is the same as that of Nv — Nu.
Properties (i) and (ii) are known respectively as increments independent of
the past, and stationary increments.
Theorem 23. Let N be a Poisson process. Then
n\
n = 0,1,2,..., for some Л > 0. That is, Nt has the Poisson distribution with
parameter Xt. Moreover, N is continuous in probability3 and does not have
explosions.
Proof. The proof of Theorem 23 is standard and is often given in more ele-
elementary courses (cf., e.g., Qinlar [33, page 71]). We sketch it here.
Step 1. For all t > 0, P(Nt = 0) = e~Xt, for some constant Л > 0.
3 N is continuous in probability means that for t > 0, limu^t Nu = Nt where the
limit is taken in probability.

14 I Preliminaries
Since {Nt = 0} = {Ns = 0} П {Nt - Ns = 0} for 0 < s < t < oo by the
independence of the increments,
P(Nt = 0) = P(NS = 0)P(Nt - NS = 0)
= P(NS = 0)P(Nt_s = 0),
by the stationarity of the increments. Let a(t) = P(Nt = 0). We have a{t) =
a(s)a(t — s), for all 0 < s < t < oo. Since a{t) can be easily seen to be right
continuous in t, we deduce that either a{t) = 0 for all t > 0 or
a(t) = e~xt for some Л > 0.
If a(t) — 0 it would follow that Nt(w) = oo a.s. for all t which would con-
contradict that N is a counting process. Note that limu^t P(\NU — Nt\ > e) =
lim«^t P(\Nu-t\ > e) = lim^^o P(NV > e) = lim^^o 1 - e~Xv = 0; hence N
is continuous in probability.
Step 2. P(Nt > 2) is o(t). (That is, limt^0 \P(Nt > 2) = 0.)
Let C{t) = P(Nt > 2). Since the paths of N are non-decreasing, C is
also non-decreasing. One readily checks that showing lim^o jP(t) — 0 is
equivalent to showing that Ит^^оо и/3(^) = 0. Divide [0, l] into n subintervals
of equal length, and let Sn denote the number of subintervals containing at
least two arrivals. By the independence and stationarity of the increments Sn is
the sum of n i.i.d. zero-one valued random variables, and hence has a Binomial
distribution (n,p), where p = /3(^). Therefore E{Sn} = np = n/3(?).
Since N is a counting process, we know the arrival times are strictly
increasing; that is, Tn < Tn+i a.s. Since Sn < N\, if E{Ni} < oo we
can use the Dominated Convergence Theorem to conclude limn_>oon/3(^) =
limn^oo E{Sn} = 0. (That E{Ni} < oo is a consequence of Theorem 34,
established in Sect. 4).
Also note that ?{./Vi} < oo implies Ni < oo a.s. and hence there are no
explosions before time 1. This implies for fixed ui, for n sufficiently large no
subinterval has more than one arrival (otherwise there would be an explosion).
Hence, limn^oo Sn(uj) = 0 a.s.
Step 3. limt^o \P{Nt = 1} = A.
Since P{Nt = 1} = 1 - P{Nt = 0} - P{Nt > 2}, it follows that
hm lP{Nt 1} lim
t-.o t t—о
Step 4- Conclusion.
We write <p(t) = E{aNt}, for 0 < a < 1. Then for 0 < s < t < oo,
the independence and stationarity of the increments implies that <p(t + s) =
<p(t)<p(s) which in turn implies that <p(t) = e*^a). But

3 The Poisson Process and Brownian Motion 15
n=0
oo
= p(Nt = 0) + aP(Nt = 1) + ^ anP(Nt = n),
n=2
and ф(а) = '/''(О), the derivative of <p at 0. Therefore
«<» - sa ^ - a {m^1+^^^+>}
= -Л + Xa.
Therefore <p(t) = e-M+Xat, hence
00
V{t) = E a"PGVt = „) =
Equating coefficients of the two infinite series yields
7*.
n=0 n=0
for n =0,1,2,.... П
Definition. The parameter Л associated to a Poisson process by Theorem 23
is called the intensity, or arrival rate, of the process.
Corollary. A Poisson process N with intensity Л satisfies
E{Nt} = Xt,
Variance^) = Var(Art) = At.
The proof is trivial and we omit it.
There are other, equivalent definitions of the Poisson process. For example,
a counting process N without explosion can be seen to be a Poisson process
if for all s, t, 0 < s < t < oo, E{Nt} < oo and
E{Nt-Ns\Ts} = X(t-s).
Theorem 24. Let N be a Poisson process with intensity X. Then Nt — Xt and
(Nt - XtJ - Xt are martingales.
Proof. Since Xt is non-random, the process Nt — Xt has mean zero and inde-
independent increments. Therefore
E{Nt -Xt- (Ns - Xs)\Ts} = E{Nt -Xt- (Ns - As)} = 0,
for 0 < s < t < oo. The analogous statement holds for (Nt - XtJ - Xt. ?

16 I Preliminaries
Definition. Let if be a stochastic process. The natural filtration of H,
denoted F° = (^?)o<t<oo, is denned by T% = a{Hs\ s < t}. That is, F$ is the
smallest filtration that makes H adapted.
Note that natural nitrations are not assumed to contain all the P-null sets
Theorem 25. Let N be a counting process. The natural filtration of N is
right continuous.
Proof. Let E = [0, oo] and В be the Borel sets of E, and let Г be the path
space given by
Г = ( П Еа, ® Bt).
s€[0,oo) s€[0,oo)
Define the maps щ : О —> Г by
щ{ш) = s ^ NsM(lj).
Thus the range of Tit is contained in the set of functions constant after t. The
сг-algebra ff is also generated by the single function space-valued random
variable щ.
Let A be an event in Пп>1 ^t+i • Then there exists a set An € ® se[o oo) ^s
such that Л = {?rt+i. G An}. Next set Wn = {щ = 7rt+i}. For each w,
there exists an n such that s •—> Ns(lj) is constant on [t,t + ^]; therefore
Q = Un>i ^iii where TVn is an increasing sequence of events. Therefore
Л = lim(Wn П A)
П
= \im{Wn П {щ+г е Л„})
= lim(Wn П {7rt e An})
n
which implies Л G ^. We conclude Пп>1 J~t+± *- -^?) which implies they are
equal. D
We next turn our attention to the Brownian motion process. Recall that
we are assuming as given a filtered probability space (fi, T, F, P) that satisfies
the usual hypotheses.
Definition. An adapted process В = (Bt)o<t<oo taking values in Rn is called
an n-dimensional Brownian motion if
(i) for 0 < s < t < oo, Bt — Bs is independent of Ts {increments are indepen-
independent of the past);
(ii) for 0 < s < t, Bt — Bs is a Gaussian random variable with mean zero and
variance matrix (t — s)C, for a given, non-random matrix C.

3 The Poisson Process and Brownian Motion 17
The Brownian motion starts at x if P{Bq = x) = 1.
The existence of Brownian motion is proved using a path-space construc-
construction, together with Kolmogorov's Extension Theorem. It is simple to check
that a Brownian motion is a martingale as long as ?{|i?o|} < oo. Therefore
by Theorem 9 there exists a version which has right continuous paths, a.s.
Actually, more is true.
Theorem 26. Let В be a Brownian motion. Then there exists a modification
of В which has continuous paths a.s.
Theorem 26 is often proved in textbooks on probability theory (e.g.,
Breiman [23]). It can also be proved as an elementary consequence of Kol-
Kolmogorov's Lemma (Theorem 72 of Chap. IV). We will always assume that we
are using the version of Brownian motion with continuous paths. We will also
assume, unless stated otherwise, that С is the identity matrix. We then say
that a Brownian motion В with continuous paths, with С = I the identity
matrix, and with Bo = x for some x € R™, is a standard Brownian motion.
Note that for an Rn standard Brownian motion B, writing Bt = {B\, ...,B?),
0 < t < oo, then each B% is an R1 Brownian motion with continuous paths,
and the B*'s are independent.
We have already observed that a Brownian motion В with ?{|i?o|} < oo
is a martingale. Another important elementary observation is the following.
Theorem 27. Let В = (-Bt)o<t<oo be a one dimensional standard Brownian
motion with Bo = 0. Then Mt = B2 — t is a martingale.
Proof. E{Mt} = E{Bf -t}=0. Also
E{Mt - Ма\Та} = E{B2t - B2S ~{t-
and
a\F.} = BsE{Bt\fs} = B2S,
since В is a martingale with Bs, Bt € L2. Therefore
E{Mt - MS\FS} = E{B2 - 2BtBs + B2 -(t-
= E{(Bt-BsJ-(t-s)\fs}
= E{(Bt - BsJ} -(t-s)
= 0,
due to the independence of the increments from the past. D
Theorem 28. Letnn be a sequence of partitions of[a,a+t]. Suppose тгт С тгп
if m > n (that is, the sequence is a refining sequence). Suppose more-
moreover that limn^oo meshGrn) = 0. Let nnB = ^2ueVn(Bti+1 - BuJ. Then
limn_»oo "KnB = t a.s., for a standard Brownian motion B.

18 I Preliminaries
Proof. We first show convergence in mean square. We have
where Fj are independent random variables with zero means. Therefore
Е{(тгпВ - tJ} = E{(? YiJ} =
Next observe that (Bt.+1 — BtiJ/(ti+\ -1{) has the distribution of Z2, where
Z is Gaussian with mean 0 and variance 1. Therefore
E{(nnB - tJ} = E{{Z2 - IJ}
U 67Г
which tends to 0 as n tends to oo. This establishes L2 convergence (and hence
convergence in probability as well).
To obtain the a.s. convergence we use the Backwards Martingale Conver-
Convergence Theorem (Theorem 14). Define
for n — —1,-2,-3, Then it is straightforward (though notationally
messy) to show that
E{Nn\Nn.1,Nn-2,...}=Nn-1.
Therefore Nn is a martingale relative to Qn = a{Nk, к < n}, n = —1, —2,
By Theorem 14 we deduce limn^_oo Nn = linin-.oo nnB exists a.s., and since
ттпВ converges to t in L2, we must have limn^oo жпВ = t a.s. as well. D
Comments. As noted in the proofs, the proof is simple (and half as long) if we
conclude only L2 convergence (and hence convergence in probability), instead
of a.s. convergence. Also, we can avoid the use of the Backwards Martingale
Convergence Theorem (Theorem 14) in the second half of the proof if we
add the hypothesis that Yln meshGrn) < oo. The result then follows, after
having proved the L2 convergence, by using the Borel-Cantelli Lemma and
Chebsyshev's inequality. Furthermore to conclude only L2 convergence we do
not need the hypothesis that the sequence of partitions be refining.
Theorem 28 can be used to prove that the paths of Brownian motion
are of unbounded variation on compacts. It is this fact that is central to
the difficulties in defining an integral with respect to Brownian motion (and
martingales in general).

4 Levy Processes 19
Theorem 29. For almost all ш, the sample paths t >-* Bt(oj) of a standard
Brownian motion В are of unbounded variation on any interval.
Proof. Let A = [a,b] be an interval. The variation of paths of В is defined to
be
VA{w) = sup V \Bti+1 -Bti\
*€Vue*
where V are all finite partitions of [a,b]. Suppose P(VA < oo) > 0. Let nn
be a sequence of refining partitions of [a, b] with limn meshGrn) = 0. Then by
Theorem 28 on {VA < oo},
< lim sup \Bti+1-Bti\ V \Bti+1-Bt.\
< lim sup \Bt -Bti\VA
= o,
since suptie7rn \Bti+1 — Bti\ tends to 0 a.s. as meshGrn) tends to 0 by the a.s.
uniform continuity of the paths on A. Since b — a < 0 is absurd, by Theorem 27
we conclude VA = oo a.s. Since the null set can depend on the interval [a, b],
we only consider intervals with rational endpoints a, b with a < b. Such а
collection is countable, and since any interval (a,b) = U^Li[an>M with an,
bn rational, we can omit the dependence of the null set on the interval. D
We conclude this section by observing that not only are the increments of
standard Brownian motion independent, they are also stationary. Thus Brow-
Brownian motion is a Levy process (as is the Poisson process), and the theorems
of Sect. 4 apply tor it. In particular, by Theorem 31 of Sect. 4, we can con-
conclude that the completed natural filtration of standard Brownian motion is
right continuous.
4 Levy Processes
The Levy processes, which include the Poisson process and Brownian motion
as special cases, were the first class of stochastic processes to be studied in
the modern spirit (by the French mathematician Paul Levy). They still pro-
provide prototypic examples for Markov processes as well as for semimartingales.
Most of the results of this section hold for W1 -valued processes; for notational
simplicity, however, we will consider only R-valued processes.4 Once again
we recall that we are assuming given a filtered probability space (п,^,?, Р)
satisfying the usual hypotheses.
4 Rn denotes n-dimensional Euclidean space. R+ = [0, oo) denotes the non-negative
real numbers.

20 I Preliminaries
Definition. An adapted process X = (Xt)t>o with Xo = 0 a.s. is a Levy
process if
(i) X has increments independent of the past; that is, Xt — Xs is independent
of Ts, 0 < s < t < oo; and
(ii) X has stationary increments; that is, Xt — Xs has the same distribution
as Xt-S, 0 < s < t < oo; and
(iii) Xt is continuous in probability; that is, limt^s Xt = Xs , where the limit
is taken in probability.
Note that it is not necessary to involve the filtration F in the definition of
a Levy process. Here is a (less general) alternative definition; to distinguish
the two, we will call it an intrinsic Levy process.
Definition. An process X = (Xt)t>o with Xo = 0 a.s. is an intrinsic Levy-
process if
(i) X has independent increments; that is, Xt — Xs is independent of Xv — Xu
if (u,v) П (s,t) = 0; and
(ii) X has stationary increments; that is, Xt — Xs has the same distribution
as Xv — Xu if t - s = v - и > 0; and
(iii) Xt is continuous in probability.
Of course, an intrinsic Levy process is a Levy process for its minimal
(completed) filtration.
If we take the Fourier transform of each Xt we get a function f(t,u) = ft(u)
given by
ft(u) = E{eiuX<},
where /o(u) = 1, and ft+s(u) = /*(«)/„(«), and ft(u) ф 0 for every (t,u).
Using the (right) continuity in probability we conclude /t(w) = exp{-?^(u)},
for some continuous function ф(и) with ф@) = О. (Bochner's Theorem can
be used to show the converse. If ip is continuous, ф@) = О, and if for all
t > 0, ft(u) = e-4i^u) satisfies J2i,j ai&jМщ - щ) > 0, for all finite
(wi,... ,un;cti,... ,an), then there exists a Levy process corresponding to
/¦)
In particular it follows that if X is a Levy process then for each t > 0, Xt
has an infinitely divisible distribution. Inversely it can be shown that for each
infinitely divisible distribution fj, there exists a Levy process X such that fj, is
the distribution of X\.
Theorem 30. Let X be a Levy process. There exists a unique modification
Y of X which is cadlag and which is also a Levy process.
Proof. Let M" = e, /I. For each fixed и in Q, the rationale in K, the process
(M"H<t<oo is a complex-valued martingale (relative to F).
We first show that the paths of X cannot explode a.s. For any real u,
(Mt")t>o is a (complex-valued) martingale and thus for a.a. w the functions

4 Levy Processes 21
t i-> Mt"(w) and ? н-> e»«-X'«M) With ? € Q+, are the restrictions to Q+ of
cadlag functions. Let
Л = {(и, и) € U х R : eiuXt^\t € Q+,
is not the restriction of a cadlag function}.
One can check that Л is a measurable set. Furthermore, we have seen that
/ 1л(и, u)P(du)) = 0, each и e R. By Fubini's Theorem
// lA(u,u)duP(duj) = lA(u,u)P{du)du = 0,
J J— oo 7-oo >/
hence we conclude that for a.a. oj the function t н-> e%uXt^\ t G Q+ is the
restriction of a cadlag function for almost all и € R. We can now conclude
that the function t н-> Xt(w), ? € Q+, is the restriction of a cadlag function
for every such w, with the help of the lemma that follows the proof of this
theorem.
Next set Yt{w) = lims€Q+]S|t Xs(oj) for all u> in the projection onto п of
{ft x Щ \ Л and Yt — 0 on Л, all ?. Since Tt contains all the P-null sets
of J- and (J-t)o<t<oo is right continuous, Yt € Tt- Since X is continuous in
probability, P{Yt -? Xt} = 0, hence У is a modification of X. It is clear that
У is a Levy process as well. D
The next lemma was used in the proof of Theorem 30. Although it is a
pure analysis lemma, we give a proof using probability theory.
Lemma. Let xn be a sequence of real numbers such that e%UXn converges as
n tends to oo for almost all tisR. Then xn converges to a finite limit.
Proof. We will verify the following Cauchy criterion: xn converges if for any
increasing sequences nk and т^, then linifc^oo ?„,. — xmk = 0. Let U be a
random variable which has the uniform distribution on [0,1]. For any real t,
by hypothesis a.s. ettUXnk and eltUXmk converge to the same limit. Therefore,
\\m eitU(xnk-xmk) = 1 &s
fc-»oo
so that the characteristic functions converge,
lim Е{еа^Хп"-х^и} = 1,
к—>oo
for all t € R. Consequently (хПк — xmk)U converges to zero in probability,
whence lim^oo xnk - xmk = 0, as claimed. D
We will henceforth always assume that we are using the (unique) cadlag
version of any given Levy process. Levy processes provide us with examples
of filtrations that satisfy the "usual hypotheses," as the next theorem shows.

22 I Preliminaries
Theorem 31. Let X be a Levy process and let Qt = J^VAf, where
is the natural filtration of X, andAf are the P-null sets of T¦ Then (Qt)o<t<oo
is right continuous.
Proof. We must show Gt+ = Gt, where Gt+ = f)u>tGu- Note that since the
filtration G is increasing, it suffices to show that Gt — C\n>1Gt+±- Thus,
we can take countable limits and it follows that if Si,..., sn < t, "then for
For ы,..., vn > t and (ui,..., un), we give the proof for n = 1 for nota-
tional convenience. Therefore let z > v > t, and suppose given Ui and «2- We
have
0 \ = li
wit
using that M = e, ? 4. is a martingale. Combining terms the above becomes
wit
and the same martingale argument yields
= Шпе^+^^Л^И! + u2)fz-v(u2)
wit
It follows that E{eiSu>x'j\gt+} = E{eiEu>x*i\gt} for all (s1,...,sn) and
all (ui,...,un), whence E{Z\gt+} = E{Z\gt} for every bounded Z G
Vo<s<oo-^s)- Tms implies gt+ = gt except possibly for events of probabil-
probability zero. However since both cr-algebras contain Af, we conclude Gt+ = Gt for
each t > 0. D
The next theorem shows that a Levy process "renews itself" at stopping
times.

4 Levy Processes 23
Theorem 32. . Let X be a Levy process and let T be a stopping time. On
the set {T < oo} the process Y = (Yt)o<t<oo defined by Yt = Хт+t ~ Хт is a
Levy process adapted to %t = Тт+ti Y is independent of Тт and Y has the
same distribution as X.
Proof. First assume T is bounded. Let A e Тт and let (щ,..., un; to,--., *n)
be given with Uj in a countable dense set (for example the rationale Q) and
tj e R+, tj increasing with j.
Recall that M = ^^ is a martingale, where /t(wj) = E{eiu*Xt}. Then
Aexp{i 2]
3
{
by applying the Optional Sampling Theorem (Theorem 16) n times. Note that
this shows the independence of Yt = Хт+t — Хт from Тт as well as showing
that Y has independent and stationary increments and that the distribution
ofY is the same as that of X.
If T is not bounded, we let Tn = min(r, n)=TAn. The formula is valid
for Лп = A(~){T < n} when A G Тт, since then An € Ттлп- Taking limits and
using the Dominated Convergence Theorem we see that our formula holds for
unbounded T as well, for events Л = А П {T < oo}, A € Тт- This gives the
result. D
Since a standard Brownian motion is a Levy process, Theorem 32 gives
us a fortiori the strong Markov property for Brownian motion. This allows
us to establish a pretty result for Brownian motion, known as the reflection
principle. Let В = (Bt)t>o denote a standard Brownian motion, Bo = 0 a.s.,
and let St = supo<s<t5s, the maximum process of Brownian motion. Since
В is continuous, St = sup0<M<t m€q.Bm, where Q denotes the rationals; hence
St is an adapted process with non-decreasing paths.
Theorem 33 (Reflection Principle for Brownian Motion). Let В =
(Bt)t>o be standard Brownian motion (Bo = 0 a.s.) and St = supo<s<t-Bs,
the Brownian maximum process. For у > 0, z > 0,
P(Bt <z-y;St>z)= P{Bt >y + z).
Proof. Let T — inf{t > 0 : Bt — z}. Then Г is a stopping time by Theorem 4,
and P(T < oo) = 1. We next define a new process X by

24 I Preliminaries
Xt = Btl{t<T} + BBT - Bt)l{t>T}-
The process X is the Brownian motion В up to time T, and after time T it is
the Brownian motion В "reflected" about the constant level z. Since В and
—В have the same distribution, it follows from Theorem 32 that X is also a
standard Brownian motion.
Next we define
R = inf{t > 0 : Xt = z}.
Then clearly
P(R <t;Xt<z-y) = P(T<t;Bt<z- y),
since (R,X) and (T,B) have the same distribution. However we also have
that R = T identically, whence
{R < t; Xt < z - y} = {T < t; Bt > z + y}
by the construction of X. Therefore
P(T<t;Bt>z + y) = P(T<t;Bt<z-y). (*)
The left side of (*) equals
P(St >z;Bt>z + y)= P(Bt >z + y),
where the last equality is a consequence of the containment {St > z) D {Bt >
z + y}. Also the right side of (*) equals P(St > z; Bt < z - y). Combining
these yields
P(St >z;Bt<z-y) = P(Bt >z + y),
which is what was to be proved. D
We also have a reflection principle for Levy processes. See Exercises 30
and 31.
Corollary. Let В = (Bt)t>o be standard Brownian motion (Bo = 0 a.s.) and
St = supo<s<t Bs. For z > 0,
P(St >z) = 2P(Bt > z).
Proof. Take у = 0 in Theorem 33. Then
P(Bt <z;St>z)=P(Bt>z).
Adding P(Bt > z) to both sides and noting that {Bt > z} = {Bt > z}(~){St >
z} yields the result since P(Bt — z) = 0. D

4 Levy Processes 25
A Levy process is cadlag, and hence the only type of discontinuities it can
have is jump discontinuities. Letting Xt- = Ит3цХ3, the left limit at t, we
define
the jump at t. If supt |AXt| < С < oo a.s., where С is a non-random constant,
then we say that X has bounded jumps.
Our next result states that a Levy process with bounded jumps has finite
moments of all orders. This fact was used in Sect. 3 (Step 2 of the proof of
Theorem 23) to show that E{N\} < oo for a Poisson process N.
Theorem 34. Let X be a Levy process with bounded jumps. Then E{\Xt\n} <
oo for all n — 1,2,3,....
Proof. Let С be a (non-random) bound for the jumps of X. Define the stop-
stopping times
T1=ini{t:\Xt\ >C}
Tn+1 = M{t > Tn : \Xt -XTn\> C}.
Since the paths are right continuous, the stopping times (Tn)n>i form
a strictly increasing sequence. Moreover |ДХт| < С by hypothesis for any
stopping time T. Therefore sups |Xjn| < 2nC by recursion. Theorem 32 im-
implies that Tn — Tn_i is independent of Ттп-г and also that the distribution
of Tn — Tn-i is the same as that of Ti.
The above implies that
for some a, 0 < a < 1. But also
P{\Xt\ > 2nC} < P{Tn
which implies that Xt has an exponential moment and hence moments of all
orders. D
We next turn our attention to an analysis of the jumps of a Levy process.
Let Л be a Borel set in R bounded away from 0 (that is, 0 ^ Л, where Л is
the closure of A). For a Levy process X we define the random variables
T\ = inf{i > 0 : AXt e A)
T2+1 = ini{t > T2 : AXt € A}.

26 I Preliminaxies
Since X has cadlag paths and 0 ф Л, the reader can readily check that {Гд >
t} G Tt+ = Ft and therefore each Tfc is a stopping time. Moreover 0 ^ Л and
cadlag paths further imply T\ > 0 a.s. and that lirrin^oo Тд = оо a.s. We
define
0<s<t n=l
and observe that NA is a counting process without an explosion. It is straight-
straightforward to check that for 0 < s < t < oo,
NA - NA € a{Xu -Xv;s<v<u<t},
and therefore NA - NA is independent of Ta; that is, NA has independent
increments. Note further that TV/1 - NA is the number of jumps that Zu =
Xs+U — Xs has in Л, 0 < и < t — s. By the stationarity of the distributions of
X, we conclude TV/1 — NA has the same distribution as iV/!s. Therefore 7УЛ is
a counting process with stationary and independent increments. We conclude
that NA is a Poisson process. Let v(A) = E{NA} be the parameter of the
Poisson process NA (г^(Л) < oo by the proof of Theorem 34).
Theorem 35. The set function Л н-> NA(u>) defines a a-finite measure on
R \ {0} for each fixed (t,oj). The set function v(A) = E{N^} also defines a
a-finite measure опЖ\ {О}.
Proof. The set function Л <-» Na(oj) is simply a counting measure: /х(Л) =
{number of s < t: AXs(u) & Л}. It is then clear that v is also a measure. D
Definition. The measure v defined by
u{A) = E{NA} = E{
0<s<l
is called the Levy measure of the Levy process X.
We wish to investigate further the role the Levy measure plays in governing
the jumps of X. To this end we establish a preliminary result. We let Nt (u>, dx)
denote the random measure of Theorem 35. Since Nt(<j,dx) is a counting
measure, the next result is obvious.
Theorem 36. Let Л be a Borel set of R, 0 ф A, f Borel and finite on A.
Then
L
f{x)Nt{u,dx)= J2
0<s<t
Just as we showed that NA has independent and stationary increments,
we have the following consequence.

4 Levy Processes 27
Corollary. Let Л be a Borel set of К with 0 <$. A, and let / be Borel and
finite on A. Then
f f(X)Nt(;dx)
JA
is a Levy process.
For a given set A (as always, 0 g A), we denned the associated jump
process to be
jtA = J2 ьх.Ыьх,).
0<s<t
By Theorem 36 and its corollary we conclude that
JA
= I xNt(;dx).
JA
Hence, J/1 is a Levy process itself, it is defined, and JA < oo a.s., each t > 0.
Theorem 37. Given A, 0 ^ A, the process Xt — J/1 is a Levy process.
Proof. It is clear that we need only check the independence and stationarity
of the increments. But
Xt- JtA-(Xs~ JA) = Xt-Xs- Y^ A^uU(AXu)
3<U<t
which is clearly cr{Xv — Xu;s < и < v < t} measurable, and due to the
stationarity of the increments of X it has the same law as Xt^a — JA_S. D
We are now in a position to consider
0<3<t
for some constant а > 0. The advantage of doing this is that Ya then has jumps
bounded by a, and hence has finite moments of all orders (Theorem 34). We
can choose any a > 0; we arbitrarily choose a = 1. Note that
¦rrl v t(-°°.-1]U[1,oo)
= Xt- I xNt(;dx).
J\x\>l
The next theorem gives an interpretation of the Levy measure as the expected
rate at which the jumps of the Levy process fall in a given set.
Theorem 38. Let A be Borel with 0 ^ A. Let v be the Levy measure of X,
and let f\A e L2(du). Then
E{ f f(x)Nt(-,dx)} = t f f{x)u{dx)
JA JA

28 I Preliminaries
and also
E{( f f(x)Nt(-,dx) - t I f{x)v{dx)J} = t I f(xJv(dx).
JA JA JA
Proof. First let / = ?) ¦ а^1ля a simple function. Then
since Nt 3 is a Poisson process with parameter v{Aj). The first equality follows
easily.
For the second equality, let Ml = N^ —tv(Ai). The Mt* are V martingales,
all p > 1, by the proof of Theorem 34. Moreover, Е{М}} ~ 0. Suppose Л*, Aj
are disjoint. We have
к е
for any partition 0 = to < t\ < ¦ ¦ ¦ < tn = t. Using the martingale property
we have
E{MiMi} = Е(?(м;к+1 - Mik)(Mik+i - Mi)}.
к
Using the inequality \ab\ < a2 + b2, we have
E - O2-
However T,k(ML+i ~ Mtj2 ^ (N^J + "(AiJt2; therefore the sums are
dominated by an integrable random variable. Since Ml and Ml have paths of
finite variation on [0, t] it is easy to deduce that if we take a sequence (?rn)n>i
of partitions where the mesh tends to 0 we have
„I™, ? (
ttG 0<s<t
Using Lebesgue's Dominated Convergence Theorem we conclude
E{MltM3t} = E{^ АМгаАМ>} = О;
0<s<t
the expectation above is 0 because Ai and Aj are disjoint, implying that Ml
and M-3 jump at different times. The second equality is now easy to verify for
simple functions. For general /, let fn be a sequence of simple functions such
that fn^A converges to / in L2{dv), and the result follows. Q

4 Levy Processes 29
Remark. The first statement in Theorem 38 remains true if /1л е Ll(dv).
See Exercise 28.
Corollary. Let / : К —> К be bounded and vanish in a neighborhood of 0.
Then
/-OO
f(AXs)} = t f{x)u{dx).
J— OO
0<s<t
Proof We need only combine Theorem 38 with Theorem 36. G
Theorem 39. Let A\, Л2 be two disjoint Borel sets with 0 ^ Л\, О ^ Л2.
Then the two processes
0<3<t
Jt2= 53
0<s<t
are independent Levy processes.
Proof. By Theorem 36 and its corollary we have that J1 and J2 are Levy
processes. To show they are independent, we begin by forming for u, v in M,
iuj\
* " Е{е™Л}
П" _ _T 1
* " Е{е^Ц
Then Cu and Dv are both martingales, with E{C?} = E{D"} = 0. As in the
proof of Theorem 38, let 7rn: 0 = to < t\ < ¦ ¦ ¦ < tn = t be a sequence of
partitions of [0, t] with limn meshGTn) = 0. Then
»+i - Dl)}
Since Cu and Dw have paths of finite variation on compacts, it follows by
letting meshGrn) tend to 0, that
0<s<t
The expectation above equals zero because Cu and Dv jump at different times,
due to the void intersection of A\ and Л2.
We conclude that Е{С^В\) = 0, and thus

30 I Preliminaries
Е{ем1емЦ = E{eiuJ!}E{eivJ?},
which in turn implies, because of the independence and stationarity of the
increments that
{i-Jl1)+---+un(j}n-jln_i))ei{vlJ2tl+---+vn(J2ln-jln_i)),
This is enough to give independence. ?
The preceding results combine to yield the following useful theorem, which
is one of the fundamental results about Levy processes.
Theorem 40. Let X be a Levy process. Then Xt = Yt + Zt, where Y, Z are
Levy processes, Y is a martingale with bounded jumps, Yt G Lp for all p > 1
and Z has paths of finite variation on compacts.
Proof. Let Jt = X^o<s<t АХа1{\лх,\>1}- Since X has cadlag paths, for each
fixed w the function s >-> Xa(w) has only finitely many jumps bigger than
one on [0, t]. Therefore J has paths of finite variation on compacts. J is also
a Levy process by Theorem 36 and its corollary. The process W = X - J
is also a Levy process (Theorem 37), and W has jumps bounded by one. We
know therefore that, n > 1, ?"{|И^|"} exists (Theorem 34), and the stationary
increments of W implies E{Wt} = at, for a = E{Wi}. (Recall E{W0} = 0.)
We set Yt — Wt — E{W{\. Then Y has independent increments and mean 0;
it is a martingale. Setting Zt — Jt + at completes the proof. ?
While Theorem 40 is the most important result about Levy processes
from the standpoint of stochastic integration, the next two theorems provide
a better understanding of Levy processes themselves.
Theorem 41. Let X be a Levy process with jumps bounded by a. That is,
sups |AXS| < a a.s. Let Zt = Xt — E{Xt}. Then Z is a martingale and Zt =
Z? + Zf where Zc is a martingale with continuous paths, Zd is a martingale,
Zf= f x(Nt(-,dx)-tv(dx)),
J{\x\<a)
and Zc and Zd are independent Levy processes.
Proof. Z has mean zero and independent increments so it is a martingale, as
well as a Levy process. For a given set Л we define
M/1 = / xNt(-,dx) - t I xu{dx)
Ja j л
= ^ ДХ81л(ДХ8) — t j xu(dx).
0<s<t ^A

4 Levy Processes 31
For this proof we take a = 1. Let Ak = {-^ < \x\ < ?}. Then МЛк
are pairwise independent Levy processes and martingales (Theorem 39). Set
M" = ]T)fc=i мЛк ¦ Then the martingales Z - Mn and M" are independent
by an argument similar to the one in the proof of Theorem 39. Moreover
Var(Zt) = Var(Zt - Mt") + Var(Mt") where Var(X) denotes the variance of a
random variable X. Therefore Var(M") < Var(Zt) < oo for all n. We deduce
that M™ is Cauchy in L2 and hence converges in L2 as n tends to oo to a
martingale Zf, and Z — M" also converges to a martingale Zc. Using Doob's
maximal quadratic inequality (Theorem 20), we can find a subsequence con-
converging a.s., uniformly in t on compacts, which permits the conclusion that
Zc has continuous paths. The independence of Zd and Zc follows from the
independence of M" and Z — Mn, for every n. ?
Note that a consequence of the convergence of Mt" to Zf in L2 in the proof
of Theorem 41 is that the integral J,x 0)и@ - x2v{dx) is finite. Note that this
improves a bit on the conclusion in Theorem 38.
We recall that for a set A, 0 ^ A, the process TV/1 = JANt(-,dx) is a
Poisson process with parameter v(A), and thus Nf - tv{A) is a martingale.
Definition. Let N be a Poisson process with parameter Л. Then Nt — \t is
called a compensated Poisson process.
Theorem 41 can be interpreted as saying that a Levy process with bounded
jumps decomposes into the sum of a continuous martingale Levy process and
a martingale which is a mixture of compensated Poisson processes. It is not
hard to show that E{eiuZ'} = e^2/2, which implies that Zc must be a
Brownian motion. The full decomposition theorem then follows easily. We
state it here without proof (consult Bertoin [12], Bretagnolle [25], Feller [72],
or Jacod-Shiryaev [110] for a proof).
Theorem 42 (Levy Decomposition Theorem). Let X be a Levy process.
Then X has a decomposition
Xt=Bt+ x(Nt(; dx) - tu(dx))
+ tE{Xi- xNx{-,dx)}+ I xNt(-,dx)
= Bt+ f x{Nt{-,dx)-tu(dx)) + at+ V Д^
l{\x\<\)
where В is a Brownian motion; for any set A, 0 ^ A, Nf = fA Nt(-, dx) is a
Poisson process independent of B; Nf is independent of Nf if A and Г are
disjoint; Nf has parameter v{A); and i/(dx) is a measure on Ш\{0} such that
oo.
Finally we observe that Theorem 42 gives us a formula (known as the
Levy-Khintchine formula) for the Fourier transform of a Levy process.

32 I Preliminaries
Theorem 43. Let X be a Levy process with Levy measure v. Then
EieiuXt\ — e~*^u)
where
ф(и) = ^-u2 -iau+ I A - eiux)v{dx) + [ A - eiux + iux)v{dx).
Moreover given v, a2, a, the corresponding Levy process is unique in distri-
distribution.
The Levy-Khintchine formula has another, more common expression. (For
a complete discussion of the various formulae of Levy and Khintchine for the
case of infinitely divisible distributions, see Feller [72, pages 558-565].) Indeed,
in the n-dimensional case we can express the Levy-Khintchine formulas as
follows (letting (•, •) denote the standard inner product for ]Rn).
Theorem 44. Let X be a Levy process in Шп, with Xo = 0. Then there exists
a convolution semigroup of probability measures on K" such that C{Xt) — /j,t.
The characteristic function of ^ is е~*^и\ where ф is given in terms of a
positive definite n x n matrix E and a Levy measure v on W1 such that
(i) v{A) < oo if Л is Borel on Шп and bounded away from 0;
(ii) fM<1 \x\2v(dx) < oo and v{Q} = 0.
Also there exists a bounded, continuous centering function rj : WLn —> Kn
such that limj_,o ^ufe* = 0» an& a constant а €Шп such that
¦ф{и) = -i(u,a) + )г{Еи,и) - f ei(u'x) - 1 - i(u,-q{x))v{dx).
A nice proof of the above theorem can be found in Bertoin [12, pages
13-15]. Several observations concerning this theorem are in order. First, one
still has (in the n-dimensional case) the key property that for / : Rn —+ Rn,
bounded, vanishing in a neighborhood of 0, that
E{ Y, /(A*-)} = * Г f(x)v(dx).
0<s<t
Second, in the case n = 1, one often takes r)(x) = j^t, possibly at the cost of
changing the constant a. In the next section (Sect. 6) we discuss semigroups
of Markov processes. The semigroup Pt for the Levy process X has the form
Ptf(x) — J f(x + y)/J,t(dy), and one can show in fact that the "infinitesimal
generator" of the semigroup Pt is, for / e bC2(JR"),

4 Levy Processes 33
/¦
f(x + y)- f(x) -
Note further that if / is constant in a neighborhood of x, then
(X) =Jf(x + y)- f{x)u{dy).
lim Pt/(X)
We next consider some examples of Levy processes. For the examples we take
n = 1.
Example. If a Levy process X has continuous paths, then v is identically
0, and X must be Brownian motion with drift. That is, Xt = crBt + at, for
constants a, a, are the only Levy processes with continuous paths.
Example. Assume that v is a finite measure. That is, и(Шп) < со, and
that a2 = 0. Then X is a compound Poisson process with jump arrival
intensity Л = v(M.n). That is, let Nt — Yll^i l{t>Ti} be a Poisson process with
arrival intensity Л. Let Zt = ]СП=1 ^jl{t>T;} where the sequence (Uj)j>i are
i.i.d., independent of the sequence (Tj)j>i, and C{U\) = ji/. Then one easily
verifies E{eiuZt} = e-*^"\ where tp(u) = /A - eiux)v(dx).
Example. Let v(dx) = ^2<^L1 OLkepk{dx), where epk(dx) denotes the positive
point mass measure at Ck G К of size one. Assume Yl'kLx Pkak < °° and &2 =
0. Let Nk be a sequence of independent Poisson processes with parameters
afc. Then Xt = YlT=i Pk(Nt — afci) has v as its Levy measure.
Example. Let X be a Levy process which is a Gamma process. That is,
C(Xt) = Г(а,р), with /3 > 0 and a = at. More precisely, the density for Xt
is ft(x) = I!?,\xta~1e~f3xl{x>0y In this case a2 = 0, and taking a = C = 1
gives <pt(u) = E{ezuZt} — A — ro)~*. (See, for example, page 43 of Jacod-
Protter [109] or p. 73 of Bertoin [12].) Then <pt(u) = e-tln(i-<«) = е*^(«)
But ^
\ = гфх{и) = iE{eiuZl } = i f
A — ги) Jo
Integrating yields
ip(u) = / e~xdx
Jo x
which implies, by the Levy-Khintchine formula, that v{dx) = ~;e~xl{x>oydx.
Example. Our last example is that of stable processes. We begin by recall-
recalling what a stable distribution is. Extending the notion of the Central Limit
Theorem, consider the normalized sums
= t
(

34 I Preliminaries
s ZtiXi _p 0
In
We say that a random variable X has a stable law if it is the limit in
distribution as n —> oo of sums of the above form. One can then prove (see,
for example, Breiman [23, page 200]) that either X has a normal distribution,
or there exists a number a, 0 < a < 2, called the index of the stable law, and
also constants mj > 0, тг > 0, and S such that E{eluX} = e^u^ and
°° 9VT 1
One says that a stable distribution is symmetric if m\ = тг. By analogy we
define a symmetric stable process (in the case n = 1) to be a Levy process
where the Levy measure has the form v(dx) = . u\+a) dx, with 0 < a < 2,
and the constant a2 in the Levy-Khintchine formula is 0, which, of course,
implies that no Brownian component is present in the process. Note that such
stable processes have nice scaling properties. If a is the index, then the law
of (Xt)t>0 is the same as the law of {fl~^Xpt)t>Q- Consult Bertoin [12] for
many other properties of stable processes.
5 Why the Usual Hypotheses?
We defined the usual hypotheses at the very beginning of this book, and we
saw with Levy processes that the natural filtration of a Levy process, once
completed, satisfies a fortiori right continuity, and thus the usual hypotheses
hold (see Theorem 31). This phenomenon is true much more generally. Indeed
it is true for all "reasonable" strong Markov processes, as well as for all count-
counting processes (cf., Theorem 25). In this section we will show that it is true for
a very large class of Markov processes. We first begin with some definitions.
Definition 1. Let X be an Rd-valued process, with X adapted to a filtration
F = (Ft)o<t<oo- X is Markov with respect to F it Ft and cr{X3; s > t} are
conditionally independent, given Xt.
Definition 2. The process X in Definition 1 is Markov with respect to
F if for any random variable Y e ha{Xs; s > t}, E{Y\Tt} = E{Y\Xt}, a.s.,
all t > 0.
The following theorem has a straightforward proof.
Theorem 45. Definitions 1 and 2 are equivalent. Moreover X is Markov with
respect toWifand only if E{f(Xu)\Ft} = E{f(Xu)\Xt} a.s. for allu>t>0.

5 Why the Usual Hypotheses? 35
Let X be a cadlag process and let X be Markov with respect to F. Since X
is Markov, we can associate with it a Markov transition function on (Rd,B),
where В denotes the Borel sets of Rd. This is a family (Ps,t)s,teR+ of Markov
kernels on (Rd, B) such that
for each triple (s, t, u) with s < t < u. We then have the fundamental rela-
relationship
E{f(X)\F} = Ps,t(XsJ) a.s.
Note that, by using indicator functions such as 1^ = /, we have
PSyt{Xs,A) a.s.
Also note that the above implies the equality
Ps,t(Xs,A) a.s.
which, of course, is weaker. That is, the converse implication is not true in
general. The most important special case is Markov processes which are time
homogeneous. In this case we have a transition semigroup Pt given by
Pt-s = Ps,t>
for s < t.
Definition. Let Co denote continuous functions vanishing at infinity. A
Markov process is a Feller process if its transition semigroup (Pt)t>o has
the following two properties.
(a) For each / e Co and each t > 0, Ptf e Co.
(b) For each / e Co and each x e Rd, Ptf(x) -> f(x) as t -> 0.
Examples of Feller processes include Brownian motion and the Poisson
process. More generally, one can show that diffusions5 and all Levy processes
are Feller processes.
Theorem 46. Let X be a cadlag Feller process for a filtration ?. Then it is
also Markov with respect to the filtration ?+ = (.Ft+)o<t<oo-
Proof. Let t > 0 be fixed, and assume / G Co. Let Yt = f(Xt). For 0 < s < t,
let
Ys = E{f(Xt)\Fs} = ??{/(*t)|*e} = Pt-.f(Xa).
Then У is a martingale for 0 < s < t, and it is cadlag, because Ptf is
continuous, and X is cadlag. Therefore, У is a martingale with respect to
F-|- too, which means that E{f(Xt)\Jrs+} — Ys, from which we deduce that
E{f{Xt)\Ts+} = E{f{Xt)\Xa}, and we are done. D
5 This depends, of course, on how one defines a diffusion; the definition is not yet
standardized in the literature.

36 I Preliminaries
We now define P^ as a probability measure such that X is Markov with
respect to P^ and also C(X0) = \i under P^, where C(X0) denotes the distri-
distribution, or "law," of Xo. Let F^ = (^f )o<t<oo be the nitration where T% de-
denotes the P» completion of F$, J^ = a{Xs : s <t}, and F° = a{Xs : s > 0}.
Also, let .Ft =
Theorem 47. If X is Markov with respect to PM, or alternatively Markov
with respect to P^ for all /j, with \\/j,\\ = 1, then F^ and F are, respectively,
right continuous.
Proof. We prove the result for FM, from which the result for F follows. Let H
denote all h e bjF^ such that
E{h\Jf} = E{h\Jf+}, P*-a.s.
Then H Э {H = f(Xs), f e ЪВ}, which is obvious for s <t, and for s > t it
is just Theorem 46. Then H contains all H of the form H = ПГ=1 /»(-^«<)> f°r
any / G ЪВ. Using the Monotone Class Theorem, H э bJ70. By completion,
П D b^. Therefore if h G Jf+, we have ?"{/i|^} = E{h\Jf+} = h, P»-
a.s. П
The property that the natural nitration of a Feller process is right con-
continuous when completed is not restricted to Feller processes. Indeed, it can
also be shown to be true for more general classes of processes, such as Hunt
processes^. However it is simply two elementary properties which yield the
right continuity of the filtration in a general setting.
Definition. Let X be a cadlag Markov process for a nitration F, which we
take to be completed and therefore can assume to be right continuous, in view
of Theorem 47. Let T be a stopping time. We say that X verifies the strong
Markov property at T if
E{f(XT+s)l{T<oo}\TT} = PS(XTJI{T<OO} a.s.
We say that X is strong Markov if the above holds for every stopping time
T.
Suppose now that the semigroup (Pj) is Borel. That is, if / e B, then
Ptf € В as well. Suppose that for every probability measure fi on (Rd,B),
there exists a probability space (fi, G, Q) and a cadlag Markov process (lt)t>o
denned on this space with C(X0) = fi. Suppose further that every such cadlag
Markov process Y is strong Markov. Then the filtrations F^ = (^r|')o<t<oo and
F = (.Ft)o<t<oo can be shown to be right continuous in a manner analogous
to the previous proofs.
See, for example, Sharpe [215, page 219] for a definition of a Hunt process.

6 Local Martingales 37
6 Local Martingales
Recall that we assume as given a filtered probability space (fi, T, F, P) satis-
satisfying the usual hypotheses. For a process X and a stopping time T we further
recall that XT denotes the stopped process
Xt = XtAr = Xtl{t<T) + ^т1{*>т>-
Definition. An adapted, cadlag process X is a local martingale if there
exists a sequence of increasing stopping times, Tn, with Нт^-юо Тп = oo a.s.
such that XtATnl{Tn>o} IS a uniformly integrable martingale for each n. Such
a sequence (Tn) of stopping times is called a fundamental sequence.
Example. Clearly any cadlag martingale is a local martingale (take Tn = n).
Example. We give an example of a local martingale which is not a martingale.
Let (.Bt)o<t<oo be a Brownian motion in M3 with B$ = x, where x / 0.
Let u(y) — \\y\\~1, a superharmonic function on M3. As a consequence of
Ito's formula (cf., Theorems 32 and 41 of Chap. II) one can show that Xt =
u(Bt) is a positive supermartingale (indeed, it is even a uniformly integrable
supermartingale). Next let Tn = inf{t > 0 : \\Bt\\ < l/nl- Outside of the
ball of radius \/n centered at the origin the function и is harmonic. Again
by Ito's formula one can show that this implies u(BtATn) is a martingale.
Since u{Bt/KTn) is bounded by n it is uniformly integrable. On the other hand,
if Bo = x ф 0 then lim^oo E{u(Bt)} = 0 as is easily seen by a calculation,
while E{u(Bo)} = \\x\\~1. Since the expectations of a martingale are constant,
u(Bt) is not a martingale. (For more on this example, see Sect. 6 of Chap. II,
following Corollary 4 of Theorem 27, as well as Exercise 20 in Chap. II.)
The reason we multiply Х4Лт„ by 1{т„>о} is to relax the integrability
condition on Xo- This is useful, for example, in the consideration of stochastic
integral equations with a non-integrable initial condition.
Definition. A stopping time T reduces a process M if MT is a uniformly
integrable martingale.
Theorem 48. Let M, N be local martingales and let S and T be stopping
times.
(a) IfT reduces M and S <T a.s., then S reduces M.
(b) The sum M + N is also a local martingale.
(c) If S, T both reduce M, then S V T also reduces M.
(d) The processes MT, Мт1{т>о} are local martingales.
(e) Let X be a cadlag process and let Tn be a sequence of stopping times
increasing to 00 a.s. such that XTjll^pn>o} is a local martingale for each
n. Then X is a local martingale.
Proof, (a) follows from the Optional Sampling Theorem (Theorem 16) and
Theorem 13. For (b), if En)n>i, (Tn)n>i are fundamental sequences for M

38 I Preliminaries
and N, respectively, then Sn /\Tn is a fundamental sequence for M + N. For
(c), let Xt = Mt - Mo. Then XSyT = Xs + XT - XSAT is a uniformly
integrable martingale. But
which is in L1, therefore XSvT+Mol{5VT>o} = -WSvTl{svT>o} is a uniformly
integrable martingale. The proof of (d) consists of the observation that if
(Tn)n>i is a fundamental sequence for M, then it is also one for MT and
Мт1{т>о} by the Optional Sampling Theorem (Theorem 16). For (e), we
have XTnl{xn>o} = Mn is a local martingale for each n. For fixed n we know
there exists a fundamental sequence Un'k increasing to сю a.s. as к tends
to oo. For each n, choose к = k(n) such that P(Un'h^ < Tn An) < 2~n.
Then limn Un'k^ = oo a.s., and [/"•*(") л Тп reduces X for each n. We
take Rm = max(!71'kA) ЛТЬ...,f/m-fe(m) Л Тт), and each Rm reduces X
by (c), the Rm are increasing, and lim Rm = oo a.s. Therefore X is a local
martingale. ?
Corollary. Local martingales form a vector space.
We will often need to know that a reduced local martingale, MT, is in Lp
and not simply uniformly integrable.
Definition. Let X be a stochastic process. A property тг is said to hold
locally if there exists a sequence of stopping times (Tn)n>i increasing to oo
a.s. such that ХТп1{т„>о} has property тг, each n > 1.
We see from Theorem 48(e) that a process which is locally a local mar-
martingale is also a local martingale. Other examples of local properties that
arise frequently are locally bounded and locally square integrable. Theorems 49
and 50 are consequences of Theorem 48(e).
Theorem 49. Let X be a process which is locally a square integrable martin-
martingale. Then X is a local martingale.
The next theorem shows that the traditional "uniform integrability" as-
assumption in the definition of local martingale is not really necessary.
Theorem 50. Let M be adapted, cadlag and let (Tn)n>i be a sequence of
stopping times increasing to oo a.s. If МТп1{тп>о} *s a martingale for each
n, then M is a local martingale.
It is often of interest to determine when a local martingale is actually a
martingale. A simple condition involves the maximal function. Recall that
XI = sups<t \XS\ and X* = sups \XS\.
Theorem 51. Let X be a local martingale such that E{X?} < oo for every
t > 0. Then X is a martingale. If E{X*} < oo, then X is a uniformly
integrable martingale.

7 Stieltjes Integration and Change of Variables 39
Proof. Let (Tn)n>i be a fundamental sequence of stopping times for X. If
s < t, then E{Xt^Tn\^s} = Xs^Tn- The Dominated Convergence Theorem
yields Е{Хг\Т8} = Xs. If E{X*} < oo, since each \Xt\ < X*, it follows that
(Xt)t>o is uniformly integrable. ?
Note that in particular a bounded local martingale is a uniformly inte-
integrable martingale. Other sufficient conditions for a local martingale to be a
martingale are given in Corollaries 3 and 4 to Theorem 27 in Chap. II, and
Kazamaki's and Novikov's criteria (Theorems 40 and 41 of Chap. Ill) establish
the result for the important special cases of continuous local martingales.
7 Stieltjes Integration and Change of Variables
Stochastic integration with respect to semimartingales can be thought of as
an extension of path-by-path Stieltjes integration. We present here the es-
essential elementary ideas of Stieltjes integration appropriate to our interests.
We assume the reader is familiar with the Lebesgue theory of measure and
integration on M_|_.
Definition. Let A = (At)t>0 be a cadlag process. A is an increasing pro-
process if the paths of A : t —> At(u) are non-decreasing for almost all u. A is
called a finite variation process (FV) if almost all of the paths of A are of
finite variation on each compact interval of K_|_.
Let A be an increasing process. Fix an и such that t i—> At(u) is right
continuous and non-decreasing. This function induces a measure ij,a{u, ds)
on K.+ . If / is a bounded, Borel function on R+, then Jo /(s)^^, ds) is
well-defined for each t > 0. We denote this integral by Jo f(s)dAs(u). If
Fs = F(s,u) is bounded and jointly measurable, we can define, и-Ъу-и, the
integral I{t,w) = J* F(s,u)dAs(u). I is right continuous in t and jointly
measurable.
Proceeding analogously for A an FV process (except that the induced
measure (j,a(<^>, ds) can have negative measure; that is, it is a signed measure),
we can define a jointly measurable integral
I(t,w)= f F(s,uj)dAs(w)
Jo
for F bounded and jointly measurable.
Let A be an FV process. We define
Then \A\t < oo a.s., and it is an increasing process.

40 I Preliminaries
Definition. For A an FV process, the total variation process, \A\ =
(|j4|t)t>o, is the increasing process defined in (*) above.
Notation. Let A be an FV process and let F be a jointly measurable process
such that Jo F(s,u>)dAs(u>) exists and is finite for all t > 0, a.s. We let
(F-A)t(u)= I F(s,u)dAs(u)
Jo
and we write F ¦ A to denote the process F ¦ A = (F ¦ At)t>0. We will also
write /0 F.s|cL4s| for (F ¦ \A\)t. The next result is an absolute continuity result
for Stieltjes integrals.
Theorem 52. Let А, С be adapted, strictly increasing processes such that
С — A is also an increasing process. Then there exists a jointly measurable,
adapted process H (defined on @, oo)J such that 0 < H < 1 and
A^H-C
or equivalently
At= I HsdCs.
Jo
Proof. If /i and v are two Borel measures on E+ with |i«i/, then we can set
otherwise.
Defining h and к by h(t) = lirninis^ta(s,t), k(t) = limsupt^sa(s,i), then h
and к are both Borel measurable, and moreover they are each versions of the
Radon-Nikodym derivative. That is,
djj, = h dv, djj, = к dv.
To complete the proof it suffices to show that we can follow the above proce-
procedure in a (t,tj) measurable fashion. With the convention jy = 0, it suffices to
define
H(t,u) = hminf —Г7 — ^-;
V ' ; rTl,r€Q+ (C(t,u)-C(rt,w))'
such an H is clearly adapted since both A and С are. ?
Corollary. Let A be an FV process. There exists a jointly measurable,
adapted process H, — 1 < H < 1, such that
\A\ = H-A and A = H-\A\
or equivalently
|Л|t = / HsdAs and At = / Hs\dAs\.
Jo Jo

7 Stieltjes Integration and Change of Variables 41
Proof. We define A+ = \{\A\t + At) and A^ = \{\A\t-At). Then A+ and A~
are both increasing processes, and \A\— A+ and \A\ — A~ are also increasing
processes. By Theorem 52 there exist processes H+ and H~ such that Л^1" =
J* Я+|йЛв|, Л" = /o H-\dA.\. It then follows that At = A+ -A; = J*(H+ -
H~)\dAs\. Let Ht = Hf — H^ and suppose H+ and Л~ are defined as in
the proof of Theorem 52. Except for a P-null set, for a given w it is clear that
\Hs(u)\ — 1 dAs(w) almost all s. Considering H ¦ A, we have
f HsdAs= f Hsd{f Hu\dAu\)
Jo Jo Jo
= I HsHs\dAs\ = I l\dAs\ = \A\t.
Jo Jo
This completes the proof. ?
When the integrand process H has continuous paths, the Stieltjes integral
Jo HsdAs is also known as the Riemann-Stieltjes integral (for fixed u>). In
this case we can define the integral as the limit of approximating sums. Such
a result is proved in elementary textbooks on real analysis (e.g., Protter-
Morrey [195, pages 316, 317]).
Theorem 53. Let A be an FV process and let H be a jointly measurable
process such that a.s. s <—> H(s,w) is continuous. Let nn be a sequence of
finite random partitions of [0,t] with limn_,oo mesh(?rn) = 0. Then for 2\ <
- ЛТк) = HsdAs a.s.
J
rt
= H
Jo
We next prove a change of variables formula when the FV process is
continuous. Ito's formula (Theorem 32 of Chap. II) is a generalization of this
result.
Theorem 54 (Change of Variables). Let A be an FV process with con-
continuous paths, and let f be such that its derivative f exists and is continuous.
Then (f(At))t>0 is an FV process and
f(At)-f(Ao)= f f'{As)dAs.
Jo
Proof. For fixed a», the function s \—> f'{As(ui)) is continuous on [0,t] and
hence bounded. Therefore the integral /0 f'(As)dAs exists. Fix t and let тг„
be a sequence of partitions of [0, t] with 1ш1п_юо meshGrra) = 0. Then
f(At)-f(Ao)= V U(Atk+1)-f(Atk)}

42 I Preliminaries
by the Mean Value Theorem, for some Sk, tk < Sk < *fc+i- The result now
follows by taking limits and Theorem 53. ?
Comment. We will see in Chap. II that the sums
f(Atk)(Atk+1 - Atk)
**€тг„[О,*]
converge in probability to
f
Jo
f f{As_)dAs
o
for a continuous function / and an FV process A. This leads to the more
general change of variables formula, valid for any FV process A, and f € C1,
namely
-/(A,)= f f\A..)dA.+ J2 {f(As)-f(As_)~f'(As_)AAs}.
0<s<t
The next corollary explains why Theorem 54 is known as a change of
variables formula. The proof is an immediate application of Theorem 54.
Corollary. Let g be continuous, and let h(t) = Jo g(u)du. Let A be an FV
process with continuous paths. Then
h(At) - h(Ao) = / g{u)du
J Aq
= I g{As)dAa.
Jo
We conclude this section with an example.
Example. Let TV be a Poisson process with parameter A. Then Mt = Nt —
Xt, the compensated Poisson process, is a martingale, as well as an FV
process. For a bounded (say), jointly measurable process H, we have
It = I HsdMs = I Hsd(Ns - Xs) = I HsdNa - X [ Hsds
Jo Jo Jo Jo
/
Hsds
0
where (Tn)n>i are the arrival times of the Poisson process N. Now suppose
the process H is bounded, adapted, and has continuous sample paths. For
0 < s < t < oo, we then have

8 Naive Stochastic Integration Is Impossible 43
Е{1г-18\Т8} = Е{ HudMu\Fs)
= 0.
The interchange of limits can be justified by the Dominated Convergence The-
Theorem. We conclude that the integral process / is a martingale. This fact, that
the stochastic Stieltjes integral of an adapted, bounded, continuous process
with respect to a martingale is again a martingale, is true in much greater
generality. We shall treat this systematically in Chap. II.
8 Nai've Stochastic Integration Is Impossible
In Sect. 7 we saw that for an FV process A, and a continuous integrand
process H, we could express the integral /0 HsdAs as the limit of sums (The-
(Theorem 53). The Brownian motion process, B, however, has paths of infinite
variation on compacts. In this section we demonstrate, with the aid of the
Banach-Steinhaus Theorem7, some of the difficulties that are inherent in try-
trying to extend the notion of Stieltjes integration to processes that have paths of
unbounded variation, such as Brownian motion. For the reader's convenience
we recall the Banach-Steinhaus Theorem.
Theorem 55. Let X be a Banach space and let Y be a normed linear space.
Let {Ta} be a family of bounded linear operators from X into Y. If for each
x e X the set {Tax} is bounded, then the set {||Ta||} is bounded.
Let us put aside stochastic processes for the moment. Let x(t) be a right
continuous function on [0,1], and let nn be a refining sequence of dyadic
rational partitions of [0,1] with lim^^oo meshGrra) = 0. We ask the question
"What conditions on x are needed so that the sums
converge to a finite limit as n —> oo for all continuous functions h?" From
Theorem 53 of Sect. 7 we know that x of finite variation is sufficient. However,
it is also necessary.
Theorem 56. If the sums Sn of (*) converge to a limit for every continuous
function h then x is of finite variation.
7 The Banach-Steinhaus Theorem is also known as the Principle of Uniform Bound-
edness.

44 I Preliminaries
Proof. Let X be the Banach space of continuous functions equipped with the
supremum norm. Let Y be M, equipped with absolute value as the norm. For
heX, let
For each fixed n it is simple to construct an h in X such that h(tk) =
sign{x(tfe+i) - x(tk)}, and ||/i|| = 1. For such an h we have
Therefore
iitji > V
each n, and supra ||ГП|| > total variation of x. On the other hand for each
h € X we have 1ш1п_юо ТП(Л) exists and therefore supra ||Tn(ft)|| < oo. The
Banach-Steinhaus Theorem then implies that supra ||Tn|| < oo, hence the total
variation of x is finite. D
Returning to stochastic processes, we might hope to circumvent the lim-
limitations imposed by Theorem 56 by appealing to convergence in probability.
That is, if X(s,u) is right continuous (or even continuous), can we have the
sums
J2 Hth(Xth+1 - Xth) (¦*)
converging to a limit in probability for every continuous process HI Unfortu-
Unfortunately the answer is that X must still have paths of finite variation, a.s. The
reason is that one can make the procedure used in the proof of Theorem 56
measurable in u, and hence a subsequence of the sums in (**) can be made
to converge a.s. to +oo on the set where X is not of finite variation. If this
set has positive probability, the sums cannot converge in probability.
The preceding discussion makes it appear impossible to develop a coherent
notion of a stochastic integral Jo HsdXs when X is a process with paths of
infinite variation on compacts; for example a Brownian motion. Nevertheless
this is precisely what we will do in Chap. II.
Bibliographic Notes
The basic definitions and notation presented here have become fundamental
to the modern study of stochastic processes, and they can be found many
places, such as Dellacherie-Meyer [45], Doob [55], and Jacod-Shiryaev [110].
Theorem 3 is true in much greater generality. For example the hitting time of
a Borel set is a stopping time. This result is very difficult, and proofs can be
found in Dellacherie [41] or [42].

Exercises for Chapter I 45
The resume of martingale theory consists of standard theorems. The reader
does not need results from martingale theory beyond what is presented here.
Those proofs not given can be found in many places, for example Breiman [23],
Dellacherie-Meyer [46], or Ethier-Kurtz [71].
The Poisson process and Brownian motion are the two most important
stochastic processes for the theory of stochastic integration. Our treatment
of the Poisson process follows Qinlar [33]. Theorem 25 is in Bremaud [24],
and is due to J. de Sam Lazaro. The facts about Brownian motion needed for
the theory of stochastic integration are the only ones presented here. A good
source for more detailed information on Brownian motion is Revuz-Yor [208],
or Hida [88].
Levy processes (processes with stationary and independent increments) are
a crucial source of examples for the theory of semimartingales and stochastic
integrals. Indeed in large part the theory is abstracted from the properties of
these processes. There do not seem to be many presentations of Levy processes
concerned with their properties which are relevant to stochastic integration
beyond that of Jacod-Shiryaev [110]. However, one can consult the books of
Bertoin [12], Rogers-Williams [210], and Revuz-Yor [208]. Our approach is
inspired by Bretagnolle [25]. Local martingales were first proposed by K. I to
and S. Watanabe [102] in order to generalize the Doob-Meyer decomposition.
Standard Stieltjes integration applied to finite variation stochastic processes
was not well known before the fundamental work of Meyer [171]. Finally, the
idea of using the Banach-Steinhaus Theorem to show that naive stochastic
integration is impossible is due to Meyer [178].
Exercises for Chapter I
For all of these exercises, we assume as given a filtered probability space
(fi, T, F, P) satisfying the usual hypotheses.
Exercise 1. Let S, T be stopping times, S <T a.s. Show Ts С Тт-
Exercise 2. Give an example where S, T are stopping times, S < T, but
T — S is not a stopping time.
Exercise 3. Let (Tn)n>i be a sequence of stopping times. Show that supra Tn,
infnTn, \imsupn_,ooTn, and liminf^-^Tn are all stopping times.
Exercise 4. Suppose (Tn)n>i is a sequence of stopping times decreasing
to a random time T. Show that T is a stopping time, and moreover that
Гт = Г\пГтп.
Exercise 5. Let p > 1 and let Mn be a sequence of continuous martingales
(that is, martingales whose paths are continuous, a.s.) with МД, e Lp, each
n. Suppose M? -» X in Lp and let Mt = E{X\Ft}-
(a) Show Mt G Lp, all t > 0.

46 Exercises for Chapter I
(b) Show M is a continuous martingale.
Exercise 6. Let TV = (Nt)t>0 be Poisson with intensity A. Suppose Xt is an
integer. Show that E{\Nt - Xt\} =
Exercise 7. Let TV = (TVt)t>o be Poisson with intensity A. Show that TV is
continuous in L2 (and hence in probability), but of course TV does not have
continuous sample paths.
Exercise 8. Let Bt = (??/,??42,.В43) be three dimensional Brownian motion
(that is, the Bl are i.i.d. one dimensional Brownian motions, 1 < г < 3). Let
та = supt{||Bt|| < a}. The time та is known as a last exit time. Show that
if a > 0, then та is not a stopping time for the natural filtration of B.
Exercise 9. Let GVtJ)i>o be a sequence of i.i.d. Poisson processes with pa-
parameter A = 1. Let Ml = \{Щ - t), and let M = (?~i M')t>0.
(a) Show that M is well-defined (i.e., that the series converges) in the I?
sense.
*(b) Show that for any t > 0, Yls<t &MS = oo a.s., where AMS — Ms - Ms_
is the jump of M at time s.
Exercise 10. Let (iVt*)t>o and (LJt)t>obe two independent sequences of i.i.d.
Poisson processes of parameters A = 1. Let Ml = \{Nl - L\).
(a) Show that M is a martingale and that M changes only by jumps.
*(b) Show that ]Ts<t |ДМа| = oo a.s., and t > 0.
Exercise 11. Show that a compound Poisson process is a Levy process.
Exercise 12. Let Z be a compound Poisson process, with i5{|[/i|} < сю.
Show that Zt — E{Ux}Xt is a martingale.
Exercise 13. Let Z = (Zt)t>o be a Levy process which is a martingale, but
Z has no Brownian component (that is, the constant a1 — 0 in the Levy-
Khintchine formula), and a finite Levy measure v. Let A = и(Ж). Show that
Z is a compensated compound Poisson process with arrival intensity A and
i.i.d. jumps with distribution /i = ji/.
*Exercise 14. Let 0 < T\ < T^ < ¦ ¦ ¦ < Tn < ¦ ¦ ¦ be any increasing sequence
of finite-valued stopping times, increasing to oo. Let Nt = SSi 1{*>T;}> and
let (Ui)i>i be independent, and also the (Ui)i>i are independent of the pro-
process TV. l3how that if supt E{\Ui\} < oo, all i, and E{Ui) = 0, all i, then
Zt = Y^i Ui\{t>Ti) ls a martingale. (Note: We are not assuming that TV
is a Poisson process here. Furthermore, the random variables (f*)i>i, while
independent, are not necessarily identically distributed.)
Exercise 15. Let Z be a Levy process which is a martingale, but with no
Brownian component and with a Levy measure of the form

Exercises for Chapter I 47
oo
v(dx) = y^akepk{dx),
fe=i
where e@k (dx) denotes point mass at /?? G R of size one, with Xjfcli /?fcafc < °°-
Show that Z is of the form
fe=i
where (Nf)t>0 is an independent sequence of Poisson processes with param-
parameters («fc)fc>i. Verify that Z is an I? martingale.
Exercise 16. Let В = (Bt)o<t<i be one dimensional Brownian motion on
[0,1]. Let Wt = Bi_t - B\, 0 < t < 1. Show that W has the same distribution
as does B, and that W is a Brownian motion for 0 < t < 1 for its natural
filtration.
Exercise 17. Let X be any process with cadlag paths. Set e > 0.
(a) Show that on [0, t] there are only a finite number of jumps larger than e.
That is, if Nt = ?s<t 1{\axb\>c}, where AXS =XS- Xs-, then P(Nt <
oo) = 1, each t > 0.
(b) Conclude that X can have only a countable number of jumps on [0,t].
Exercise 18. Let Z be a Levy process, and let e > 0. Let AZS = Zs — Zs~,
and set Jt? = X^<t ^^s^{\azs\>e}- Show that Z - J? and J? are independent
Levy processes.
Exercise 19. Let X be an adapted process with continuous paths. Show that
X is locally bounded.
*Exercise 20. Give an example of an adapted process X with Xo = 0 that
has cadlag paths which are not locally bounded.
Exercise 21. (This exercise shows that 2 is the best constant in Doob's
quadratic martingale inequality.) Let (Г2, J-, P) be Lebesgue measure on @,1)
with the Borel sets. Define the filtration
Tt = {A&T \ (t,l) cA, or Л с @,*]}, 0<*< 1,
and Tt—T for t > 1. Intuitively, with this filtration we know all about what
is going on during @, t], but nothing about what is happening during (t, 1).
(a) Show that for Y ? L1, the martingale Mt = Е{У\^} is given by
() 0<co<t,
Mt(u) = "
/ Y(s)ds, t < и < 1.
1 ~ * Jt

48 Exercises for Chapter I
(b) Let Y{uj) = A - u>)-a for some a, 0 < a < 1/2. Show that Y <E L2 and
also if Mt = E{Y\Ft), then
(c) Show supt Mt = jz^Y, и-Ъу-со, and deduce
Note that Iima_>1/2A - a) = 2.
Exercise 22. (This exercise gives an example of a martingale with each path
bounded but which nevertheless is not locally bounded.) Let (fi,.F,F, P) be
as in Exercise 21. Let Y(u>) = ^, and Mt = E{Y\Tt}.
(a) Show that
(b) Let Г be a stopping time, Г not identically 0. Show that T(w) > со on
some w-interval @, e) with e > 0.
(c) Use (b) to show E{M%.} > /Q? ^ds = oo, and deduce that M cannot be
locally an L2 martingale.
(d) Conclude that M has bounded paths but that Мт1{т>о) is unbounded
for every stopping time Г not identically 0.
*Exercise 23. Find an example of a local martingale M and finite stop-
stopping times S, T with S < T a.s. such that Е{\МТ\\Р3) < oo, but Ms ф
E{MT\Ts} a.s. {Hint: Begin by showing that the equality would imply that
every positive local martingale is a martingale.)
Exercise 24. (This exercise helps to clarify the difference between stopping
time (T-algebras of the two forms Gt- and Gt-) Let Z be a Levy process,
and let Gt — (?{Zs;s < i} V J\f, where N are all the null sets in the given
complete probability space [Q.,J-,P). For a stopping time Г define Gt- =
a{A П (t < Г); A ? Gt}; at time 0, set Go- = Go- For a process X, let
XT~ = Xtl{t<T] + XT-l{t>T}, and take X0- = 0.
(a) Show St- = cj{T, ZT~}vAf.
(b) Show Gt = v{T,Zt}vN.
(c) Show Gt = °{Gt-,Zt} if T is a.s. finite.
(d) Conclude that for a bounded time Г if Zy = Z?- a.s., then С/т = Gt--
(Note that if G = (?t)t>o is the completed natural filtration of a Brownian
motion, then Gt = Gt- for all bounded stopping times.)

Exercises for Chapter I 49
Exercise 25. Let Z be a Levy process (or any process which is continuous
in probability). Show that the probability Z jumps at a given fixed time t is
zero.
Exercise 26. Use Exercises 24 and 25 to show that with the assumptions of
Exercise 24, for any fixed time t, Qt- = Qt, giving a continuity property of
the filtration for fixed times (since by Theorem 31 we always have Qt+ — Qt),
but of course not for stopping times.
Exercise 27. Let S, T be two stopping times with S < T. Show Ts- С
Тт-- Moreover if (Tn)n>i is any increasing sequence of stopping times with
limn_,oo Tn = Г, show that
Ft- = \j Ртп--
n
Exercise 28. Prove the first equality of Theorem 38 when /1л е Ll{dv).
**Exercise 29. Show that if Z is a Levy process and a local martingale,
then Z is a martingale. That is, all Levy process local martingales are actually
martingales.
Exercise 30 (reflection principle for Levy processes). Let Z be a sym-
symmetric Levy process with Zo — 0. That is, Z and —Z have the same dis-
distribution. Let St = sups<t Zs. Show that
P(St >z-Zt<z-y)< P(Zt >z + y).
Exercise 31. Let Z be a symmetric Levy process with Zq — 0 as in Exer-
Exercise 30. Show that
P(St >z)< 2P{Zt > z).

II
Semimartingales and Stochastic Integrals
1 Introduction to Semimartingales
The purpose of the theory of stochastic integration is to give a reasonable
meaning to the idea of a differential to as wide a class of stochastic processes
as possible. We saw in Sect. 8 of Chap. I that using Stieltjes integration on a
path-by-path basis excludes such fundamental processes as Brownian motion,
and martingales in general. Markov processes also in general have paths of
unbounded variation and are similarly excluded. Therefore we must find an
approach more general than Stieltjes integration.
We will define stochastic integrals first as a limit of sums. A priori this
seems hopeless, since even by restricting our integrands to continuous pro-
processes we saw as a consequence of Theorem 56 of Chap. I that the differential
must be of finite variation on compacts. However an analysis of the proof of
Theorem 56 offers some hope. In order to construct a function h such that
h(tk) = sign(x(ifc+i) — x(tk)), we need to be able to "see" the trajectory of x
on (tk,tk+i]- The idea of K. Ito was to restrict the integrands to those that
could not see into the future increments, namely adapted processes.
The foregoing considerations lead us to define the stochastic processes that
will serve as differentials as those that are "good integrators" on an appropri-
appropriate class of adapted processes. We will, as discussed in Chap. I, assume that
we are given a filtered, complete probability space (fi,.F,F, P) satisfying the
usual hypotheses.
Definition. A process H is said to be simple predictable if H has a rep-
representation
Ht =
where 0 = T\ < ¦ ¦ ¦ < Tn+i < oo is a finite sequence of stopping times,
Hi G TTi with \Ht\ < oo a.s., 0 < i < n. The collection of simple predictable
processes is denoted S.

52 II Semimartingales and Stochastic Integrals
Note that we can take T\ = To = 0 in the above definition, so there is
no "gap" between To and T\. We can topologize S by uniform convergence in
(t,co), and we denote S endowed with this topology by Su. We also write L°
for the space of finite-valued random variables topologized by convergence in
probability.
Let X be a stochastic process. An operator, Ix, induced by X should
have two fundamental properties to earn the name "integral." The operator
Ix should be linear, and it should satisfy some version of the Bounded Con-
Convergence Theorem. A particularly weak form of the Bounded Convergence
Theorem is that the uniform convergence of processes Hn to H implies only
the convergence in probability of Ix{Hn) to Ix(H).
Inspired by the above considerations, for a given process X we define a
linear mapping Ix '¦ S —> L° by letting
n
Ix{H) = ЩХо + 2_^ Hi(Xri+1 — Xti),
2=1
where H e S has the representation
Ht = ffol{o} '
Since this definition is a path-by-path definition for the step functions
H(u>), it does not depend on the choice of the representation of H in S.
Definition. A process X is a total semimartingale if X is cadlag, adapted,
and Ix '¦ Su —> L° is continuous.
Recall that for a process X and a stopping time Г, the notation XT denotes
the process (XtAT)t>o-
Definition. A process X is called a semimartingale if, for each t G [0, oo),
X1 is a total semimartingale.
With our definition of a semimartingale, the second fundamental property
we want (bounded convergence) holds. We postpone consideration of examples
of semimartingales to Sect. 3.
2 Stability Properties of Semimartingales
We state a sequence of theorems giving some of the stability results which are
particularly simple.
Theorem 1. The set of (total) semimartingales is a vector space.
Proof. This is immediate from the definition. D

2 Stability Properties of Semimartingales 53
Theorem 2. If Q is a probability and absolutely continuous with respect to
P, then every (total) P semimartingale X is a (total) Q semimartingale.
Proof Convergence in P-probability implies convergence in Q-probability.
Thus the theorem follows from the definition of X. D
Theorem 3. Let (Pk)k>i be a sequence of probabilities such that X is a Pk
semimartingale for each к. Let R = ^ZfeLi ^кРк> where Xk > 0, each k, and
Sfet=i -^fc = 1- Then X is a semimartingale under R as well.
Proof. Suppose Hn e S converges uniformly to H G S. Since X is a Pk
semimartingale for all Pk, Ix{Hn) converges to Ix(H) in probability for every
Pk- This then implies Ix(Hn) converges to Ix(H) under R. О
Theorem 4 (Strieker's Theorem). Let X be a semimartingale for the
filtration F. Let G be a subfiltration of F, such that X is adapted to the G
filtration. Then X is a G semimartingale.
Proof. For a filtration И, let S(H) denote the simple predictable processes for
the filtration Ш = (Tit)t>o- In this case we have S(G) is contained in S(F).
The theorem is then an immediate consequence of the definition. G
Theorem 4 shows that we can always shrink a filtration and preserve the
property of being a semimartingale (as long as the process X is still adapted),
since we are shrinking as well the possible integrands; this, in effect, makes
it "easier" for the process X to be a semimartingale. Expanding the filtra-
filtration, therefore, should be—and is—a much more delicate issue. Expansion
of nitrations is considered in much greater detail in Chap. VI. We present
here an elementary but useful result. Recall that we are given a filtered space
{?l,J-, F, P) satisfying the usual hypotheses.
Theorem 5 (Jacod's Countable Expansion). Let A be a collection of
events in T such that if Aa, Ap e A then Aa П A$ = 0, (а ф /3). Let Tit be
the filtration generated by Ft and A. Then every (F, P) semimartingale is an
(H, Р) semimartingale also.
Proof. Let An G A. If P{An) = 0, then An and An are in J-q by hypothesis.
We assume, therefore, that P{An) > 0. Note that there can be at most a
countable number of An e A such that P(An) > 0. If Л = \Jn>1 An is the
union of all An € A with P(An) > 0, we can also add Ac to A without loss
of generality. Thus we can assume that A is a countable partition of Q with
P(An) > 0 for every An G A. Define a new probability Qn by Qn{-) = P(-\An),
for An fixed. Then Qn <C P, and X is a (F, Qn) semimartingale by Theorem 2.
If we enlarge the filtration F by all the T measurable events that have Qn-
probability 0 or 1, we get a larger filtration Jn = (J")t>o, and X is a (J™, Qn)
semimartingale. Since Qn(Am) = 0 or 1 for m ф n, we have Tt С Ht С J™,
for t > 0, and for all n. By Strieker's Theorem (Theorem 4) we conclude that
X is an (И, Qn) semimartingale. Finally, we have dP = J2n>i P(An)dQn,

54 II Semimartingales and Stochastic Integrals
where X is an (H, Qn) semimartingale for each n. Therefore by Theorem 3
we conclude X is an (H, P) semimartingale. D
Corollary. Let Л be a finite collection of events in F, and let H = (Ht)t>o
be the filtration generated by Tt and A. Then every (F, P) semimartingale is
an (H, P) semimartingale also.
Proof. Since A is finite, one can always find a (finite) partition П of п such
that Ht= f(V П. The corollary then follows by Theorem 5. ?
Note that if В = (Bt)t>o is a Brownian motion for a filtration {J~t)t>o,
by Theorem 5 we are able to add, in a certain manner, an infinite number of
"future" events to the filtration and В will no longer be a martingale, but it
will stay a semimartingale. This has interesting implications in finance theory
(the theory of continuous trading). See for example Duffie-Huang [60].
The corollary of the next theorem states that being a semimartingale is a
"local" property; that is, a local semimartingale is a semimartingale. We get
a stronger result by stopping at Tn— rather than at Tn in the next theorem.
A process X is stopped at Г— if Xj~ = Xfl{o<t<T} + ^T-l{t>T}> where
Xo- = 0.
Theorem 6. Let X be a cadlag, adapted process. Let (Tn) be a sequence of
positive r.v. increasing to oo a.s., and let (Xn) be a sequence of semimartin-
semimartingales such that for each n, XTn~ = (Xn)Tn~. Then X is a semimartingale.
Proof. We wish to show X* is a total semimartingale, each t > 0. Define
Rn = Tnl{Tn<t} + °°l{Tn>t}- Then
P{\Ix*{H)\ >c}< P{\I{xn)t(H)\ >c} + P(Rn < oo).
But P(Rn < oo) = P(Tn < t), and since Tn increases to oo a.s., P(Tn < t) —> 0
as n —> oo. Thus if Hk tends to 0 in Su, given e > 0, we choose n so that
P(Rn < oo) < e/2, and then choose к so large that P{\I(X"y(Hk)\ > с} <
e/2. Thus, for к large enough, P{\IXt(Hk)\ > c} < e. ?
Corollary. Let X be a process. If there exists a sequence (Tn) of stopping
times increasing to oo a.s., such that XTn (or XTnl{Tn>0}) is a semimartin-
semimartingale, each n, then X is also a semimartingale.
3 Elementary Examples of Semimartingales
The elementary properties of semimartingales established in Sect. 2 will allow
us to see that many common processes are semimartingales. For example, the
Poisson process, Brownian motion, and more generally all Levy processes are
semimartingales.

3 Elementary Examples of Semimartingales 55
Theorem 7. Each adapted process with cadlag paths of finite variation on
compacts (of finite total variation) is a semimartingale (a total semimartin-
semimartingale).
Proof. It suffices to observe that \IX{H)\ < \\H\\U /0°° \dXs\, where /0°° \dX3\
denotes the Lebesgue-Stieltjes total variation and ||#||u — sup(t№) \H(t,w)\.
П
Theorem 8. Each L2 martingale with cadlag paths is a semimartingale.
Proof Let X be an I? martingale with Xo = 0, and let H G S. Using Doob's
Optional Sampling Theorem and the L2 orthogonality of the increments of
L2 martingales, it suffices to observe that
E{{lX{n)) ) = Ь{{2^Нг(Лт1+1 -XTi)) f = b{2_^HiixTi+1 ~ XTi) }
г=0 г=0
n n
У (A^-.j —X r?i) / — ll^llu-^l/ ( T' — Ту/
i-0 i=0
" J<||ff||^{^}. D
Corollary 1. Each cadlag, locally square integrable local martingale is a
semimartingale.
Proof. Apply Theorem 8 together with the corollary to Theorem 6. ?
Corollary 2. A local martingale with continuous paths is a semimartingale.
Proof. Apply Corollary 1 together with Theorem 51 in Chap. I. G
Corollary 3. The Wiener process (that is, Brownian motion) is a semimartin-
semimartingale.
Proof. The Wiener process Bt is a martingale with continuous paths if So is
integrable. It is always a continuous local martingale. ?
Definition. We will say an adapted process X with cadlag paths is decom-
decomposable if it can be decomposed Xt = Xo + Mt + At, where Mo = Л о = О,
M is a locally square integrable martingale, and A is cadlag, adapted, with
paths of finite variation on compacts.
Theorem 9. A decomposable process is a semimartingale.
Proof. Let Xt — Xo + Mt + At be a decomposition of X. Then M is a semi-
semimartingale by Corollary 1 of Theorem 8, and Л is a semimartingale by The-
Theorem 7. Since semimartingales form a vector space (Theorem 1) we have the
result. D
Corollary. A Levy process is a semimartingale.

56 II Semimaxtingales and Stochastic Integrals
Proof. By Theorem 40 of Chap. I we know that a Levy process is decompos-
decomposable. Theorem 9 then gives the result. ?
Since Levy processes are prototypic strong Markov processes, one may well
wonder if all R"-valued strong Markov processes are semimartingales. Simple
1 /я
examples, such as Xt = Bt , where В is standard Brownian motion, show
this is not the case (while this example is simple, the proof that X is not a
semimartingale is not elementary1). However if one is willing to "regularize"
the Markov process by a transformation of the space (in the case of this
example using the "scale function" S{x) = x3), "most reasonable" strong
Markov processes are semimartingales. Indeed, Dynkin's formula, which
states that if / is in the domain of the infinitesimal generator G of the strong
Markov process Z, then the process
Mt = f(Zt) - f(Z0) -
is well-defined and is a local martingale, hints strongly that if the domain of
G is rich enough, the process Z is a semimartingale. In this regard see Sect. 7
of Cinlar, Jacod, Protter, and Sharpe [34].
4 Stochastic Integrals
In Sect. 1 we defined semimartingales as adapted, cadlag processes that acted
as "good integrators" on the simple predictable processes. We now wish to
enlarge the space of processes we can consider as integrands. In Chap. IV we
will consider a large class of processes, namely those that are "predictably
measurable" and have appropriate finiteness properties. Here, however, by
keeping our space of integrands small—yet large enough to be interesting—
we can keep the theory free of technical problems, as well as intuitive.
A particularly nice class of processes for our purposes is the class of
adapted processes with left continuous paths that have right limits (the French
acronym would be caglad).
Definition. We let D denote the space of adapted processes with cadlag
paths, L denote the space of adapted processes with caglad paths (left con-
continuous with right limits) and bL denote processes in L with bounded paths.
We have previously considered Su, the space of simple, predictable pro-
processes endowed with the topology of uniform convergence; and L°, the space
of finite-valued random variables topologized by convergence in probability.
We need to consider a third type of convergence.
1 See Theorem 71 of Chap. IV which proves a similar assertion for Xt = |St|°
0 < a < 1/2.

4 Stochastic Integrals 57
Definition. A sequence of processes {Hn)n>i converges to a process H uni-
uniformly on compacts in probability (abbreviated ucp) if, for each t > 0,
sup0<s<t |ff™ — Hs\ converges to 0 in probability.
We write H* = supo<s<t \HS\. Then if Yn G В we have Yn -> Г in ucp
if (Fra - У)* converges to 0 in probability for each t > 0. We write Ducp,
LUcp> Sucp to denote the respective spaces endowed with the ucp topology.
We observe that Ducp is a metrizable space; indeed, a compatible metric is
given, for X, Y e Ю>, by
n=l
The above metric space Bucp is complete. For a semimartingale X and a
process H 6 S, we have defined the appropriate notion Ix(H) of a stochastic
integral. The next result is key to extending this definition.
Theorem 10. The space S is dense in L under the ucp topology.
Proof. Let Y 6 L. Let Rn = inf{t: \Yt\ > n}. Then Rn is a stopping time and
yn _ YRnl{Rn>oy are in bL and converge to Y in ucp. Thus bL is dense in
L. Without loss we now assume Y 6 bL. Define Z by Zt = limu_»t Yu. Then
u>t
Z 6 D. For ? > 0, define
T?+1 = inf{t :t>T^ and \Zt - ZT*\ > e}.
Since Z is cadlag, the T^ are stopping times increasing to oo a.s. as n increases.
Let Z? = VV Zt'Iit' tc ь for each e > 0. Then Ze are bounded and con-
verge uniformly to Z as ? tends to 0. Let Ce = lol{o} + Sn^r«l(Te,Te J-
Then Us 6 bL and the preceding implies U? converges uniformly on compacts
Finally, define
*yn,e ~y -i _j_
i=l
and this can be made arbitrarily close to У 6 bL by taking e small enough
and n large enough. D
We defined a semimartingale X as a process that induced a continuous
operator Ix from Su into L°. This Ix maps processes into random variables
(note that X^ is a random variable). We next define an operator (the stochas-
stochastic integral operator), induced by X, that will map processes into processes
(note that XTi is a process).

58 II Semimaxtingales and Stochastic Integrals
Definition. For H 6 S and X a cadlag process, define the (linear) mapping
Jx : S -> В by
JX(H) = H0X0 +
г=1
for H in S with the representation
n
H = tfol{o} +^ЯД(ТгХ+1],
#» G -^ and 0 = To < Ti < ¦ ¦ ¦ < Tn+1 < oo stopping times.
Definition. For HeS and X an adapted cadlag process, we call Jx {H) the
stochastic integral of H with respect to X.
We use interchangeably three notations for the stochastic integral:
Jx{H)= fHsdXs=H-X.
Observe that Jx(H)t = Ix*(H). Indeed, Ix plays the role of a definite inte-
integral. For H e S, IX(H) = /0°° HsdXs.
Theorem 11. Let X be a semimartingale. Then the mapping Jx '¦ Sucp —>
Ducp is continuous.
Proof. Since we are only dealing with convergence on compact sets, without
loss of generality we take X to be a total semimartingale. First suppose Hk
in S tends to 0 uniformly and is uniformly bounded. We will show Jx(Hk)
tends to 0 ucp. Let 5 > 0 be given and define stopping times Tk by
Tk =ini{t: \(Hk-X)t\ >5}.
Then fffel[0>Tfc] 6 S and tends to 0 uniformly as к tends to oo. Thus for every
t,
P{(Hk ¦ X)* >S}< P{\Hk ¦ XTkAt\ > 5}
= P{\(Hkl[OtTkyX)t\>6}
= P{\Ix(H%,TkAt])\>6}
which tends to 0 by the definition of total semimartingale.
We have just shown that Jx '¦ Su —» Ducp is continuous. We now use this
to show Jx : Sucp —> Ducp is continuous. Suppose Hk goes to 0 ucp. Let
S > 0, e > 0, t > 0. We have seen that there exists rj such that \\H\\U < r)
implies P(Jx(H)*t > S) < §. Let Rk = inf{s : \Hk\ > 77}, and set Hk =
Hkl[0,Rk]l{Rk>0}. Then ff'eS and \\Hk\\u < rj by left continuity. Since
Rk > t implies (Hk ¦ X)*t = (Hk ¦ X)*, we have

4 Stochastic Integrals 59
P((Hk ¦ Xft >5)< P((Hk ¦ X)* >S) + P(Rk < t)
< e
if к is large enough, since lim/t^oo P((Hk)l > rj) = 0. ?
We have seen that when X is a semimartingale, the integration operator
Jx is continuous on SUcp, and also that Sucp is dense in Lucp. Hence we are
able to extend the linear integration operator Jx from S to L by continuity,
since Ducp is a complete metric space.
Definition. Let X be a semimartingale. The continuous linear mapping Jx '¦
LUCp —> ^>ucp obtained as the extension of Jx : S —> Ю> is called the stochastic
integral.
The preceding definition is rich enough for us to give an immediate example
of a surprising stochastic integral. First recall that if a process (At)t>o has
continuous paths of finite variation with Ao — 0, then the Riemann-Stieltjes
integral of /0 AsdAs yields the formula (see Theorem 54 of Chap. I)
/
Jo
¦t I
AsdAs = ~At.
о 2
Let us now consider a standard Brownian motion В = (Bt)t>o with Bq = 0.
The process В does not have paths of finite variation on compacts, but it is
a semimartingale. Let (тг„) be a refining sequence of partitions of [0, oo) with
limn^oo meshGrn) - 0. Let Btn = 52гк€ъп B*k 1(tk,tk+i]- Then Bn € L for each
n. Moreover, Bn converges to В in ucp. Fix t > 0 and assume that t is a
partition point of each тг„. Then
jB{Bn)t = J2
tk <i
and
JB(B)t= lim JB(Bn)t = lim
{\(Btk+l + Btk)(Btk+1 - Btk)
tk?irn,tk<t
-\(Btk+1-Btk)(Btk+1-Btk)}
where we have used telescoping sums. The last term on the right, however,
converges a.s. to t, by Theorem 28 of Chap. I. We therefore conclude that

60 II Semimartingales and Stochastic Integrals
i:
BsdBs = -B* - -t, (**)
a formula distinctly different from the Riemann-Stieltjes formula (*). The
change of variables formula (Theorem 32) presented in Sect. 7, will give a
deeper understanding of the difference in formulas (*) and (**).
5 Properties of Stochastic Integrals
Throughout this paragraph X will denote a semimartingale and H will denote
an element of L. Recall that the stochastic integral defined in Sect. 4 will be
denoted by the three notations Jx(H) = H ¦ X = J HsdXs. Evaluating these
processes at t, we have
= / HsdXs = f
Jo J[o,t
HXt= [ HsdXs = / HsdXs.
Ml
To exclude 0 in the integral we write
/ HsdXs = / HsdXs.
o+ J(o,t]
The integral /0°° HsdXs is defined to be limt-юо /0* HsdXs when the limit
exists. Note that f* HsdXs = H0X0 + f*+HsdXs. For a process Y e Ю>,
we recall that AYt = Yt - Y"t_, the jump at t. Also since Yo- = 0 we have
AY0 = Yo. Recall further that for a process Z and stopping time T, we let
ZT denote the stopped process {Zj) = ZthT- We will establish in this section
several elementary properties of the stochastic integral. These properties will
help us to understand the integral and to examine examples at the end of
the section. Recall that two processes Y and Z are indistinguishable if
P{oj : t >—> Xt{u>) and t >—> Yt(oj) are the same functions} = 1.
Theorem 12. Let T be a stopping time. Then (H ¦ X)T = i/l[Or] • X =
н-{хт).
Theorem 13. The jump process A(H-X)S is indistinguishable from Hs (AXS).
Proofs. Both properties are clear when ffeS, and they follow when H e L
by passing to the limit. Indeed, we take convergence in ucp for each t, then
a.s. convergence by taking a subsequence, and then we choose the rationale
(which of course is countable and dense, so the union of null sets is still a null
set), and then use the path regularity to get indistinguishability. ?
Let Q denote another probability law, and let Hq ¦ X denote the stochastic
integral of H with respect to X computed under the law Q.
Theorem 14. Let Q <?C P. Then Hq ¦ X is Q indistinguishable from HP ¦ X.

5 Properties of Stochastic Integrals 61
Proof. Note that by Theorem 2, X is a Q semimartingale. The theorem is clear
if H 6 S, and it follows for H G L by passage to the limit in the ucp topology,
since convergence in P-probability implies convergence in Q-probability. ?
Theorem 15. Let Pk be a sequence of probabilities such that X is a Pk
semimartingale for each к. Let R = YlT=i ^кРк where Xk > 0, each к, and
Efeli Afc = !• Then HR-X = HPk ¦ X, Pk-a.s., for all к such that Xk > 0.
Proof. If Afc > 0 then Pk <€. R, and the result follows by Theorem 14. Note
that by Theorem 3 we know that X is an R semimartingale. ?
Corollary. Let P and Q be any probabilities and suppose X is a semimartin-
semimartingale relative to both P and Q. Then there exists a process H ¦ X which is a
version of both Hp ¦ X and Hq ¦ X.
Proof. Let R = (P + Q)/2. Then Hr-X is such a process by Theorem 15. ?
Theorem 16. Let G = (Qt)t>o be another filtration such that H is in both
L(G) and L(F), and such that X is also a G semimartingale. Then Hq • X =
H?-X.
Proof. L(G) denotes left continuous processes adapted to the filtration G. As
in the proof of Theorem 10, we can construct a sequence of processes Hn
converging to H where the construction of the Hn depends only on H. Thus
Hn 6 S(G) П S(F) and converges to H in ucp. Since the result is clear for S,
the full result follows by passing to the limit. ?
Remark. While Theorem 16 is a simple result in this context, it is far from
simple if the integrands are predictably measurable processes, rather than
processes in L. See the comment following Theorem 33 of Chap. IV.
The next two theorems are especially interesting because they show—at
least for integrands in L—that the stochastic integral agrees with the path-
by-path Lebesgue-Stieltjes integral, whenever it is possible to do so.
Theorem 17. If the semimartingale X has paths of finite variation on com-
compacts, thenH-X is indistinguishable from the Lebesgue-Stieltjes integral, com-
computed path-by-path.
Proof. The result is evident for H 6 S. Let Hn e S converge to H in ucp.
Then there exists a subsequence n^ such that limnfc_)OO(i/™'c — HI = 0 a.s.,
and the result follows by interchanging limits, justified by the uniform a.s.
convergence. D
Theorem 18. Let X, X be two semimartingales, and let H, H G L. Let
A = {lj : H.(w) = Я.(ы) andX.(tj) = X.(w)}, andletB = {u : t н-> Xt(u)
is of finite variation on compacts}. Then H ¦ X = H ¦ X on A, and H ¦ X is
equal to a path-by-path Lebesgue-Stieltjes integral on B.

62 II Semimartingales and Stochastic Integrals
Proof. Without loss of generality we assume P(A) > 0. Define a new prob-
probability law Q by Q{A) = Р{Л\А). Then under Q we have that Я and Я as
well as X and X are indistinguishable. Thus Hq ¦ X = Hq • X, and hence
Я • X = Я • X P-a.s. on A by Theorem 14, since Q <C P.
As for the second assertion, if В = Q the result is merely Theorem 17.
Define R by Д(Л) = Р(Л\В), assuming without loss that P(B) > 0. Then
Д<Р and В = Q, i?-a.s. Hence Яд ¦ X equals the Lebesgue-Stieljes integral
-R-a.s. by Theorem 17, and the result follows by Theorem 14. D
The preceding theorem and following corollary are known as the local
behavior of the integral.
Corollary. With the notation of Theorem 18, let 5, T be two stopping times
with S <T. Define
С = {cj : Ht(w) = Ht(w); Xt(w) = Xt(w); S(w) < t < T(cj)}
D = {u> : t н-> Xt(oj) is of finite variation on S(oj) < t < T(oj)}.
Then H ¦ XT - H ¦ Xs =H -XT -H -Xs on С and H ¦ XT - H ¦ Xs equals
a path-by-path Lebesgue-Stieltjes integral on D.
Proof. Let Yt=Xt- XtAS. Then Я • Y = H ¦ X - H ¦ Xs, and Y does not
change the set [0, S], which is evident, or which—alternatively—can be viewed
as an easy consequence of Theorem 18. One now applies Theorem 18 to YT
to obtain the result. D
Theorem 19 (Associativity). The stochastic integral process Y = Я • X is
itself a semimartingale, and for G ? L we have
GY = G(H-X) = (GH) ¦ X.
Proof. Suppose we know Y = H-X is a semimartingale. Then G-Y = Jy(G).
If G, H are in S, then it is clear that Jy(G) = J\(GH). The associativity
then extends to L by continuity.
It remains to show that Y = H ¦ X is a semimartingale. Let (Я") be in S
converging in ucp to Я. Then Hn ¦ X converges to Я ¦ X in ucp. Thus there
exists a subsequence (nk) such that НПк ¦ X converges a.s. to Я • X.
Let G 6 S and let Ynk = НПк-Х, Y = HX. The Ynk are semimartingales
converging pointwise to the process Y. For G 6 S, Jy(G) is defined for any
process Y; so we have
JY(G)=G-Y= lim G-Ynk= lim G • (НПк ¦ X)
= lim (GHnk)-X
nk~*oo
which equals \imnk^O0Jx{GHnk) = Jx(GH), since X is a semimartingale.
Therefore JY(G) = JX(GH) for GeS.

5 Properties of Stochastic Integrals 63
Let Gn converge to G in Su. Then GnH converges to GH in Lucp, and since
X is a semimartingale, hin^^oo Jy(G") = Нт^^оо Jx(GnH) = J\{GH) =
Jy{G). This implies Yl is a total semimartingale, and so Y = H ¦ X is a
semimartingale. ?
Theorem 19 shows that the property of being a semimartingale is preserved
by stochastic integration . Also by Theorem 17 if the semimartingale X is an
FV process, then the stochastic integral agrees with the Lebesgue-Stieltjes
integral, and by the theory of Lebesgue-Stieltjes integration we are able to
conclude the stochastic integral is an FV process also. That is, the property
of being an FV process is preserved by stochastic integration for integrands in
L.2
One may well ask if other properties are preserved by stochastic integra-
integration; in particular, are the stochastic integrals of martingales and local mar-
martingales still martingales and local martingales? Local martingales are indeed
preserved by stochastic integration, but we are not yet able easily to prove it.
Instead we show that locally square integrable local martingales are preserved
by stochastic integration for integrands in L.
Theorem 20. Let X be a locally square integrable local martingale, and let
Яб1. Then the stochastic integral H ¦ X is also a locally square integrable
local martingale.
Proof. We have seen that a locally square integrable local martingale is a
semimartingale (Corollary 1 of Theorem 8), so we can formulate H ¦ X. With-
Without loss of generality, assume Xo = 0. Also, if Tk increases to oo a.s. and
(H -X)Tk is a locally square integrable local martingale for each k, it is simple
to check that H ¦ X itself is one. Thus without loss we assume X is a square
integrable martingale. By stopping H, we may further assume H is bounded,
by ?. Let Hn^G S be such that Hn converges to H \n ucp. We can then modify
Hn, call it Hn, such that Hn is bounded by ix Hn 6 S, and Hn converges
uniformly to H in probability on [0,4]. Since Hn 6 bS, one can check that
Hn • X is a martingale. Moreover
¦ XJt} - ^
and hence (Hn • X)t are uniformly bounded in L2 and thus uniformly inte-
integrable. Passing to the limit then shows both that H ¦ X is a martingale and
that it is square integrable. ?
See Exercise 43 in Chap. IV which shows this is not true in general.

64 II Semimartingales and Stochastic Integrals
In Theorem 29 of Chap. Ill we show the more general result that if M is
a local martingale and ff e L, then H ¦ M is again a local martingale.
A classical result from the theory of Lebesgue measure and integration
(on R) is that a bounded, measurable function / mapping an interval [0,6]
to R is Riemann integrable if and only if the set of discontinuities of / has
Lebesgue measure zero (e.g., Kingman and Taylor [127, page 129]). Therefore
we cannot hope to express the stochastic integral as a limit of sums unless
the integrands have reasonably smooth sample paths. The spaces Ю> and L
consist of processes which jump at most countably often. As we will see in
Theorem 21, this is smooth enough.
Definition. Let a denote a finite sequence of finite stopping times:
0 = To < Ti < ¦ ¦ ¦ < Tfe < 00.
The sequence a is called a random partition. A sequence of random parti-
partitions <7„,
on : To™ < T? < • ¦ • < T?n
is said to tend to the identity if
(i) HmrasupfeT^ = 00 a.s.; and
(ii) \\an\\ = supfe \Tg+1 - T?\ converges to 0 a.s.
Let У be a process and let a be a random partition. We define the process
Y sampled at a to be
Ya =
It is easy to check that
YfdX. = Y0X0 + Y^YTi{XT^ - XT'),
!•¦
for any semimartingale X, any process Y in S, Ю>, or L.
Theorem 21. Let X be a semimartingale, and let Y be a process in D or in L.
Let (an) be a sequence of random partitions tending to the identity. Then the
processes /0*+ Y°ndXs = J2i Yr?(Xt i+1 —Xt*) tend to the stochastic integral
(Y-) • X in ucp.
Proof (The notation У_ means the process whose value at s is given by
(Y-)s — limu_)SU<syu; also, (YL)o = 0, by convention.) We prove the the-
theorem for the case where Y is cadlag, the other case being analogous. Also
without loss of generality we can take Xo = 0. Y cadlag implies F_ e L. Let
Yk 6 S such that Yk converges to Y_ (ucp). We have

5 Properties of Stochastic Integrals 65
f(Y_ - Y"")adXa = f(Y_ - Yk)sdXs + f(Yk - {Yk)°")sdXs
+ J((?{)"»-Ya»)adX.
where Yk denotes the cadlag version of Yk. The first term on the right side
equals J\ (Y- -Yk), and since Jx is continuous in Lucp and since У_ — Yk —> 0,
we have f(Y- —Yk)sdXs —» 0 (ucp). The same reasoning applies to the third
term, for fixed n, as к —> oo. Indeed, the convergence to 0 of {(y?)"n - Y"n)
as к —> оо is uniform in n.
It remains to consider the middle term on the right side above. Since the Yk
are simple predictable we can write the stochastic integrals in closed form, and
since X is right continuous the integrals (for fixed (к, ш)) J(Yk - (Y+)"nKdX3
tend to 0 as n —> oo. Thus one merely chooses к so large that the first and
third terms are small, and then for fixed k, the middle term can be made
small for large enough n. ?
We consider here another example. We have already seen at the end of
Sect. 4 that if Б is a standard Wiener process, then
i:
showing that the semimartingale calculus does not formally generalize the
Riemann-Stieltjes calculus.
We have seen as well that the stochastic integral agrees with the Lebesgue-
Stieltjes integral when possible (Theorems 17 and 18) and that the stochastic
integral also preserves the martingale property (Theorem 20; at least for lo-
locally square integrable local martingales). The following example shows that
the restriction of integrands to L is not as innocent as it may seem if we want
to have both of these properties.
Example. Let Mt = Nt — Xt, a compensated Poisson process (and hence a
martingale with Mt 6 Lp for all t > 0 and all p > 1). Let (Ti)i>\ be the jump
times of M. Let Ht = l[0)Tl)(t). Then Я G D. The Lebesgue-Stieltjes integral
is
/ HsdMs = f HsdNs -X f Hsds
Jo Jo Jo
oo л
= ^tfTil{t>Ti}-A/
ds
Jo
= -A(iATi).

66 II Semimartingales and Stochastic Integrals
This process is not a martingale. We conclude that the space of integrands
cannot be expanded even to Ю>, in general, and preserve the structure of the
theory already established.3
6 The Quadratic Variation of a Semimartingale
The quadratic variation process of a semimartingale, also known as the bracket
process, is a simple object that nevertheless plays a fundamental role.
Definition. Let X, Y be semimartingales. The quadratic variation pro-
process of X, denoted [X, X] = ([X,X]t)t>o, is defined by
[X,X] =X2 -2 f X-dX
(recall that Xo_ = 0). The quadratic covariation of X, Y, also called the
bracket process of X, Y, is defined by
[X,Y] = XY- f X-dY - /Y_
dX.
It is clear that the operation (X, Y) —> [X, Y] is bilinear and symmetric.
We therefore have a polarization identity
[X,Y] = \({X + YX + Y] - [X,X] - [Y,Y]).
The next theorem gives some elementary properties of [X, X]. (X is assumed
to be a given semimartingale throughout this section).
Theorem 22. The quadratic variation process of X is a cadlag, increasing,
adapted process. Moreover it satisfies the following.
(i) [X,X}0 = X2 and A[X,X] = (AXJ.
(ii) If an is a sequence of random partitions tending to the identity, then
with convergence in ucp, where an is the sequence 0 = T$ < T" < • • • <
T" < • • • < T? and where T" are stopping times.
(Hi) If T is any stopping time, then [XT,X] = [X,XT] = [XT,XT] =
[X,Xf.
3 Ruth Williams has commented to us that this example would be more convincing
if M were itself a semimartingale, H bounded and in D, and J* HsdMs were not
a semimartingale. Such a construction is carried out in [1].

6 The Quadratic Variation of a Semimartingale 67
Proof. X is cadlag, adapted, and so also is / X-dX by its definition; thus
[X, X] is cadlag, adapted as well. Recall the property of the stochastic integral:
. Then
(AXJS = (Xs - Xs_J = X2S - 2XSXS_ + X2
= X2S - X2_ + 2XS_(XS_ - X.
from which part (i) follows.
For part (ii), by replacing X with X = X — Xo, we may assume Xq — 0.
Let Rn — supjT". Then Rn < oo a.s. and limni?n — oo a.s., and thus by
telescoping series
converges ucp to X2. Moreover, the series J^ Ij»(ITi+i —XT") converges in
ucp to f X^dX by Theorem 21, since X is cadlag. Since b2 - a2 — 2a(b - a) =
(b-aJ, and since XTr(XT^ -XT") = XT"(XT^ -XT"), we can combine
the two series convergences above to obtain the result. Finally, note that if
s < t, then the approximating sums in part (ii) include more terms (all non-
negative), so it is clear that [X, X] is non-decreasing. (Note that, a priori, one
only has [X,X]S < [X,X]t a.s., with the null set depending on s and t; it is
the property that [X, X] has cadlag paths that allows one to eliminate the
dependence of the null set on s and t.) Part (iii) is a simple consequence of
part (ii). D
An immediate consequence of Theorem 22 is the observation that if В is a
Brownian motion, then [B, B]t = t, since in Theorem 28 of Chap. I we showed
the a.s. convergence of sums of the form in part (ii) of Theorem 22 when the
partitions are refining.
Another consequence of Theorem 22 is that if X is a semimartingale with
continuous paths of finite variation, then [X, X] is the constant process equal
to Xq. To see this one need only observe that
^ -XT"J< sup |XT"+l - XT" | ? |XT"+l - XT" |
where V is the total variation. Therefore the sums tend to 0 as \\crn\\ —> 0.
Theorem 22 has several more consequences which we state as corollaries.
Corollary 1. The bracket process [X, Y] of two semimartingales has paths
of finite variation on compacts, and it is also a semimartingale.

68 II Semimartingales and Stochastic Integrals
Proof. By the polarization identity [X, Y] is the difference of two increas-
increasing processes, hence its paths are of finite variation. Moreover, the paths are
clearly cadlag, and the process is adapted. Hence by Theorem 7 it is a semi-
martingale. D
Corollary 2 (Integration by Parts). Let X, Y be two semimartingales.
Then XY is a semimartingale and
XY = IX_dY+ f Y-dX + [X,Y].
Proof. The formula follows trivially from the definition of [X,Y]. That XY
is a semimartingale follows from the formula, Theorem 19, and Corollary 1
above. D
In the integration by parts formula above, we have (X-)o = (Y-)o = 0.
Hence evaluating at 0 yields
X0Y0 = (Х_)оУо + (K-)o*o + [X, Y}0.
Since [X, Y]o = AXqAYq = X0Yq, the formula is valid. Without the conven-
convention that (X_)o = 0, we could have written the formula
XtYt= f Xs^dYs+ f
Corollary 3. All semimartingales on a given filtered probability space form
an algebra.
Proof. Since semimartingales form a vector space, Corollary 2 shows they
form an algebra. D
A theorem analogous to Theorem 22 holds for [X,Y] as well as [X, X]. It
can be proved analogously to Theorem 22, or more simply by polarization.
We omit the proof.
Theorem 23. Let X and Y be two semimartingales. Then the bracket process
[X, Y] satisfies the following.
(i) [X, Y]o = X0Y0 and A[X, Y] = AX AY.
(ii) If an is a sequence of random partitions tending to the identity, then
[X,Y] ^
where convergence is in ucp, and where an is the sequence 0 = Tft < T" <
• • • < IT < ¦ ¦ ¦ < T?n, with T? stopping times,
(in) If T is any stopping time, then{XT,Y] = [X,YT] = [XT,YT] = [X,Y)T.

6 The Quadratic Variation of a Semimartingale 69
We next record a real analysis theorem from the Lebesgue-Stieltjes theory
of integration. It can be proved via the Monotone Class Theorem.
Theorem 24. Let a, C, 7 be functions mapping [0,00) to M with a(Q) =
/3@) = 7@) = 0. Suppose a, C, 7 are all right continuous, a is of finite
variation, and C and 7 are each increasing. Suppose further that for all s, t
with s < t, we have
dau\<([ dCu)
s Js
Then for any measurable functions f', g we have
\fg\\da\<{f f2dC
s Js
In particular, the measure da is absolutely continuous with respect to both dC
and dj.
Note that \da\ denotes the total variation measure corresponding to the
measure da, the Lebesgue-Stieltjes signed measure induced by a. We use this
theorem to prove an important inequality concerning the quadratic variation
and bracket processes.
Theorem 25 (Kunita-Watanabe Inequality). Let X and Y be two semi-
martingales, and let H and К be two measurable processes. Then one has
a.s.
/¦OO ЛОО ЛОО
/ \Hs\\Ks\\d[X,Y}s\<( H2d[X,X}s)H K2sd[Y,Y}s)K
Jo Jo Jo
Proof. By Theorem 24 we only need to show that there exists a null set N,
such that for ш ? N, and (s, t) with s < t, we have
d[X,X]u)?( d[Y,Y]u)?. (*)
J s
Let N be the null set such that ifw<?N, then 0 < /g d[X + rY,X + rY}u, for
every r, s, t; s < t, with r, s, t all rational numbers. Then
0 < [X + rY, X + rY]t - [X + rY, X + rY]s
= r\[Y, Y]t - [Y, Y}s) + 2r([X, Y)t - [X, Y]s) + ([X, X}t - [X, X}3).
The right side being positive for all rational r, it must be positive for all real
r by continuity. Thus the discriminant of this quadratic equation in r must be
non-negative, which gives us exactly the inequality (*). Since we have, then,
the inequality for all rational (s,t), it must hold for all real (s,t), by the right
continuity of the paths of the processes. ?

70 II Semimartingales and Stochastic Integrals
Corollary. Let X and Y be two semimartingales, and let H and К be two
measurable processes. Then
лОО лОО лОО
E{ [Hs[[Ks[\d[X,Y]s\}<\[{ Н2АХ,Х)а)Ц\ь,\\{1 K2td[Y,Y]t)l\\L4
Jo Jo Jo
Proof. Apply Holder's inequality to the Kunita-Watanabe inequality of The-
Theorem 25. D
Since Theorem 25 and its corollary are path-by-path Lebesgue-Stieltjes
results, we do not have to assume that the integrand processes H and К be
adapted.
Since the process [X, X] is non-decreasing with right continuous paths, and
since A[X,X]t = (AXtJ for all t > 0 (with the convention that Xo_ = 0),
we can decompose [X, X] path-by-path into its continuous part and its pure
jump part.
Definition. For a semimartingale X, the process [X, X]c denotes the path-
by-path continuous part of [X, X].
We can then write
[X,X]t = [X,X}$ +X2o+
0<s<t
= [x,x}c + J2 (AxsJ-
0<s<t
Observe that [X, X]q = 0. Analogously, [X, Y]c denotes the path-by-path
continuous part of [X, Y].
Comment. In Chap. IV, Sect. 7, we briefly discuss the continuous local mar-
martingale part of a semimartingale X satisfying an auxiliary hypothesis known
as Hypothesis A. This (unique) continuous local martingale part of X is de-
denoted Xе. It can be shown that a unique continuous local martingale part, Xе,
exists for every semimartingale X.4 (If X is an FV process, then Xе = 0, as
we shall see in Chap. III.) It is always true that [Xе, Xе] = [X,X)C. Although
we will have no need of this result in this book, this notation is often used in
the literature.
We remark also that the conditional quadratic variation of a semimartin-
semimartingale X, denoted (X, X), is defined in Chap. Ill, Sect. 4. It is also true
that {Xе, Xе) = [Xе, Xе] = [X,X]C, and these notations are used inter-
interchangeably in the literature. // X is already continuous and Xq = 0, then
{X,X) = [X,X] = [X,Xf.
4 See, for example, Dellacherie-Meyer [46, page 355]. See also in this regard Exer-
Exercise 6 of Chap. IV which proves the result for L2 martingales.

6 The Quadratic Variation of a Semimartingale 71
Definition. A semimartingale X will be called quadratic pure jump if
[X, X]c = 0.
If X is quadratic pure jump, then [X,X]t = X$ + J2o<s<t(AXsJ- The
Poisson process N is an obvious example of a quadratic pure jump semi-
semimartingale. Prom the definition or immediately from Theorem 22, we see that
[N, N]t = Nf. More generally it can be shown that if X is a Levy process with
a Levy decomposition Xt = Bt + Yt as in Theorem 42 of Chap. I, where В is
a Brownian motion and
Yt= f x{Nt{-,dx)-tv{dx))
J{\*\<1}
0<s<t
(using the notation of Theorem 42 of Chap. I), then У is a quadratic pure
jump semimartingale. Last note that the trivial continuous process Xt = t is
quadratic pure jump since [X, X}^ = [X, X]t = 0.
The next theorem gives a simple criterion for a process X to be a quadratic
pure jump semimartingale.
Theorem 26. If X is adapted, cadlag, with paths of finite variation on com-
compacts, then X is a quadratic pure jump semimartingale.
Proof. Without loss of generality we assume Xo = 0. We have already seen
that such an X is a semimartingale (Theorem 7), and that the stochastic inte-
integral with respect to X is nothing more than a path-by-path Lebesgue-Stieltjes
integral (Theorem 17). The integration by parts formula for Lebesgue-Stieltjes
differentials applied to X times itself yields X2 = f X-dX + JXdX, com-
computed path-by-path. The semimartingale integration by parts formula (Corol-
(Corollary 2 of Theorem 22), on the other hand, yields X2 = 2 f X^dX + [X,X].
Moreover
XdX = /(X_ + AX)dX = I X_dX + f AXdX,
and
rt
f X-dXs + f AXsdXs = f
Jo Jo Jo
s<t
Equating the two formulas we deduce [X,X]t = J2s<t(^XsJ, yielding the
theorem. D
Note in particular that if X is adapted with continuous paths of finite
variation, then [X,X]t = X$, all t > 0.
Theorem 27. Let X be a local martingale with continuous paths that are
not everywhere constant. Then [X,X] is not the constant process Xq, and
X2 — [X, X] is a continuous local martingale. Moreover if [X, X]t = 0 for all
t then Xt=0 for all t.

72 II Semimartingales and Stochastic Integrals
Proof. Note that a continuous local martingale is a semimartingale (Corol-
(Corollary 2 of Theorem 8). We have X2 - [X,X] = 2jX_dX, and by the mar-
martingale preservation property (Theorem 20) we have that 2 f X-dX is a local
martingale. Moreover Д2/X-dX — 2(X-)(AX), and since X is continu-
continuous, AX — 0, and thus 2 JX-dX is a continuous local martingale, hence
locally square integrable. Thus X2 — [X, X] is a locally square integrable local
martingale.
By stopping, we can suppose X is a square integrable martingale. As-
Assume further Xo = 0. Next assume that [X, X] actually were constant. Then
[X,X]t = [X,X]0 = X§ = 0, for all t. Since X2 - [X,X] is a local mar-
martingale, we conclude X2 is a non-negative local martingale, with Xq = 0.
Thus X2 = 0, all t. This is a contradiction. If Xo is not identically 0, we set
Xt = Xt — Xo and the result follows. ?
The following corollary is of fundamental importance in the theory of
martingales.
Corollary 1. Let X be a continuous local martingale, and S < T < oo be
stopping times. If X has paths of finite variation on the stochastic interval
(S, T), then X is constant on [S, T]. Moreover if [X, X] is constant on [S, T] П
[0, oo), then X is constant there too.
Proof. M = XT — Xs is also a continuous local martingale, and M has
finite variation on compacts. Moreover [M, M] = \XT — Xs, XT — Xs] =
[X, X]T — [X, X]s, and therefore [M, M] is constant everywhere, hence by
Theorem 27, M must be constant everywhere; thus X is constant on [S, T). D
Observe that if t > 0 is arbitrary and we take 5 = 0 and T = t in Corol-
Corollary 1, then we can conclude that a continuous local martingale with paths
of finite variations on compacts is a.s. constant. While non-trivial continu-
continuous local martingales must therefore always have paths of infinite variation
on compacts, they are not the only such local martingales. For example, the
Levy process martingale Zd of Theorem 41 of Chap. I will have paths of infi-
infinite variation if the Levy measure v has infinite mass in a neighborhood of the
origin. Another example is Azema's martingale, which is presented in detail
in Sect. 8 of Chap. IV.
Corollary 2. Let X and Y be two locally square integrable local martin-
martingales. Then [X, Y] is the unique adapted cadlag process A with paths of finite
variation on compacts satisfying the two properties:
(i) XY — A is a local martingale; and
(ii) AA = AX AY, Ao = X0Y0.
Proof Integration by parts yields
XY= f X_dY+ fY-dX + [X,Y],

6 The Quadratic Variation of a Semimartingale 73
but the martingale preservation property tells us that both stochastic integrals
are local martingales. Thus XY — [X, Y] is a local martingale. Property (ii)
is simply an application of Theorem 23. Thus it remains to show uniqueness.
Suppose А, В both satisfy properties (i) and (ii). Then A — В = {XY —
B) — (XY — A), the difference of two local martingales which is again a local
martingale. Moreover,
A(A -B) = AA-AB = AX AY - AX AY = 0.
Thus A — В is a continuous local martingale, Aq — Bq = 0, and it has paths
of finite variation on compacts. Corollary 1 yields At — Bt — Aq + Bq = 0 and
we have uniqueness. D
Corollary 2 can be useful in determining the process [X, Y]. For example
if X and Y are locally square integrable martingales without common jumps
such that XY is a martingale, then [X, Y] = XoYo. One can also easily verify
(as a consequence of Theorem 23 and Corollaries 1 and 2 above, for example)
that if X is a continuous square integrable martingale and У is a square
integrable martingale with paths of finite variation on compacts, then [X, Y] =
XoYo, and hence XY is a martingale. (An example would be X a Brownian
motion and Y a compensated Poisson process.) Corollary 2 is true as well for
X, Y local martingales, however we need Theorem 29 of Chap. Ill to prove
the general result.
In Chap. Ill we show that any local martingale is a semimartingale (see the
corollary of Theorem 26 of Chap. Ill), and therefore if M is a local martingale
its quadratic variation [M, M]t always exists and is finite a.s. for every t > 0.
We use this fact in the next corollary.
Corollary 3. Let M be a local martingale. Then M is a martingale with
E{M?} < oo, all t > 0, if and only if E{[M,M}t} < oo, all t > 0. If
E{[M,M]t] < oo, then E{M?} = E{[M,M}t}.
Proof. First assume that M is a martingale with E{M?} < oo for all t > 0.
Then M is clearly a locally square integrable martingale. Let Nt = Mt2 -
[M, M]t = 2 JQ Ms-dMs, which is a locally square integrable local martingale
by Theorem 20. Let (Tn)n>i be stopping times increasing to oo a.s. such
that Nfn is a square integrable martingale. Then E{N^n} = E{Nq} = 0, all
t > 0. Therefore
Doob's maximal quadratic inequality gives E{(M^J} < 4Е{М?} < oo.
Therefore by the Dominated Convergence Theorem
E{M?}= lim E{M?ATn}
n—*oo
= lim E{[M,M}tATn}
n—>-oo
= E{[M,M]t},

74 II Semimartingales and Stochastic Integrals
where the last result is by the Monotone Convergence Theorem. In particular
we have that E{[M, M]t} < oo.
For the converse, we now assume E{[M,M]t} < oo, all t > 0. Define
stopping times by
T" = inf{* >0: \Mt\ >n}An.
Then T" increase to oo a.s. Furthermore (MT )* < n + |ДМ<гп| < n +
[M,M]n' , which is in L2. By Theorem 51 of Chap. I, MT" is a uniformly
integrable martingale for each n. Also we have that
for all t > 0. Therefore MT satisfies the hypotheses of the first half of this
theorem, and E{(MfnJ} = E{[MTn,MTn]t}. Using Doob's inequality we
have
E{(Mt\TnJ} < AE{{Mff} = T\T\
<4E{[M,M}t}.
The Monotone Convergence Theorem next gives
E{(Mt*J}= lim E{(Mt\TnJ}
n—*oo
<4E{[M,M]t} < oo.
Therefore, again by Theorem 51 of Chap. I, we conclude that M is a martin-
martingale. The preceding gives E{M?} < oo. D
For emphasis we state as another corollary a special case of Corollary 3.
Corollary 4. If M is a local martingale and E{[M, M\oo} < oo, then M is a
square integrable martingale (that is supt E{M?} = E{M%O\ < oo). Moreover
E{M?} = E{[M, M]t} for all t, 0 < t < oo.
Example. Before continuing we consider again an example of a local martin-
martingale that exhibits many of the surprising pathologies of local martingales. Let
В be a standard Brownian motion in M3 with Bo = A,1,1). Let Mt = Ц-Е^Ц,
where \\x\\ is standard Euclidean norm in K.3. (We previously considered this
example in Sect. 6 of Chap. I.) As noted in Chap. I, the process M is a contin-
continuous local martingale; hence it is a locally square integrable local martingale.
Moreover E{M2} < oo for all t. However instead of i i—> E{M2} being an
increasing function as it would if M were a martingale, lim^oo E{M2} = 0.
Moreover E{[M,M}t} > E{[M,M}0} = 1 since [M,M]t is increasing. There-
Therefore we cannot have E{M2} = E{[M,M]t} for all t. Indeed, by Corollary 3
and the preceding we see that we must have E{[M, M]t} = oo for all t > 0. In
conclusion, M = \\B\\~1 is a continuous local martingale with E{M2} < oo
for all t which is both not a true martingale and for which E{M2} < oo while
E{[M,M]t} = oo for all t > 0. (Also refer to Exercise 20 at the end of this
chapter.)

6 The Quadratic Variation of a Semimartingale 75
Corollary 5. Let X be a continuous local martingale. Then X and [X, X]
have the same intervals of constancy a.s.
Proof. Let r be a positive rational, and define
Tr = inf{* > r : Xt ф Xr}.
Then M = XTr — XT is a local martingale which is constant. Hence [M, M] =
[X, X]Tr - [X, X]r is also constant. Since this is true for any rational r a.s.,
any interval of constancy of X is also one of [X, X].
Since X is continuous, by stopping we can assume without loss of generality
that X is a bounded martingale (and hence square integrable). For every
positive, rational r we define
Sr = mf{t>r:[X,X}t>[X,X}r}.
Then
E{(XSr - Xrf} = E{Xl) - E{X2}
by Doob's Optional Sampling Theorem. Moreover
E{X2Sr} - E{X2} = E{[X,X}Sr - [X,X}r} = 0,
by Corollary 3. Therefore E{(XSr - XrJ} = 0, and XSr = Xr a.s. Moreover
this implies Xq = Xsq a.s. on {Sq = Sr} for each pair of rationale (r, q), and
therefore we deduce that any interval of constancy of [X, X] is also one of
X. О
Note that the continuity of the local martingale X is essential in Corol-
Corollary 5. Indeed, let Nt be a Poisson process, and let Mt—Nt— t. Then M is a
martingale and [M, M]t = Nt; clearly M has no intervals of constancy while
N is constant except for jumps.
Theorem 28. Let X be a quadratic pure jump semimartingale. Then for any
semimartingale Y we have
[X,Y}t = X0Y0+ ? AXSAYS.
0<s<t
Proof The Kunita-Watanabe inequality (Theorem 25) tells us d[X, Y]s is a.s.
absolutely continuous with respect to d[X, X] (path-by-path). Thus [X, X]c =
0 implies [X, Y]c = 0, and hence [X, Y] is the sum of its jumps, and the result
follows by Theorem 23. D
Theorem 29. Let X and Y be two semimartingales, and let H, К € L. Then
[H-X,K-Y]t= f HsKsd[X,Y]s
Jo
and, in particular,

76 II Semimartingales and Stochastic Integrals
[H-X,H-X]t= H2sd[X,X}s.
Jo
Proof. First assume (without loss of generality) that Xq = Yq = 0. It suffices
to establish the following result
[H-X,Y]t= [ Hsd[X,Y]s, (*)
Jo
and then apply it again, by the symmetry of the form [-,-], and by the asso-
associativity of the stochastic integral (Theorem 19).
First suppose H is the indicator of a stochastic interval. That is, H =
l[o,T]i where T is a stopping time. Establishing (*) is equivalent in this case
to snowing [XT, Y] = [X, Y]T, a result that is an obvious consequence of
Theorem 23, which approximates [X, Y] by sums.
Next suppose H = UI^st], where S, T are stopping times, S <T a.s.,
and U e Fs. Then jHsdXs'= U(XT - Xs), and by Theorem 23
= U{[XT,Y}-[XS,Y]}
= U{[X,Y}T - [X,Y]3} = jHsd[X,Y}s.
The result now follows for H € S by linearity. Finally, suppose H € L and
let Hn be a sequence in S converging in ucp to H. Let Zn = Hn-X, Z = H-X.
We know Zn, Z are all semimartingales. We have fH^d[X,Y]s = [Zn,Y],
since Hn € S, and using integration by parts
[Zn,Y] - YZn - JY..dZn - I Zn_dY
= YZn
By the definition of the stochastic integral, we know Zn —> Z in мер, and
since Hn —> H (ucp), letting n —> oo we have
lim [Zn,Y] = YZ- fy-HdX- f Z-dY
n^oo1 J J
fy^dZ- IZ^dY
= YZ-
again by integration by parts. Since limn^oo f H™d[X, Y]s = f Hsd[X,Y]s,
we have [Z, Y] = [H -X,Y}= J Hsd[X, Y)s, and the proof is complete. D
Example. Let Bt be a standard Wiener process with Bq = 0, (i.e., Brownian
motion). B% — t is a continuous martingale by Theorem 27 of Chap. I. Let
H € L be such that E{f* H^ds} < oo, each t > 0. By Theorem 28 of Chap. I

6 The Quadratic Variation of a Semimartingale 77
we have [B,B]t = t, hence [H ¦ B,H ¦ B]t = J*H^ds. By the martingale
preservation property, J HsdBs is also a continuous local martingale, with
(# • B)o = 0. By Corollary 3 to Theorem 27
f
Jo
HsdBsJ} = E{\H B,H- B}t}
o
It was this last equality,
E{(fHsdBsf} = E{fH2sds),
Jo Jo
that was crucial in K. Ito's original treatment of a stochastic integral.
Theorem 30. Let H be a cadlag, adapted process, and let X, Y be two
semimartingales. Let an be a sequence of random partitions tending to the
identity. Then
-YT")
converges in ucp to JHs^d[X,Y]s (ff0- = 0). Here an = @ < To™ < Tf <
¦¦ <Тгп <¦¦¦ <T?J.
Proof. By the definition of quadratic variation, [X, Y] = ХУ-Х_ -Y-Y--X,
where X- Y denotes the process (/0 Xs-dYs)t>o. By the associativity of the
stochastic integral (Theorem 19)
tf_ ¦ [X,Y] = Я_ • (XY) - tf_ • (X_ -Y)-H-' (У_ • X)
= tf_ • (XY) - (tf_X_) ¦ Y - (Я_У_) • X
= tf_ • (XY) - (HX)- ¦ Y - (HY)- ¦ X.
By Theorem 21 the above is the limit of
_ XT")
- YT"). D

78 II Semimartingales and Stochastic Integrals
7 Ito's Formula (Change of Variables)
Let Л be a process with continuous paths of finite variation on compacts.
If H € L we know by Theorem 17 that the stochastic integral H ¦ A agrees
a.s. with the path-by-path Lebesgue-Stieltjes integral JHsdAs. In Sect. 7 of
Chap. I (Theorem 54) we proved the change of variables formula for / ? C1,
namely
f
f(At)-f(A0)= ff'{As
Jo
)dAs
We also saw at the end of Sect. 4 of this chapter that for a standard Wiener
process В with Bo = 0,
/
Jo
* 1 1
BsdBs = -В2 - -t. (**)
Taking f(x) = x2/2, the above formula is equivalent to
f(Bt) - /(Bo) = f f'(Bs)dBs + \ f f"(Bs)ds,
Jo z Jo
which does not agree with the Lebesgue-Stieltjes change of variables formula
(*). In this section we will state and prove a change of variables formula valid
for all semimartingales.
We first mention, however, that the change of variables formula for con-
continuous Stieltjes integrals given in Theorem 54 of Chap. I has an extension
to right continuous processes of finite variation on compacts. We state this
result as a theorem but we do not prove it here because it is merely a special
case of Theorem 32, thanks to Theorem 26.
Theorem 31 (Change of Variables). Let V be an FV process with right
continuous paths, and let f be such that f exists and is continuous. Then
(f(Vt))t>0 is an FV process and
f(Vt)
- f(V0) = f !'{V.-)dVB + ? {/(Ve) - /(Ve_) - f'(Vs.)AVs}.
J°+ 0<s<t
0<s<t
Recall that the notation JQ+ = f,Q f, denotes the integral over the half
open interval @, t]. We wish to establish a formula analogous to the above,
but for the stochastic integral; that is, when the process is a semimartingale.
The formula is different in this case, as we can see by comparing equation (*)
with equation (**); we must add an extra term!
Theorem 32 (Ito's Formula). Let X be a semimartingale and let f be
a C2 real function. Then f(X) is again a semimartingale, and the following
formula holds:

7 Ito's Formula (Change of Variables) 79
f(Xt) - f(X0) = /* f(Xs.)dXs + \ f f"(Xs
Jo+ z Jo+
0+ ^ ./0 +
{f{Xs)-f{Xa-)-f'{Xe-)AXs}.
0<s<t
Proof Note that the jump part of the stochastic integral J f"(Xs-)d[X, X]s is
given by J2s<t f"(Xs-)(AXsJ, and this is a convergent series. By adding and
subtracting 1/2 of this series, we can rewrite Ito's formula in the equivalent
form
f(Xt)-f(X0)= f f'{Xs_)dXs + \ f f"(Xs_)d[X,X]s
+ ? {f(Xs)~ f(Xs.)- f'{Xs-)AXs-\f"{Xs-){AXsJ}
0<s<t
which is perhaps less obviously a generalization of the "classical" case, but
notationally simpler to prove. The proof rests, of course, on Taylor's Theorem
which says
fiv) - fix) = f(x)(y -x) + \f"{x){y - xJ + R(x,y)
where \R(x,y)\ < r(\y — x\)(y — xJ, such that r : R+ —> R+ is an increasing
function with limujo f(u) = 0, which is valid for f ? C2 defined on a compact
set. We now prove separately the continuous case and the general case.
Proof for the continuous case. We first restrict our attention to a continuous
semimartingale X, since the proof is less complicated but nevertheless gives
the basic idea. Without loss of generality we can take Xo = 0. Define stopping
times
Rm = inf{i : \Xt\ > m}.
Then the stopped process XRm is bounded by m, and if Ito's formula is valid
for XRm for each ra, it is valid for X as well. Therefore we assume that X
takes its values in a compact set. We fix a t > 0, and let an be a refining
sequence of random partitions5 of [0, t] tending to the identity [an = @ =
To" < T2n < ¦ • • < T? = t)}. Then
f(Xt) - /(Xo) = E {/ (X^+1) " / (Х?Г)} (* * *)
5 Note that it would suffice for this proof to restrict attention to deterministic
partitions.

80 II Semimartingales and Stochastic Integrals
The first sum converges in probability to the stochastic integral JQ f'(Xs-)dXs
by Theorem 21; the second sum converges to \ /0 f"(Xs)d[X, X]s in probabil-
probability by Theorem 30. It remains to consider the third sum ^Tli R (Xyn, X^v- J •
But this sum is majorized, in absolute value, by
supr(|XTn+i - XTr\){^2(XTr+i -XTrJ},
T+i
and since J2i(Xr?+1 — ХтрJ converges in probability to [X,X]t (Theo-
(Theorem 22), the last term will tend to 0 if limn^oosupir(\X'X'r' t — Xtp\) = 0.
However s i—» Xs(w) is a continuous function on [0,t], each fixed ш, and
hence uniformly continuous. Since limn^ooSup, \T™+1 — T"| = 0 by hypoth-
hypothesis, we have the result. Thus, in the continuous case, f(Xt) — f(Xo) =
/0* f'(Xs-)dXs + \ /0* f"{Xs-)d[X, Xs], for each t, a.s. The continuity of the
paths then permits us to remove the dependence of the null set on (, giving
the complete result in the continuous case.
Proof for the general case. X is now given as a right continuous semimartin-
gale. Once again we have a representation as in (***), but we need a closer
analysis. For any t > 0 we have ^20<s<t(AXsJ < [X,X]t < oo a.s., hence
^0<s<((AIsJ is convergent. Given e > 0 and t > 0, let A = A{e,t) be
a set of jump times of X that has a.s. a finite number of times s, and let
В = B(e,t) be such that Х^ев(А^J < ?2> where A and В are disjoint and
Аи В exhaust the jump times of X on @, t]. We write
f(Xt) - f(X0) =
= E {/(%.) - явд}+E {
i,A i,B
where ?\>A denotes ?\ 1{Ап(тр,щг№}- Then
{ }
j,A s€A
and by Taylor's formula
- XTrJ}

7 Ito's Formula (Change of Variables) 81
As in the continuous case, the first two sums on the right side above converge
J*+f'(Xs-)dXs and \ /0*+
ous case, the fist to sus o
respectively to J*+f'(Xs-)dXs and \ /0*+ f"(Xs^)d[X,X}s. The third sum
converges to
]T !f'{X3.)AX3
Assume temporarily that \XS\ < k, some constant k, all s < t. Then /" is
uniformly continuous, and using the right continuity of X we have
,ХТп+1) < r(e+)[X,X]t,
where r(e+) is Iimsup5|?r(E). Next let e tend to 0. Then r(e+)[X,X]t tends
to 0, and
seA(e,t)
\f(Xs) - f(Xt.) - f(Xs_)AXs - У"(
tends to the series in (* * *), provided this series is absolutely convergent.
Let Vk = inf{t > 0 : \Xt\ > к}, with Xo = 0. By first establishing (***) for
X\\pyk), which is a semimartingale since it is the product of two semimartin-
gales (Corollary 2 of Theorem 22), it suffices to consider semimartingales
taking their values in intervals of the form [—k, k]. For / restricted to [—fc, k]
we have \f(y) - f(x) - (y - x)f'(x)\ < C(y - xf. Then
\f(Xs)-f(Xs_)-f'(Xs_)AXs\<C
0<s<t 0<s<t
< <2 < K[X,X)t < oo a.s.
Thus the series is absolutely convergent and this completes the proof. D
Corollary (Ito's Formula). Let X be a continuous semimartingale and let
/ be a C2 real function. Then f(X) is again a semimartingale and the following
formula holds:
f(Xt) - /(Xo) = f f'(Xs)dXs + \ f f"(Xs)d[X,X]s.
Jo+ z Jo+
Theorem 32 has a multi-dimensional analog. We omit the proof.
Theorem 33. Let X = (X1,..., Xn) be an n-tuple of semimartingales, and
let f : Rn —> M. have continuous second order partial derivatives. Then f(X)
is a semimartingale and the following formula holds:

82 II Semimartingales and Stochastic Integrals
f(Xt) - f(X0)
0<s<t i=l '
The stochastic integral calculus, as revealed by Theorems 32 and 33, is
different from the classical Lebesgue-Stieltjes calculus. By restricting the class
of integrands to semimartingales made left continuous (instead of L), one can
define a stochastic integral that obeys the traditional rules of the Lebesgue-
Stieltjes calculus.
Definition. Let X, Y be semimartingales. Define the Fisk-Stratonovich
integral of Y with respect to X, denoted /0 Vs_ о dXs, by
[ Ув_ о dXs = / Ys.dXs + \[Y,X)ct.
Jo Jo l
The Fisk-Stratonovich integral is often referred to as simply the Stratono-
vich integral. The notation "o" is called Ito's circle. Note that we have
defined the Fisk-Stratonovich integral in terms of the semimartingale integral.
With some work one can slightly enlarge the domain of the definition and we
do so in Sect. 5 of Chap. V. In particular, Theorem 34 below is proved with
the weaker hypothesis that / G C2 (Theorem 20 of Chap. V). We will write
the F-S integral as an abbreviation for the Fisk-Stratonovich integral.
Theorem 34. Let X be a semimartingale and let f be C3. Then
f(Xt)-f(X0) = f f'(Xs.)odXs+ J2 if(Xs) - f(Xs-) - f'(Xs.)AXs}.
J0+ ns.<+
0<s<t
Proof. Note that /' is C2, so that f'{X) is a semimartingale by Theorem 32
and in the domain of the F-S integral. By Theorem 32 and the definition, it
suffices to establish \\f'(X),X}c = \ \l f"(XsJ)d{X,X\cs. However
f'(Xt)-f'(X0)= f f"(Xs.)dXs + l f
0+ ^ ./0+
У] {/'(*.) - f'(Xs-) - f"(Xs..)AXs}.
0<s<t
Thus
[f'{X),X}c = [f"(X-) -X,X}C + [i/<3>(X_) • [X,X},X}C.
The first term on the right side above is /„* f"{Xs.)d[X,X]cs by Theorem 29;
the second term can easily be seen, as a consequence of Theorem 22 and the

7 Ito's Formula (Change of Variables) 83
fact that [X, X] has paths of finite variation, to be ?0<s<t f{3){Xs-){^Xsf.
That is, zero, and the theorem is proved. D
Note that if X is a semimartingale with continuous paths, then Theorem 34
reduces to the classical Riemann-Stieltjes formula f(Xt)—f(Xo) = /0 f'(Xs)o
dXs. This is, of course, the main attraction of the Fisk-Stratonovich integral.
Corollary (Integration by Parts). Let X, Y be semimartingales, with at
least one of X and Y continuous. Then
XtYt - X0Y0 = f Xs- о dYs + [ Fs_ о dXs.
J Jo
0Y0 = f
Jo
Proof. The standard integration by parts formula is
XtYt=
o
However [X, Y}t = [X, Y}$ + X0Y0 if one of X or Y is continuous. Thus adding
\\X, Y]^ to each integral on the right side yields the result. ?
One can extend the stochastic calculus to complex-valued semimartingales.
Theorem 35. Let X, Y be continuous semimartingales, let Zt = Xt + iYt,
and let f be analytic. Then
f(Zt)=f(Z0)+ f f'{Zs)dZs + \f f"(Zs)d[Z,Z]s.
Jo+ z Jo+
Proof. Using Ito's formula for the real and imaginary parts of / yields
Since / is analytic by the Cauchy-Riemann equations ff = /' and |^ =
if. Differentiating again gives 0 = % = /" and ?^ = if" and 0 =
—/". The result now follows by collecting terms and observing that dZs =
dXs + idYs and [Z, Z] = [X + iY, X +iY} = [X, X] + 2i[X, Y] - [Y, Y]. О
Theorem 35 also has a version for Zt = Xt+iYt, with both X and Y cadlag
semimartingales. The proof is almost the same as the proof of Theorem 35.

84 II Semimartingales and Stochastic Integrals
Theorem 36. Let X, Y be cadlag semimartingales, let Zt = Xt + iYt, and
let f be analytic. Then
+ f f(Zs.)dZs + l [ f\ZsJ)d\Z,Z\\
Observe that for a complex-valued semimartingale the process [Z, Z] is in
general a complex-valued process. For many applications, it is more appropri-
appropriate to use the non-negative increasing process
[Z,Z} = {X,X} + {Y,Y]
to play the role of the quadratic variation.
8 Applications of Ito's Formula
As an application of the change of variables formula, we investigate a simple,
yet important and non-trivial, stochastic differential equation. We treat it, of
course, in integral form.
Theorem 37. Let X be a semimartingale, Xq = 0. Then there exists a
(unique) semimartingale Z that satisfies the equation Zt = 1 + JQ Zs_dXs. Z
is given by
Zt=exj>(xt-hx,X]t\ П (
^ ' 0<s<t
where the infinite product converges.
Proof. We will not prove the uniqueness here since it is a trivial consequence of
the general theory to be established in Chap. V. (For example, see Theorem 7
of Chap. V.) Note that the formula for Zt is equivalent to the formula
Zt = exp \xt - 1[Х,Х]Л ТТA + AXs)exp{-AXs}.
Since Xt — \[X,X]t is a semimartingale, and ex is C2, we need only show
that Ils<t(l + AXs)exp{-AXs} is cadlag, adapted, and of finite variation.
It will then be a semimartingale too, and thus Z will be a semimartingale.
The product is clearly cadlag, adapted; it thus suffices to show the product
converges and is of finite variation.
Since X has cadlag paths, there are only a finite number of s such that
|AXsj > 1/2 on each compact interval (fixed w). Thus it suffices to show

8 Applications of Ito's Formula 85
Vt = П A
0<s<t
converges and is of finite variation. Let Us = AXs1{|axs|<i/2}- Then we have
log Vt — X^s<t{'°e(^ +^*) ~~ ^*}> which is an absolutely convergent series a.s.,
since So<s«(^*J — Р^-^Ь < °° a-s-' because | log(l + x) - x\ < x2 when
|ж| < 1/2. Thus log(Vt) is a process with paths of finite variation, and hence
so also is expjlogVj} = Vj.
To show that Z is a solution, we set Kt = Xt — | [X, X]%, and let f(x, y) =
yex. Then Z* = f(Kt, St), where 5t = По<*<*A + Ax*) exp{-AXs}. By the
change of variables formula we have
Zt-1= f Zs_dKs+ f eK°-dSs + \ f Zs-d[K,K]cs (*)
./0+ Jo+ ^ ^o+
0<s<t
t 1 ft Л
= / Zs-dXs-\ [ Zs_d[X,X]cs+ f eK^
Jo+ *¦ Jo+ Jo+
+ \ f Za-d[X,X)ce+ Y. (Zs-Zs--Z)
2 J°+ 0<s<t
since [K, S]c — [S, S]c = 0. Note that S is a pure jump process. Hence
/0*+ eK-dSs = J20<s<teK>-ASs. Also Zs = Ze_(l + AXS), and ZS_
AX, so the last sum on the right side of equation (*) becomes
-eK-
0<s<t 0<s<t
Thus equation (*) simplifies due to cancellation to Zt — 1 = Jo Zs-dXs, and
we have the result. ?
Definition. For a semimartingale X, Xo = 0, the stochastic exponential
of X, written ?(X), is the (unique) semimartingale Z that is a solution of
The stochastic exponential is also known as the Doleans-Dade expo-
exponential. Theorem 37 gives a general formula for ?{X). This formula simplifies
considerably when X is continuous. Indeed, let X be a continuous semimartin-
semimartingale with Xo = 0. Then
?{X)t=exp{Xt-±[X,X]t}.
An important special case is when the semimartingale X is a multiple Л of a
standard Brownian motion В = (Bt)t>0. Since AB has no jumps we have

86 II Semimartingales and Stochastic Integrals
?(XB)t = exp{ABt - y[B,B]t} = exp{ABt - y*
Moreover, since ?(XB)t = 1 + XjQ?(XB)s-dBs we see that ?(XB)t =
eXBt--2-t js a continuous martingale. The process ?{XB) is sometimes referred
to as geometric Brownian motion. Note that the previous theorem gives us
?{X) in closed form. We also have the following pretty result.
Theorem 38. Let X and Y he two semimartingales with Xq = Yq = 0. Then
?{X)?(Y) = ?{X + Y + [X,Y]).
Proof. Let Ut = ?{X)t and Vt = ?(Y)t. Then the integration by parts formula
gives that UtVt - 1 = /0* Us_dVs + /0*+ Vs_dUs + [U, V)t. Since U and V are
exponentials, this is equivalent to
/ U8-V.-dYa+ [ Us.Vs.dXs+ [ Us-V
o+ Jo+ Jo+
Letting Wt = UtVt, we deduce that Wt = 1 + /0* Ws-d(X + Y+ [X,Y])S so
0*
that W = ?{X +Y+[X,Y\), which was to be shown. ?
Corollary. Let X be a continuous semimartingale, Xo = 0. Then ?(X)~1 =
Proof. By Theorem 38,
?{X)?(-X + [X, X}) = ?{X + {-X + [X, X}) + [X, -X}),
since [—X, [X, X]] = 0. However ?@) = 1, and we are done. D
In Sect. 9 of Chap. V we consider general linear equations. In particular,
we obtain an explicit formula for the solution of the equation
./o
Xt=Ht+ XsdZs,
Jo
where Z is a continuous semimartingale. We also consider more general in-
inverses of stochastic exponentials. See, for example, Theorem 63 of Chap. V.
Another application of the change of variables theorem (and indeed of
the stochastic exponential) is a proof of Levy's characterization of Brownian
motion in terms of its quadratic variation.
Theorem 39 (Levy's Theorem). A stochastic process X — (^t)t>o *s a
standard Brownian motion if and only if it is a continuous local martingale
with [X,X]t = t.

8 Applications of Ito's Formula 87
Proof. We have already observed that a Brownian motion В is a continuous
local martingale and that [B,B]t = t (see the remark following Theorem 22).
Thus it remains to show sufficiency. Fix tieR and set F(x, t) — ехр{шх +
Ц^-t}. Let Zt = F(Xt,t) = exp{iuXt + (^-) t}. Since F e C2 we can apply
Ito's formula (Theorem 33) to obtain
/"* и2 Гг и2 /"*
/ ZsdXs + \ / Zads-\ / Zsd{X,X}s
Jo l Jo l Jo
i:
= 1 +ra / ZsdX
su,^i.s,
which is the exponential equation. Since X is a continuous local martingale,
we now have that Z is also one (complex-valued, of course) by the martingale
preservation property. Moreover stopping Z at a fixed time to, Z*°, we have
that Zta is bounded and hence a martingale. It then follows for 0 < s < t that
E{exp{iu(Xt - X3)}\F.} = exp |~(t -
Since this holds for any и G M we conclude that Xt - Xs is independent of
Ts and that it is normally distributed with mean zero and variance (t — s).
Therefore X is a Brownian motion. D
Observe that if M and N are two continuous martingales such that MN
is a martingale, then [M, N] — 0 by Corollary 2 of Theorem 27. Therefore if
Bt = (Bl,... ,-B") is an n-dimensional standard Brownian motion, B\B\ is
a martingale for i ^ j, and we have that
Theorem 39 then has a multi-dimensional version, which has an equally simple
proof.
Theorem 40 (Levy's Theorem: Multi-dimensional Version). LefX. =
(X1,... ,Xn) be continuous local martingales such that
Then X is a standard n-dimensional Brownian motion.
As another application of Ito's formula, we exhibit the relationship between
harmonic and subharmonic functions and martingales.
Theorem 41. Let X = (X1,... ,Xn) be an n-dimensional continuous local
martingale with values in an open subset D o/M™. Suppose that [Xl,Xi] = 0
if i Ф 3i and [Хг,Хг] = A, 1 < г < п. Let и : D —> R be harmonic (resp.
subharmonic). Then u(X) is a local martingale (resp. submartingale).

88 II Semimartingales and Stochastic Integrals
Proof. By Ito's formula (Theorem 33) we have
u(Xt) - u(Xq) = f Vw(Xs) • dXs + i f Au(Xs)dA
Jo+ * Jo+
f f
o+ * Jo+
where the "dot" denotes Euclidean inner product, V denotes the gradient,
and Д denotes the Laplacian. If и is harmonic (subharmonic), then Аи =
0(Au > 0) and the result follows. D
If В is a standard n-dimensional Brownian motion, then В satisfies the
hypotheses of Theorem 41 with the process At = t. That u(Bt) is a submartin-
submartingale (resp. supermartingale) when и is subharmonic (resp. superharmonic) is
the motivation for the terminology submartingale and supermartingale. The
relationship between stochastic calculus and potential theory suggested by
Theorem 41 has proven fruitful (see, for example, Doob [56]).
Levy's characterization of Brownian motion (Theorem 39) allows us to
prove a useful change of time result.
Theorem 42. LetM = (Mt)t>o be a continuous local martingale with Mo — 0
and such that Нгщ-юо \M, M]t = oo a.s. Let
Ts =inf{i>0: [M,M]t>s}.
Define Qs = Ттв and Bs = Мтв. Then (Bs,Qs)s>0 is a standard Brownian
motion. Moreover ([M,M]t)t>o are stopping times for (Qs)s>o and
Mt = B[M,M]t a-s- 0 < i < oo.
That is, M can he represented as a time change of a Brownian motion.
Proof. The (Ts)s>o are stopping times by Theorem 3 of Chap. I. Each Ts
is finite a.s. by the hypothesis that limt_,.oo [M, M]t = oo a.s. Therefore the
<t-fields Qs = Ft, are well-defined. The filtration {Qs)s>o need not be right
continuous, but one can take 7is = Gs+ = ^тв+ to obtain one. Note further
that {[M,M]t < s} = {Ts > t}, hence ([M,M]t)t>o are stopping times for
the filtration Q = (Gs)s>o-
By Corollary 3 of Theorem 27 we have E{MTs} = E{[M, M\Ts} = s < oo,
since [M, M]tb = s identically because [M, M] is continuous. Thus the time
changed process is square integrable. Moreover
E{BU - Bs\gs} = E{MTu - MT. \Ft,} = 0
by the Optional Sampling Theorem. Also
e{bI- в1\да) = e{(bu- B,J\g.}
= E{{MTu -MTaf\FTa}
= E{{M, M)Tu - [M,M}t, \Ft,}
= и — s.

8 Applications of Ito's Formula 89
Therefore .B2 — s is a martingale, whence [B, B]s = s provided В has contin-
continuous paths, by Corollary 2 to Theorem 27.
We want to show that Bs = Mts has continuous paths. However by Corol-
Corollary 5 of Theorem 27 almost surely all intervals of constancy of [M, M\ are also
intervals of constancy of M. It follows easily that В is continuous. It remains to
show that Mt = В[М<щ±. Since Bs = Mts, we have that В[М<щг = Mt[MiMU,
a.s. Since (Ts)s>o is the right continuous inverse of [M,M], we have that
T[m,m\i — *> with equality holding if and only if t is a point of right in-
increase of [M,M]. (If (Ts)s>0 were continuous, then we would always have
that T[M,M]t = t.) However TjM>M]t > t implies that t >-> [M,M]t is constant
on the interval (?, TjMiM]J; thus by Corollary 5 of Theorem 27 we conclude
M is constant on (?,TjMjM]t). Therefore B[M,M]t = Мт[м M]t = Mt a.s., and
we are done. D
Another application of the change of variables formula is the determination
of the distribution of Levy's stochastic area process. Let Bt = (Xt,Yt) be an
M2-valued Brownian motion with (Xq,Yq) = @,0). Then during the times s
to s + ds the chord from the origin to В sweeps out a triangular region of area
^RsdNs, where
R3 = y/X* + У82
and
dNs = -^-dXs + ф-dY,.
Therefore the integral At = /0 RsdNs = /0 (—YsdXs+XsdYs) is equal to twice
the area swept out from time 0 until time t. Paul Levy found the characteristic
function of At and therefore determined its distribution. Theorem 43 is known
as Levy's stochastic area formula.
Theorem 43. Let Bt = (Xt,Yt) be an M?-valued Brownian motion, Bo =
@,0),u6R. Let At = /0* XsdYs - /0* YsdXs. Then
E{eluAt} = \—, 0<i<oo,-oo<u<oo.
cosh(ui)
Proof. Let a(t), C(t) be C1 functions, and set
Then
dVt — iudAf —(X? + Y?)dt
- a{t){XtdXt + YtdYt + dt} + C'(t)dt
= (-iuYt - a(t)Xt)dXt + (iuXt - a{t)Yt)dYt
- l-dt{a'{t)X2t + a'(t)Yt2 + 2a(t) -

90 II Semimartingales and Stochastic Integrals
Next observe that, from the above calculation,
d[V, V]t = (-iuYt - a{t)XtJdt + (iuXt - a(t)YtJdt
= (a(tJ - u2)(X2 + Y2)dt.
Using the change of variables formula and the preceding calculations
dev'=ev<(dVt+±d[V,V}t)
= eVt(-iuYt - a(t)Xt)dXt + eVt(iuXt - a(t)Yt)dYt
+ -eVtdt{(a(tJ - u2 - a'(t))(X2 + Y2) + 2C'(t) - 2a(t)}.
2*
Therefore eVt is a local martingale provided
a'{t) = a{tf - u2
13'(t) = a(t).
Next we fix t0 > 0 and solve the above ordinary differential equations with
a(t0) = /3(t0) = 0. The solution is
a(t) = и tanh(u(i0 - t))
/3(t) = -lQgcoeh(u(i0-t)),
where tanh and cosh are hyperbolic tangent and hyperbolic cosine, respec-
respectively. Note that for 0 < t < t0,
\eVt\ = exp{—a(t)
Thus eVt, 0 < t < to is bounded and is therefore a true martingale, not just a
local martingale, by Theorem 51 of Chap. I. Therefore,
However, Vt0 = iuAt0 since a(to) = /3(io) = 0. As Ao — Xq — Yq = 0, it
follows that Vo = —log cosh(ui0)- We conclude that
E{eiuAt°} = exp{-log cosh(ui0)}
1
cosh(uio)'
and the proof is complete. ?
There are of course other proofs of Levy's stochastic area formula (e.g.,
Yor [246], or Levy [150]). As a corollary to Theorem 43 we obtain the density
for the distribution of At.

8 Applications of Ito's Formula 91
Corollary. Let Bt = (Xt, Yt) be an M2-valued Brownian motion, Bo = @, 0),
and set At = /0 XsdYs — JQ YadXs. Then the density function for the distri-
distribution of At is
2t coshGrx/2i)'
-00 < X < 00.
Proof. By Theorem 43, the Fourier transform (or characteristic function) of
At is E{eiuAt} = cosh'(Mt). Thus we need only to calculate ? / ±^du. The
z0 = Щ_,
Next we
given by
as shown
1 1O Ul tilt
we have
integrate
* 1U11
1 J
Res(/, z0) -
along the closed
Cr- \
in Fig. 1
2/ = 0,
X — Г,
7Г
У=1'
x= -r,
below.
X
c$
= —r
coshBt) • O1
РЫ
Q'(zo)
curve Cr
-r <
0 <y
r > x
IT
net
e-KT/2t
it
traversed
x < r,
IT
1
> -r,
>o,
a;
у
= r
^;uoll^^>^ над a, puic at
counter clockwise, and
cr
c2
cr
cr
Fig. 1. The closed curve of integration Cr = C^ + C^ + C* + C*.
Therefore Jc f(z)dz — 2ni Res(/, Zq). Along C% the integral is
— / —— du =
Jr cosh(ui)
l_r cosh(ui)
The integrands on C2 and Cf are dominated by 2e~r, and therefore

92 II Semimartingales and Stochastic Integrals
lim / f(z)dz = A + era/t) /
•^°° Jcr J-
= 2m Res(/,
.-«
lim / f(z)dz = A + e™/*) / .. ..du
J J cosh(ui)
-2тг-
Finally we can conclude
2ttJ_
g — ШХ J
A11
Loo cosh(ui) 2тг
2t cosh(^f)'
П
The stochastic area process A shares some of the properties of Brownian
motion, as is seen by recalling that At = /0 RsdNs, where TV is a Brownian
motion by Levy's Theorem (Theorem 39), and N and R are independent (this
must be proven, of course). For example A satisfies a reflection principle. If
one changes the sign of the increments of A after a stopping time, the process
obtained thereby has the same distribution as that of A. One can use this
fact to show, for example, that if St = supo<s<tAs, then St has the same
distribution as \At\, for t > 0.
Bibliographic Notes
The definition of semimartingale and the treatment of stochastic integration
as a Riemann-type limit of sums is in essence new. It has its origins in the
fundamental theorem of Bichteler [13, 14], and Dellacherie [42]. The peda-
pedagogic approach used here was first suggested by Meyer [176], and it was then
outlined by Dellacherie [42]. Dellacherie's outline was further expanded by
Lenglart [145] and Protter [201, 202]. These ideas were also present in the
work of M. Metivier and J. Pellaumail, and to a lesser extent Kussmaul, al-
although this was not appreciated at the time by us. See for example [160], [162],
and [138]. A similar idea was developed by Letta [148]. A recent treatment is
in Bichteler [15].
We will not attempt to give a comprehensive history of stochastic inte-
integration here, but rather just a sketch. The important early work was that of
Wiener [228, 229], and then, of course, Ito [93, 94, 95, 97, 98]. However, it
has recently been discovered, by B. Bru and M. Yor, that the young French
mathematician W. Doeblin6 had essentially established Ito's formula in early
6 Wolfgang Doeblin (Doblin) was born in 1915 in Berlin, and after acquiring French
citizenship in 1936 was known as Vincent Dobbin. He was killed in June 1940.

Bibliographic Notes 93
1940, but was killed later that year in action in the Second World War, and
thus his results were lost until recently [26, 27]. Doob stressed the martingale
nature of the Ito integral in his book [55] and proposed a general martingale
integral. Doob's proposed development depended on a decomposition theo-
theorem (the Doob-Meyer decomposition, Theorem 8 of Chap. Ill) which did not
yet exist. Meyer proved this decomposition theorem in [163, 164], and com-
commented that a theory of stochastic integration was now possible. This was
begun by Courrege [36], and extended by Kunita and Watanabe [134], who
revealed an elegant structure of square integrable martingales and established
a general change of variables formula. Meyer [166, 167, 168, 169] extended
Kunita and Watanabe's work, realizing that the restriction of integrands to
predictable processes is essential. He also extended the integrals to local mar-
martingales, which had been introduced earlier by Ito and Watanabe [102]. Up
to this point, stochastic integration was tied indirectly to Markov processes,
by the assumption that the underlying filtration of cr-algebras be "quasi-left
continuous." This hypothesis was removed by Doleans-Dade and Meyer [53],
thereby making stochastic integration a purely martingale theory. It was also
in this article that semimartingales were first proposed in the form we refer
to as classical semimartingales in Chap. III.
A different theory of stochastic integration was developed independently
by McShane [154, 155], which was close in spirit to the approach given here.
However it was technically complicated and not very general. It was shown in
Protter [200] (building on the work of Pop-Stojanovic [194]) that the theory
of McShane could for practical purposes be viewed as a special case of the
semimartingale theory.
The subject of stochastic integration essentially lay dormant for six years
until Meyer [171] published a seminal "course" on stochastic integration. It
was here that the importance of semimartingales was made clear, but it was
not until the late 1970's that the theorem of Bichteler [13, 14], and Del-
lacherie [42] gave an a posteriori justification of semimartingales. The seem-
seemingly ad hoc definition of a semimartingale as a process having a decomposi-
decomposition into the sum of a local martingale and an FV process was shown to be
the most general reasonable stochastic differential possible. (See also Kuss-
maul [138] in this regard, and the bibliographic notes in Chap. III.)
Most of the results of this chapter can be found in Meyer [171], though they
are proven for classical semimartingales and hence of necessity the proofs are
much more complicated. Theorem 4 (Strieker's Theorem) is (of course) due to
Strieker [218]; see also Meyer [173]. Theorem 5 is due to Meyer [175]. There
are many other methods of expanding a filtration and still preserving the
semimartingale property. For further details, see Chap. VI on expansion of
filtrations.
Theorem 14 is originally due to Lenglart [142]. Theorem 16 is known to
be true only in the case of integrands in L. The local behavior of the integral
(Theorems 17 and 18) is due to Meyer [171] (see also McShane [154]). The a.s.

94 Exercises for Chapter II
Kunita-Watanabe inequality, Theorem 25, is due to Meyer [171], while the ex-
expected version (the corollary to Theorem 25) is due to Kunita-Watanabe [134].
That continuous martingales have paths of infinite variation or are con-
constant a.s. was first published by Fisk [73] (Corollary 1 of Theorem 27). The
proof given here of Corollary 4 of Theorem 27 (that a continuous local martin-
martingale X and its quadratic variation [X, X] have the same intervals of constancy)
is due to Maisonneuve [151]. The proof of Ito's formula (Theorem 32) is by
now classic; however we benefited from Follmer's presentation of it [75]. A
popular alternative proof which basically bootstraps up form the formula for
integration by parts, can be found (for example) on pages 57-58 of J. Jacod
and A. N. Shiryaev [110].
The Fisk-Stratonovich integral was developed independently by Fisk [73]
and Stratonovich [217], and it was extended to general semimartingales by
Meyer [171]. Theorem 35 is inspired by the work of Getoor and Sharpe [215].
The stochastic exponential of Theorem 37 is due to Doleans-Dade [49]. It
has become extraordinarily important. See, for example, Jacod-Shiryaev [110].
The pretty formula of Theorem 38 is due to Yor [237]. Exponentials have of
course a long history in analysis. For an insightful discussion of exponentials
see Gill-Johansen [82]. That every continuous local martingale is the time
change of a Brownian motion is originally due to Dubins-Schwarz [59] and
Dambis [37]. The proof of Levy's stochastic area formula (Theorem 43) is
new and is due to Janson. See Janson-Wichura [113] for related results. The
original result is in Levy [150], and another proof can be found in Yor [246].
Exercises for Chapter II
Exercise 1. Let В be standard Brownian motion and let / be a function
mapping M —> M which is continuous except for one point where there is a
jump discontinuity. Show that Xt = f(Bt) cannot be a semimartingale.
Exercise 2. Let Q « P (Q and P are both probability measures and Q
is absolutely continuous with respect to P). Show that if Xn —> X in im-
improbability, then Xn —> X in Q-probability.
Exercise 3. Let (X, Y) be standard two dimensional Brownian motion. (That
is, X and Y are independent one dimensional standard Brownian motions.)
Let 0 < a < 1. Set
Bt = aXt + \/l - a2Yt.
Show that В is a standard one dimensional Brownian motion. Compute [X, B]
and [Y,B].
Exercise 4. Let / : M+ —> K+ be non-decreasing and continuous. Show that
there exists a continuous martingale M such that [M, M]t = /(?)¦
Exercise 5. Let В be standard Brownian motion and let H € L be such that
Ht\ = 1, all t. Let Mt = Jo HsdBs. Show that M is also a Brownian motion.

Exercises for Chapter II 95
Exercise 6. Give an example of semimartingales Xn, X, Y such that
limn-HXjX™ = Xt a.s., all t, but limn_>oo[Xn,F]i ф [X,Y\t a.s., for some
t.
Exercise 7. Let Y, Z be semimartingales, and let Hn, H e L. Suppose
Нт„^оо Я" = Я in ucp. Let X" = / H?dYs and X = / HsdYs. Show that
UmT,_>oo[A-n,Z]t = [X,Z]t in ucp, all i.
Exercise 8. Suppose / and all fn are C2, and that fn and /^ converge
uniformly on compacts to / and /', respectively. Let X, Y be continuous
semimartingales and show that Шпп_>оо[/п(.Х'),У] = [f(X),Y].
Exercise 9. Let M be a continuous local martingale and let A be a continu-
continuous, adapted process with paths of finite variation on compact time sets, with
Ao = 0. Let X = M + A. Show that [X,X] = [M, M) a.s.
Exercise 10. Let X be a semimartingale and let Q be another probability
law, with Q « P. Let [X, X]p denote the quadratic variation of X considered
as a P semimartingale. Show that \X,X}Q = [X,X]P, Q-a.s.
Exercise 11. Let В be standard Brownian motion, and let T = inf{i > 0 :
Bt = lorBt = -2}.
(a) Show that T is a stopping time.
(b) Let M = BtAr and let N = —M. Show that M and N are continuous
martingales and that [M,M]t = [N,N]t = t AT. Show that nevertheless
M and N have different distributions.
Exercise 12. Let X be a semimartinagle such that Ylo<s<t l^-^-sl < °° a-s-'
each t > 0. Let f e C2 and show that J2o<s<t \&f(Xs)\ < oo a.s., each t > 0
as well.
Exercise 13. Let X be a semimartingale and suppose Hn and H are in L.
Further suppose that Hn -> H in ucp. Show that f H?dXs -+ / HsdXs in
ucp.
* Exercise 14. Let X be a semimartingale, and let A be a non-negative,
continuous, increasing process with А^, = oo a.s. Suppose Нт^-юо /0 A +
As)~ldXs exists and is finite a.s. Show that lim^oo ^ = 0 a.s.
Exercise 15. Let M be a continuous local martingale and assume [M, M}^, =
oo a.s. Show that limj-K» [д/мЬ = ^ a's'
Exercise 16. Let В be a standard Brownian motion and let H be adapted
and continuous. Show that for fixed i,
lim -= 5- / HsdBs = Hu
h-*0 Bt+h — Bt Jt
with convergence in probability.

96 Exercises for Chapter II
Exercise 17. Let Y be a continuous, adapted process with Yq = О and with
Y constant on [l,oo). Let
t
^ Ysdsl{t>x).
n
Show that Xn is a semimartingale, each n, but that linin^oo Xn = Y need
not be a semimartingale.
Exercise 18. (Continuation of Exercise 17.) Let В be a standard Brownian
motion and let Xn = n Jt_x Bsdsl^t>x}- Solve
t = s s ' 0 = '
and show that Нт„_юо X™ = Bt a.s., each t, but that linin^oo Zn ф Z, where
Z solves the equation dZt — ZtdBt.
Exercise 19. (Related to Exercises 17 and 18.) Let A? = ? sin(ni), 0 < t <
ж/2. Show that An is a semimartingale for each n, and that /0T \dA™\ = 1,
each n (this means the total variation process, path-by-path, of A), but that
lim sup \At\ =0.
*Exercise 20. (This exercise refers to an example briefly considered in Sect. 6
of Chap. I on local martingles, and again in Sect. 6 of Chap. II.) Let В be
a standard Brownian motion in M3, and let и : К3 \ {0} —> R be given by
u(y) — \\y\\~1 ¦ Assume Bq = x a.s., where x ф 0, show the following.
(a) Mt = u(Bt) is a local martingale.
(b) limt^ooEx{u(Bt))=0.
(c) u{Bt) e L2, each t > 0.
(d) supt>0E*(u(BtJ)<oo.
Conclude that u(B) is not a martingale.
Exercise 21. Let А, С be two non-decreasing cadlag processes with Aq =
Co = 0. Assume that both A^ = lim^oo At and C^ = Цщ^а, Ct are finite.
(a) Show that ^ooCoo = /0°°(^oo - A,)dC, + ^(C» - Cs-)dAs.
(b) Deduce from part (a) the general formula
/•OO POO
= / AsdCs + / Cs_dAs.
Jo Jo
Note that the above formulae are not symmetric. See also Theorem 3 in
Chap. VI.

Exercises for Chapter II 97
Exercise 22. Let A be non-decreasing and continuous, and assume A^ =
lim^oo At is finite, Ao — 0. Show that for integer p > 0,
ЛОО />0O ЛОО
(Aoo)"=p\ / dASl / dAS2-- dASp.
Jo Jsi Jsp-i
*Exercise 23 (expansion of nitrations). Let Б be a standard Brownian
motion on a filtered probability space (O, J7, F, P), and define a new filtration
G by S° = Tt V a{Bx) and Qt = f]u>t G°.
(a) Show that E{Bt - BS\QS} = |5f (Bi - Bs) for 0 < s < t < 1.
(b) Show that
' Вл-Вч ,
I
Jo
1-
where /? is a G-Brownian motion. (Note in particular that the process
(Bt - Jo Bilf"ds)o<t<i is a G martingale on [0,1).)
(c) Show that
f1 Bi-B8l,
-\ds < oo a.s.
/
./o
1-s
and conclude that В is a G semimartingale.
This is a special case of Theorem 3 in Chap. VI.
Exercise 24. With the notation of Exercise 23, let Xt = Bt -?Bi, 0 < t < 1.
(a) Show that
) f
Jo
f ^ps
Jo *¦ ~ s
(b) Deduce from (a) that X satisfies the stochastic differential equation
Xt=b+
Уо 1 ~s
(X is called a Brownian bridge from 0 to 0.)
*Exercise 25. Let {Bt,Tt)t>o be a standard Brownian motion. Let a,b > 0
and T = inf{t > 0 : Bt e (-a,bH}.
(a) Show that
is a local martingale,
(b) Show that

98 Exercises for Chapter II
(c) For 0 < 9 < -^, show that XT is a positive supermartingale, and deduce
(d) Use (c) to show that Xj, = sups<r \XS\ G L1, and conclude that X is a
martingale.
(e) Conclude that
Exercise 26. Let {Bt,Tt)t>o be standard Brownian motion, and let T =
infji > 0 : Bt e (-d,dH}. Let M = BT. Show that
(a) if d < f, then E{exp{±[M,M]T}} < oo; but
(b) if d = f, then Е{ещ>{\[М,M]T}} = oo. (#m?; Use Exercise 25.)
Exercise 27. Let {Bt,Tt)t>o be standard Brownian motion, and let Xt =
e"at(Xo+<J /0 easdBs). Show that X is a solution to the stochastic differential
equation
dXt — -aXtdt + adBt.
Exercise 28. Let В be a standard Brownian motion and let ?(B) denote the
stochastic exponential of B. Show that lim^oo ?(B)t = 0 a.s.
Exercise 29. Let X be a semimartingale. Show that
1 = -dX + d[X,X]
[?(Xy ?(X)
Exercise 30. Let Bbea standard Brownian motion.
(a) Show that M is a local martingale, where
Mt = exp{Bt - -t}.
(b) Calculate [M,M]t, and show that M is a martingale.
(c) Calculate E{eBt}.
The next eight problems involve a topic known as changes of time. For
these problems, let (Q,,J-,?,P) satisfy the usual hypotheses. A change of
time R = (Rt)t>o 1S a family of stopping times such that for every wef!, the
function R.(w) is non-decreasing, right continuous, Rt < oo a.s., and Ro — 0.
Let Qt = Tnt. Change of time is discussed further in Sect. 3 of Chap. IV.
Exercise 31. Show that G = (?t)t>o satisfies the usual hypotheses.
Exercise 32. Show that if M is an F uniformly integrable martingale and
Mt := Mnt, then MisaG martingale.

Exercises for Chapter II 99
Exercise 33. If M is an F (right continuous) local martingale, show that M
is a G semimartingale.
*Exercise 34. Construct an example where M is an F local martingale, but
M is not a G local martingale. (Hint: Let (Хп)пе® be an adapted process. It
is a local martingale if and only if \Xn\dP is a сг-finite measure on Tn-i-, and
E{Xn\Tn-\} = Xn-i, each n > 1. Find Xn where \Xn\dP is not сг-finite on
Tn-i, any n, and let Rn = 2n.)
*Exercise 35. Let R be a time change, with s »-> Rs continuous, strictly
increasing, Ro = 0, and Rt < oo, each t > 0. Show that for a continuous
semimartingale X,
rRt rt
/ HsdXs = / HRudXRu
Jo Jo
for bounded Яб1.
*Exercise 36. Let R and X be as in Exercise 35. No longer assume that
Rt < oo a.s., each t > 0, but instead assume that X is a finite variation
process. Let At — infjs > 0 : Rs > t}.
(a) Show that R strictly increasing implies that A is continuous.
(b) Show that R continuous implies that A is strictly increasing.
(c) Show that for general R, RAt > t, and if R is strictly increasing and
continuous then R^t = t.
(d) Show that for bounded ffeLwe have
Rt rt rtAAoo
HsdXs = / l{R.<O0)HR.dXR. = / HR.dXR..
o Jo Jo
(e) Show that for bounded H e L we have
- Г
'Jo
See in this regard Lebesgue 's change of time formula, given in Theorem 45 of
Chap. IV.
*Exercise 37. Let R be a change of time and let G be the filtration given by
Qt = TRt. Let At = infjs > 0 : Rs > t}. Show that A = (At)t>o is a change of
time for the filtration G. Show also that iit—*Rt is continuous a.s., Ro = 0,
and Roo = oo, then RAt = t a.s., t > 0.
*Exercise 38. Let A, G, be as in Exercise 37 and suppose that RAt = t a.s.,
t > 0. Show that QM С Tt, each t>0.
*Exercise 39. A function is Holder continuous of order a if \f(x) — f(y)\ <
K\x—y\a. Show that the paths of a standard Brownian motion are a.s. nowhere
locally Holder continuous of order a for any a > 1/2. (Hint: Use the fact that

ш
Semimartingales and Decomposable Processes
1 Introduction
In Chap. II we defined a semimartingale as a good integrator and we developed
a theory of stochastic integration for integrands in L, the space of adapted
processes with left continuous, right-limited paths. Such a space of integrands
suffices to establish a change of variables formula (or "Ito's formula"), and it
also suffices for many applications, such as the study of stochastic differential
equations. Nevertheless the space L is not general enough for the consider-
consideration of such important topics as local times and martingale representation
theorems. We need a space of integrands analogous to measurable functions
in the theory of Lebesgue integration. Thus defining an integral as a limit of
sums—which requires a degree of smoothness on the sample paths—is inade-
inadequate. In this chapter we lay the groundwork necessary for an extension of our
space of integrands, and the stochastic integral is then extended in Chap. IV.
Historically the stochastic integral was first proposed for Brownian motion,
then for continuous martingales, then for square integrable martingales, and
finally for processes which can be written as the sum of a locally square
integrable local martingale and an adapted, cadlag processes with paths of
finite variation on compacts; that is, a decomposable process. Later Doleans-
Dade and Meyer [53] showed that the local square integrability hypothesis
could be removed, which led to the traditional definition of a semimartingale
(what we call a classical semimartingale). More formally, let us recall two
definitions from Chaps. I and II and then define classical semimartingales.
Definition. An adapted, cadlag process Л is a finite variation process
(FV) if almost surely the paths of A are of finite variation on each compact
interval of [0, сю). We write /0°° \dAs\ or \A\qo for the random variable which
is the total variation of the paths of A.
Definition. An adapted, cadlag process X is decomposable if there exist
processes N, A such that

102 III Semimartingales and Decomposable Processes
Xt = Xo + Nt + At
with iVo = A) = 0, N a locally square integrable local martingale, and A an
FV process.
Definition. An adapted, cadlag process У is a classical semimartingale
if there exist processes N, В with No — -Bo = 0 such that
Yt = Yo + Nt + Bt
where N is a local martingale and В is an FV process.
Clearly an FV process is decomposable, and both FV processes and de-
decomposable processes are semimartingales (Theorems 7 and 9 of Chap. II).
The goal of this chapter is to show that a process X is a classical semimartin-
semimartingale if and only if it is a semimartingale. To do this we have to develop a small
amount of "the general theory of processes." The key result is Theorem 25
which states that any local martingale M can be written
M = N + A
where N is a local martingale with bounded jumps (and hence locally square
integrable), and A is an FV process. An immediate consequence is that a
classical semimartingale is decomposable and hence a semimartingale by The-
Theorem 9 of Chap. II. The theorem of Bichteler and Dellacherie (Theorem 43)
gives the converse: a semimartingale is decomposable.
We summarize the results of this chapter, that are important to our treat-
treatment, in Theorems 1 and 2 which follow.
Theorem 1. Let X be an adapted, cadlag process. The following are equiva-
equivalent:
(i) X is a semimartingale;
(ii) X is decomposable;
(Hi) given /3 > 0, there exist M, A with Mq = Aq = 0, M a local martingale
with jumps bounded by C, A an FV process, such that Xt = Xo + Mt + At;
(iv) X is a classical semimartingale.
Definition. The predictable cr-algebra V on M+ x О is the smallest cr-
algebra making all processes in L measurable. We also let V (resp. bV) denote
the processes (resp. bounded processes) that are predictably measurable.
The next definition is not used in this chapter, except in the Exercises,
but it is natural to include it with the definition of the predictable cr-algebra.
Definition. The optional cr-algebra О on R+ x О is the smallest cr-algebra
making all cadlag, adapted processes measurable. We also let О (resp. ЪО)
denote the processes (resp. bounded processes) that are optional.

2 The Classification of Stopping Times 103
Theorem 2. Let X be a semimartingale. If X has a decomposition Xt =
Xo + Mt + At with M a local martingale and A a predictably measurable FV
process, Mo = Aq = 0, then such a decomposition is unique.
In Theorem 1, clearly (ii) or (iii) each imply (iv), and (iii) implies (ii),
and (ii) implies (i). That (iv) implies (iii) is an immediate consequence of
the Fundamental Theorem of Local Martingales (Theorem 25). While Theo-
Theorem 25 (and Theorems 3 and 22) is quite deep, nevertheless the heart of Theo-
Theorem 1 is the implication (i) implies (ii), essentially the theorem of K. Bichteler
and C. Dellacherie, which itself uses the Doob-Meyer decomposition theorem,
Rao's Theorem on quasimartingales, and the Girsanov-Meyer Theorem on
changes of probability laws. Theorem 2 is essentially Theorem 30.
We have tried to present this succession of deep theorems in the most
direct and elementary manner possible. In the first edition we were of the
opinion that Meyer's original use of natural processes was simpler than the
now universally accepted use of predictability. However, since the first edition,
R. Bass has published an elementary proof of the key Doob-Meyer decompo-
decomposition theorem which makes such an approach truly obsolete. We are pleased
to use Bass' approach here; see [11].
2 The Classification of Stopping Times
We begin by denning three types of stopping times. The important ones are
predictable times and totally inaccessible times.
Definition. A stopping time T is predictable if there exists a sequence of
stopping times (Tn)n>! such that Tn is increasing, Tn < T on {T > 0}, all n,
and limn^oo Tn = T a.s. Such a sequence (Tn) is said to announce T.
If X is a continuous, adapted process with Xo = 0, and T = ini{t : \Xt\ >
c}, for some с > 0, then T is predictable. Indeed, the sequence Tn = ini{t :
\Xt\ >c—^}Anisan announcing sequence. Fixed times are also predictable.
Definition. A stopping time T is accessible if there exists a sequence
(Tk)k>i of predictable times such that
oo
: Tk(u>) = T(w) < oo}) = P{T < oo).
Such a sequence (Tk)k>i is said to envelop T.
Any stopping time that takes on a countable number of values is clearly ac-
accessible. The first jump time of a Poisson process is not an accessible stopping
time (indeed, any jump time of a Levy process is not accessible).
Definition. A stopping time T is totally inaccessible if for every pre-
predictable stopping time S,

104 III Semimaxtingales and Decomposable Processes
P{w : Т{ш) = S{uj) < 00} = 0.
Let T be a stopping time and Л е Тт- We define
if шеЛ,
It is simple to check that since Л е Тт, Т\ is a stopping time. Note further
that T = хпт(Тл,ТЛо) =ТЛЛ ТЛс.
A simple but useful concept is that of the graph of a stopping time.
Definition. Let Г be a stopping time. The graph of the stopping time
T is the subset of R+ x О given by {(t,u:) : 0 < t = T(w) < 00}; the graph
of T is denoted by [T].
Theorem 3. Let T be a stopping time. There exist disjoint events А, В such
that A Li В = {Т < сю} a.s., Та is accessible and Тв is totally inaccessible,
and T = Та Л Тв a.s. Such a decomposition is a.s. unique.
Proof. If T is totally inaccessible there is nothing to show. So without loss of
generality we assume it is not. We proceed with an inductive construction:
Let R±=T and take
c*i = sup{PE = Ri < 00) : Sis predictable}.
Choose Si predictable such that P(Si = R\ < 00) > ^- and set Vx = S±.
Define R2 = Ri^y^Ri}- F°r the inductive step let
a, = sup{PE = Ri < сю) : S is predictable}
If a, — 0 we stop the induction. Otherwise, choose Si predictable such that
P(Si = Ri < сю) > ^ and set Vi = Si^s. ^oes not equai any of vjti<j<i-i}-
(
^ q y j
Let Ri+i = Ri^Vi^Ri}- The graphs of the Vi are disjoint by construction (the
finite valued parts, of course). Moreover the sets {Ri+i Ф Ri, Ri < сю} are
disjoint. Thus E ^ < T,p(Ri+i Ф Ri, Ri < 00) < 1, which implies that
ai —> 0. The Ri form a non-decreasing sequence, so set U = lim^oo Ri.
If U is not totally inaccessible, there exists a predictable time W such that
P(W — U) > 0 and hence P(W — U) > c^ for some i. This contradicts how
we chose Si at step i of the induction. Therefore U is totally inaccessible, and
В = {U = T < сю}. ?
A beautiful application of Theorem 3 is Meyer's Theorem on the jumps
of Markov processes, a special case of which we give here without proof. (We
use the notation established in Chap. I, Sect. 5 and write FM = (.T^tKt^oo-)
Theorem 4 (Meyer's Theorem). LetX be a (strong) Markov Feller process
for the probability PM, where the distribution of Xq is given by fx, and with its
natural completed filtration FM. Let T be a stopping time with P^(T > 0) — 1.
Let A = {lj : Xt{oj) ф ХТ-(ш) and T{uj) < сю}. Then Т = ТлЛТЛс, where
T/i is totally inaccessible and Тдс is predictable.

3 The Doob-Меуег Decompositions 105
A consequence of Meyer's Theorem is that the jump times of a Poisson
process (or more generally a Levy process) are all totally inaccessible.
We will need a small refinement of the concept of a stopping time cr-algebra.
This is particularly important when the stopping time is predictable.
Definition. Let Г be a stopping time. The ст-algebra Тт- is the smallest
cr-algebra containing To and all sets of the form А Л {t < T}, t > 0 and
ieft.
Observe that Тт- С TT, and also the stopping time T is Тт- measurable.
We also have the following elementary but intuitive result. We leave the proof
to the reader.
Theorem 5. Let T be a predictable stopping time and let (Tn)n>i be an
announcing sequence for T. Then Тт- = c{Un>i ^тп} = Vn>i ^тп ¦
3 The Doob-Meyer Decompositions
We begin with a definition. Let N denote the natural numbers.
Definition. An adapted, cadlag process X is a potential if it is a non-
negative supermartingale such that lim^oo E{Xt} = 0. A process {Хп)пе^
is also called a potential if it is a nonnegative supermartingale for N and
limn^oo E{Xn} = 0.
Theorem 6 (Doob Decomposition). A potential (Хп)пе® has a decom-
decomposition Xn = Mn — An, where An+\ > An a.s., A$ — 0, An e J-n-i, and
Mn = ^{^ool^n}. Such a decomposition is unique-
Proof. Let Mo = Xo and Ao = 0. Define Mx = Mo + {X± - ?{Xi|JF0}), and
Ai = Xo - E{X1\Tq). Define Mn, An inductively as follows:
Mn = Mn-! + {Xn -
An = An_! + (Xn_! - E{Xn\Tn-i}).
Note that E{An] = E{X0} - E{Xn} < E{X0} < oo, as is easily checked by
induction. It is then simple to check that Mn and An so defined satisfy the
hypotheses.
Next suppose Xn = Nn — Bn is another such representation. Then Mn —
Nn = An - Bn and in particular Mx - N± = Ax - Вг ef0- Thus Mx - Nx =
E{M± - Nil^o} = Mo - No = Xo - Xo = 0, hence Mx = Nx. Continuing
inductively shows Mn = Nn, all п. П
We wish to extend Theorem 6 to continuous time supermartingales. Note
the unusual measurability condition that An e J-n-i which of course is
stronger than simply being adapted. The continuous time analog is that the
process A be natural or what turns out to be equivalent, predictably measur-
measurable.

106 III Semimartingales and Decomposable Processes
Throughout this paragraph we assume given a filtered probability space
(fi, J-, F, P) satisfying the usual hypotheses. Before we state the first decom-
decomposition theorem in continuous time, we establish a simple lemma.
Lemma. Let (Yn)n>i be a sequence of random variables converging to 0 in
L2. Then 8uptE{Yn\ft} -* 0 in 1?.
Proof. Let M" = E{Yn\Tt} which is of course a martingale for each fixed n.
Using Doob's maximal quadratic inequality, I?{supt(M"J} < 4jB{(M^,J} =
4?{Yn2} -> 0. D
Definition. We will say a cadlag supermartingale Z with Z$ — 0 is of
Class D if the collection {Zt ¦ T a finite valued stopping time} is uniformly
integrable.
The name "Class D" was given by P. A. Meyer in 1963. Presumably he
expected it to come to be known as "Doob Class" at some point, but it has
stayed Class D for 40 years now, so we see no point in changing it. (There are
no Classes A, B, and C.) We now come to our first version of the Doob-Meyer
decomposition theorem. It is this theorem that gives the fundamental basis
for the theory of stochastic integration.
Theorem 7 (Doob-Meyer Decomposition: Case of Totally Inacces-
Inaccessible Jumps). Let Z be a cadlag supermartingale with Z$ = 0 of Class D,
and such that all jumps of Z occur at totally inaccessible stopping times. Then
there exists a unique, increasing, continuous, adapted process A with Aq = 0
such that Mt = Zt + At is a uniformly integrable martingale.
We first give the proof of uniqueness which is easy. For existence, we will
first establish three lemmas.
Proof of uniqueness. Let Z — M - A and Z = TV — С be two decompositions
of Z. Subtraction yields M - N — A - C, which implies that M — TV is a
continuous martingale with paths of finite variation. We know however by the
Corollary of Theorem 27 of Chap. II that M —TV is then a constant martingale
which implies Mt - Nt = Mo - TV0 = 0 - 0 = 0 for alH. Thus M — TV and
A = C. D
Lemma 1. Let F be a discrete time filtration and let С be a non-decreasing
process with Co = 0, and С к G J-k-i- Suppose there exists a constant TV > 0
such that E{Coo - Ck\fk} < TV a.s. for all k. Then E{C^} < 2TV2.
Proof of Lemma 1. First observe -EjCoo} = E{E{Coo - С0\Т0}} < TV. Let-
Letting Cfe = Cfc+i — Cfe > 0, we obtain by rearranging terms:
k>0 k>0
Thus it follows that

3 The Doob-Meyer Decompositions 107
- Ck\Tk}ck} < 2NE{^ck} < 2NE{C00}
fe>0 fe>0
< 2N2. П
Choose and fix a constant v G Z+ and let Dn = {k2~n : 0 < k2~n < v).
Lemma 2. Let T be a totally inaccessible stopping time. For 5 > 0, let
RE) = suPt<,, P(t < T < t + 6\Ft). Then RF) -> 0 in probability as 6 -> 0.
Proof of Lemma 2. Let о > 0 and Sn(S) = inft{i e Dn : P(t < T < t +
5\Tt) > а} Аи. First we assume Sn(S) is less than T. Since Sn(E) is countably
valued, it is accessible, and since T is totally inaccessible, P(SnE) = T) = 0.
Suppose Г с {Т <t}, and also Г е Tt. Then
?{?{l{t<T<t+5}|jrt}lr} = E{l{t<T<t+s]lr} = 0.
Suppose now P(T < Sn(S)) > 0. Then for some t e Dn, P(T < t, SnF) =
t) > 0. Let Г = {T < t, Sn(S) = t}. Then from the definition of SnF) we
obtain
E{l(t<T<t+s)\ft}lr >alr, (*)
a contradiction. Thus we conclude SnF) < T a.s.
Next we define a stopping time S as follows. Let S(S) = infn Sn(S) and
S = suPn S(?). Fix n, so that S(i) = inffe Sfc(?). Thus on {5 = T}, S(±) <
S. Hence since S = supn5(^), we have S is accessible on {S — T}, which
implies that T is accessible on {S = T}, which in turn implies P(S = T) = 0.
Consequently E{1^s=t}\^s-} — 0 a.s.
Suppose now that the result is not true. That is suppose there exists an
e > 0 such that P(R(S) > a) > e, for all 6 tending to 0. Let /3 > 0 and S </3.
For n sufficiently large
p(E{^{sn(s)<T<sn(S)+s}\^snE)} > a) > e.
We also have P(T — SF)+6) = 0, since if not SF)+6-^ could announce part
of T which would contradict that T is totally inaccessible. Thus P({Sn{S) <
T < Sn(S)+S} A {S(S) <T< SF)+6}) -> 0 as n -л оо where the symbol Д
denotes the symmetric difference set operation. Since the symmetric difference
tends to 0, if P(Sn(S) < T < Sn(S) + 6\Т3п{8) > a) > e for any 6 > 0, then
we must also have
since otherwise were it to tend to 0 along with the symmetric difference tend-
tending to 0, then we would have (*) —> 0, a contradiction. Thus we have P(SE) <
T < S(S) + ftfsiS)) > a on a set Л, with P(A) > e. From the definition of S
and working on the set Л, this implies P(E{l{S<T<s+p}\^s-} > a) > e. Let
C —> 0 to obtain P(T = S\Fs-) > a, and we have a contradiction. ?

108 III Semimartingales and Decomposable Processes
Lemma 3. Let the hypotheses of Theorem 7 hold and also assume \Z\ < N
where N > 0 is a constant, and further that the paths of Z are constant after
a stopping time v. Let W(S) - snpt<u<t+s E{Zt - Zu\Ft}. Then W{8) -> 0
in L2 as 5 -> 0.
Proof of Lemma 3. Since \Z\ < N we have |W(<5)| < 2N. Thus it suffices
to show W(S) —* 0 in probability. Let e, a > 0 and b = a^fe. Define
and W\S) = suPt<u<t+, \E{Zb - z\\Tt\V with W+(S) and W~(S) defined
analogously. Then
sup \Zb-Zbs\\Tt],
which by Doob's inequality implies
P(Wb(S) > a) < \E{ sup \Zb-Zb\2}.
Я r<«<r+«
However since Zb is cadlag and ]AZb| < b, Fatou's Lemma implies
limsup \e{ sup \Zb - Zb\2} <%=e.
s->o a" r<s<r+s a
Also, again using that Z is cadlag, for each fixed w the number of jumps larger
than b is finite. Define inductively
Ti = inf{i > 0 : AZt > b}, Ti+1 = inf{i > Tt : AZt > b}.
Since \Z\ < N be hypothesis, \AZTi\ < 2N. Choose к such that P(Tk < v) <
e. Then
P(W+(S) >a)< P(Tk <
г=1
By the previous lemma we know that R(S) —> 0 in L2, so by taking E small
enough, we get the above expression less than 2e. The reasoning for W~ is
analogous. We achieve W{5) < WbE)+ W+F) +W~F), which gives W{5) ->
0 in L2. D
We return to the proof of Theorem 7.
Proof of existence. First suppose the jumps of Z are bounded by a constant
с Let TN = inf{* > 0 : |Zt| > ЛГ - с} Л ЛГ, and Z? = ZMTjv. Then |Ztw| <

3 The Doob-Meyer Decompositions 109
\ZtL | + с < N, and ZN is constant after T^. Thus we are now reduced to the
case considered by the Lemma 2.
Fix n and let F? = T к . Define
All the a? > 0 since Z is a supermartingale, and also a? s F^-x- Let A%!
Z^=i aj • Then Lk = Z_fc_ + Л^ is an Tfc discrete time martingale. Define
We want to show that ?{supt |Bf — BJ™|2} —> 0 as n, m —> oo. Suppose m>n.
Since Bm and Bn are constant on intervals of the form (k2~m, (k + lJ"m],
the sup of the difference occurs at some k2~m.
Fix t and let и = inf{s : s € Dn and s > t}. Observe that
AT\?T} = ^{^t - Zoolft} is bounded by 2N. Also
= E{E{ZU -
which is bounded by 2N. This implies
with the right side being nonnegative and bounded by WB~n), and of course
also by E{W{2~n)\Tt}. Using Lemmas 1 and 2 we get that ?{supt \Bf -
¦S*"|2} -+ 0 as n, m —> oo. Therefore the sequence B™ is a Cauchy sequence
and we denote its limits by At.
Next we show that the process A has continuous paths. We do this by
analyzing the jumps of the approximating processes Bn. Note that for t =
k2~n,
which are bounded by WB"n). But then supt |AB"| -> 0 in L2. Thus there
exists a subsequence (rij) such that sup4 |AB1 —> 0 a.s., and hence the limit
Л is continuous.
It remains to show that Zt +At is a uniformly integrable martingale. Both
are square integrable, so we only need concern ourselves with the martingale
property. Let s,t e Dn, s<t, and Л e Ts. Then E{(Zt + At)lA} = E{(ZS +
А3Iл}- The result follows by taking limits of Zt+B™.
At the beginning of the proof we made the simplifying assumption that
Z had bounded jumps. We now treat the general case. Choose N and let
T = inf{* > 0 : \Zt\ > N} A N. Then ZT (that is, Z stopped at the time T)
has at most one jump bigger than 2N. By localization and the uniqueness of
the decomposition, it suffices to prove the result for this case. Thus without

110 III Semimartingales and Decomposable Processes
loss of generality we assume Z has at most one jump greater than or equal
to 2N in absolute value, that it occurs at the time T, and that Z is constant
after the time T. We let
s<t
s<t
Since — Z+ and Z~ both have decreasing paths, they are both supermartin-
gales. Suppose we can show the theorem is true for — Z+ and Z~. Let — Z+ —
M+ — A+ and Z~ = M~ — A~ be the two decompositions, with A+ and A~
both continuous. Then Z = Z + (-Z+ + A+) ~(Z~ +A~) = Z + M+-M~ is
a supermartingale with jumps bounded by 2N. Let Z = M - A be the unique
decomposition of Z which we now know exists. Then Z = Z + M+ — M~,
and therefore Z = (M + M+ — M~) — A is the desired (unique) Doob-Meyer
decomposition.
Thus it remains to show that — Z+ and Z~ both have Doob-Meyer de-
decompositions. First observe that \AZT\ < \%т-\ + \Zt\ < N + \ZT\ e L1, and
hence ^{|AZr|} < oo. Choose a,e > 0. Next choose R > N so large that
.E{|Ai?r|l{|A.zT|>fl}} — ?a- The cases for when the jump is positive and when
the jump is negative being exactly the same except for a minus sign, we only
treat the case where the jump is negative. Let Z^ = AZTl{t>T}l{-AZT>R}i
and Zf = Zt - Zf. We define Bn~,BnR,Bnd analogously to the way we
defined Bn in equation (*) above.
We first show 5"~ converges uniformly in t in probability. We have:
-йГ~1>а) (**)
t
< P(sup \Bf - B?d\ > ?) + P(sup |BfR| > ^) + P(sup \B?R\ > ^)
The second term on the right side of (**) above is less than 3e:
P(sup\B?R\ > ?) = P(B^ > \) < *E{B?}
t о о а
< ?-E{\ZR\} < ^-E{\AZT\l{lAZTl>R}}
< 3e.
The third term on the right side of (**) is shown to be less than 3e similarly.
Since \ZR\ is bounded by R, the first term on the right side of (**) can be
made arbitrarily small by taking m and n large enough, analogous to what
we did at the beginning of the proof for Bn.
Thus as we did before in this proof, we can conclude that B"+ converges
uniformly in t in probability as n —> oo, and we denote its limit process by Af.

3 The Doob-Meyer Decompositions 111
We prove continuity of A exactly as before. Finally by taking a subsequence
tij such that Bnj+ converges almost surely, we get
+ E{ lim B?+} +
{
Tlj
by Fatou's Lemma. From this it easily follows that Z+ + A+ is a uniformly
integrable martingale, and the proof is complete. ?
While Theorem 7 is sufficient for most applications, the restriction to su-
permartingales having jumps only at totally inaccessible stopping times can
be insufficient for some needs. When we move to the general case we no
longer have that A is continuous in general (however see Exercises 24 and 26
for supplemental hypotheses that assure that the increasing process A of the
Doob-Meyer decomposition is in fact continuous). Without the continuity we
lose the uniqueness of the decomposition, since there exist many martingales
of finite variation (for example, the compensated Poisson process) that we can
add to the martingale term and subtract from the finite variation term of a
given decomposition, to obtain a second, new decomposition. Instead of the
continuity of A we add the condition that A be predictably measurable.
Theorem 8 (Doob-Meyer Decomposition: General Case). Let Z be a
cadlag supermartingale with Zo = 0 of Class D. Then there exists a unique,
increasing, predictable process A with Aq = 0 such that Mt = Zt + At is a
uniformly integrable martingale.
Before proving Theorem 8 let us introduce the concept of a natural process.
We introduce two definitions and prove the important properties of natural
processes in the next three theorems.
Definition. An FV process A with Aq = 0 is of integrable variation if the
expected total variation is finite: E{fo°° \dAs\} < oo. A shorthand notation for
this is .Ed^loo} < oo. An FV process A is of locally integrable variation
if there exists a sequence of stopping times (T")n>i increasing to oo a.s. such
that E{Jq" \dAs\} = E{\A\Tn} < oo, for each n.
Definition. Let A be an (adapted) FV process, Aq = 0, of integrable varia-
variation. Then Л is a natural process if
for all bounded martingales M.
Here is the key theorem about natural processes. This use of natural pro-
processes was Meyer's original insight that allowed him to prove Doob's conjec-
conjecture, which became the Doob-Meyer decomposition theorem.
Theorem 9. Let A be an FV process, Aq = 0, and Е{\А\Ж} < oo. Then A
is natural if and only if

112 III Semimartingales and Decomposable Processes
/•OO
( f
Me-dAt\ =
/0
for any bounded martingale M.
Proof. By integration by parts we have
/•OO /-00
/ Ms-dAs = MqoAx, - MqAq — / As_dMs — [M, A]oo-
Jo Jo
Then Mo A) = 0 and letting Nt = /0 As-dMs, we know that N is a local
martingale (Theorem 20 of Chap. II). However using integration by parts we
see that JSfiV^} < oo, hence N is a true martingale (Theorem 51 of Chap. I).
Therefore E {/0°° As_dMs} = ElN^,} - E{N0} =0, since N is a martingale,
so that the equality holds if and only if E{\M, Ajoo} = 0. ?
Theorem 10. Let A be an FV process of integrable variation which is natural.
If A is a martingale then A is identically zero.
Proof. Let T be a finite stopping time and let H be any bounded, nonnegative
rp
martingale. Then E{JQ Hs^dAs} = 0, as is easily seen by approximating
sums and the Dominated Convergence Theorem, since /0 |cL4,,| e L1 and
E{AT} = 0. Using the naturality of A, E{HTAT} = E{f* Hs_dAs} = 0,
and letting Ht = Е{1^т>о^\Тг} then shows that Р(Ат > 0) = 0. Since
E{AT} = 0, we conclude AT = 0 a.s., hence A = 0. ?
Theorem 11. Let A be an FV process of integrable variation with AQ = 0.
// A is predictable, then A is natural.
The proof of this theorem is quite intuitive and natural provided we accept
a result from Chap. IV. (We do not need this theorem to prove the theorem
we are using from Chap. IV.)
Proof. Let M be a bounded martingale. First assume A is bounded. Then
the stochastic integral fQ AsdMs exists, and it is a martingale by Theo-
Theorem 29 in Chap. IV combined with, for example, Corollary 3 of Theorem 27
of Chap. II. Therefore E{f?° AsdMs} = E{A0M0} = 0, since Ao = 0. How-
However -Б{/0°° As_dMs} = E{Ao_Mo} = 0 as well, since A0- = 0. Further,
/»oo /»oo /»oo
/ AsdMs- Aa-dMa = (As-As_)dMs
Jo Jo Jo
= f
AAsdMs
]T AASAMS
0<«<oo

3 The Doob-Meyer Decompositions 113
since Л is a quadratic pure jump semimartingale. Therefore
/.OO ;-OO
E{[A,M]oo} = E{ AsdMs- As_dMs} = 0.
Jo Jo
Since M was an arbitrary bounded martingale, A is natural by definition.
Finally we remove the assumption that A is bounded. Let An = nA(AV(-n)).
Then An is bounded and still predictable, hence it is natural. For a bounded
martingale M
E{[M,AU) = limSflM,^"],»} = О,
n
by the Dominated Convergence Theorem. Therefore A is natural. ?
The next theorem is now obvious and should be called a Corollary at best.
Because of its importance, however, we give it the status of a theorem.
Theorem 12. Let M be a local martingale with paths of finite variation on
compact time sets. If M is predictably measurable, then M is constant. That
is, Mt = Mo for all t, almost surely.
Proof. This theorem is a combination of Theorems 10 and 11. ?
Proof of Theorem 8 (Doob-Meyer: General Case). We begin with the unique-
uniqueness. Suppose Z = M - A and Z = N - С are two decompositions. By sub-
subtraction we have M — N = А — С is a martingale with paths of finite variation
which is predictable. By Theorem 12 we have that M — N is constant and
since Mo — No = 0, it is identically zero. This gives the uniqueness.
The existence of the decomposition is harder. We begin by defining stop-
stopping times Tnj, where Tnj is the j-th time \AZt\ is in the half-open interval
bounded by 2~™ and 2~(n~l\ where -oo < n < oo. We decompose Tnj
into its accessible and totally inaccessible parts. Since we can cover the ac-
accessible part with a sequence of predictable times, we can thus separate each
Tnj into a totally inaccessible time and a sequence of predictable times, with
disjoint (stopping time) graphs. Therefore, by renumbering, we can obtain a
sequence of stopping times Ej)j>i with disjoint graphs, each one of which is
either predictable or totally inaccessible, which exhaust the jumps of Z (in
the sense that the jump times of Z are contained in the union of the graphs
of the Si) and are such that for each i there exists a constant 6, such that
bi < \AZSi\ <2bi.
We define Zo(t) = Zt and inductively define Zi+\{t) = Zi{t) and Ai{t) =
0 if Si is totally inaccessible; whereas Ai(t) = — ElAZsil^Si-^iSi^} and
then Zi+i(t) = Zi(t) + Ai(t) in the case where Si is predictable (note that
AZSi G Ll).
We will show that: each At is increasing; Z± is a supermartingale for each
i; and -E{]>Zj=i A,-(oo)} < C, for each i, where С is a constant not depending
on i. We will prove these three properties by induction. We begin by showing
that each At is increasing. This is trivial if Si is totally inaccessible. Let then

114 III Semimartingales and Decomposable Processes
Si be a predictable time. Let S™ be an announcing sequence for Sj. Since each
Zi is a supermartingale, and using the Martingale Convergence Theorem, we
have EiAZ^SJlfs,-} = \\m.nE{AZi{Si)\Ts?}- Now fix n:
and thus Ai is increasing. To see that Zi+\ is a supermartingale, it will suffice
to show that whenever XJ\ and U2 are stopping times with Ui < U2 then
E{Zi+i(Ui)} > E{Zi+i(U2)}. Again letting SJ1 be an announcing sequence
for Si we have
E{Zi(U!)} - E{Zi(U2)} = E{Zi(U{)} - EiZidUi V S?) Л U2)}
V Я?) л U2)} - E{Zi((U! V SO Л U2)}
V SO Л U2)} - E{Zi(U2)}
where each of the summands on the right side above are nonnegative. Then
let n -> 00 to get E{Zi(Ui)} - E{Zi(U2)} > 0. Finally we want to show that
E{Y^l-i Aj(oo)} < C. It is easy to check (see Exercise 1 for example) that
the minimum of predictable stopping times is predictable, hence we can order
the predictable times so that Si < S2 < ¦ ¦ ¦ on the set where they are all
finite. Let Ai be the collection of all j such that j < г and such that the Sj
are all predictable. Then
AZSj} = lira ]T E{ZS) - ZSj}
Ai
+E{ZSt -Z00}+ E{Z0 - ZSk})
= E{Z0-Zoo},
which is bounded by a constant not depending on i.
To complete the proof, we note that because the processes Ai are in-
increasing in t and the expectation of their sum is bounded independently of
г, we have that X)»=i-^*@ converges uniformly in t a.s., as h —> oo, and
we call the limit Aoo(t). Fatou's Lemma gives us that Aoo(t) is integrable.
Each Ai(t) is easily seen to be predictable, and hence A^ is the almost sure
limit of predictable measurable processes, so it too is predictable. Next set
Z(x>(t) = Zt + Aoo(t) = ]haiZi(t). Since each Z+ is a supermartingale, cou-
coupled with the uniform convergence in t of the partial sums of the A+ pro-
processes, we get that Дх>(?) is a supermartingale. Moreover since each Zi is
cadlag, again using the uniform convergence of the partial sums we obtain

3 The Doob-Meyer Decompositions 115
that Zqo is also cadlag. Since the partial sums are uniformly bounded in ex-
expectation, Zqo (t) is of Class D. Finally by our construction of the stopping
times Si, -E{AZoo(T)|:Fr_} = 0 for all predictable times T. We can then
show as before that WF) —> 0 in probability as 6 —> 0 for Z^, and we ob-
obtain Zooty) = Mt - A(t). The process At — A»(?) + A(t) is then the desired
increasing predictable process. ?
The next theorem can also be considered a Doob-Meyer decomposition
theorem. It exchanges the uniform integrability for a weakening of the con-
conclusion that M be a martingale to that of M being a local martingale.
Theorem 13 (Doob-Meyer Decomposition: Case Without Class D).
Let Z be a cadlag supermartingale. Then Z has a decomposition Z — Z§ +
M — A where M is a local martingale and A is an increasing process which is
predictable, and Mq = Aq =0. Such a decomposition is unique. Moreover if
limt^oo E{Zt} > —oo, then ElA^} < oo.
Proof. First consider uniqueness. Let Z = Zq + M — A and Z = Zq + N — С
be two decompositions. Then M — N = A — С by subtraction. Hence A — С
is a local martingale. Let MT" be a uniformly integrable martingale. Then
E{ZQ - ZtAT»} = E{-Mf} + E{AT} = E{Af}
and therefore E{Aj"} < E{Z0 - Zt}, using Theorem 17 of Chap. I. Letting
n tend to oo yields E{At} < E{Zq - Zt}. Thus A is integrable on [0, to], each
to, as is С. Therefore A — С is of locally integrable variation, predictable,
and a local martingale. Since Aq — Cq = 0, by localization and Theorems 10
and 11, A — С = 0. That is, A = C, and hence M = N as well and we have
uniqueness.
Next we turn to existence. Let Tm = mi{t : \Zt\ > m} A m. Then Tm in-
increase to oo a.s. and since they are bounded stopping times Zt™ € L1 each m
(Theorem 17 of Chap. I). Moreover the stopped process ZTm is dominated by
the integrable random variable max(|Zrm|, m). Hence if X = Z7"™ for fixed m,
then H = {Xt',T a stopping time} is uniformly integrable. Let us implicitly
stop Z at the stopping time Tm and for n > 0 define Ytn = Zt - E{Zn\Tt},
with Ytn = l^"n = 0 when t > n. Then Yn is a positive supermartingale
of Class D and hence Ytn = Yon + Mt" - A?. Letting iVtn = E{Zn\Ft}, a
martingale, we have on [0, n] that
Zt=Z0+M?+N?-A?.
To conclude, therefore, it suffices to show that A™ — A™ on [0, n], for m > n.
This is a consequence of the uniqueness already established. The uniqueness
also allows us to remove the assumption that Z is stopped at the time Tm.
Finally, note that since E{At} < E{Z0 — Zt}, and since A is increasing,
we have by the Monotone Convergence Theorem

116 III Semimartingales and Decomposable Processes
EiA^} = lim E{At} < lim E{Z0 - Zt}
t—ЮО t—>OO
which is finite if lim^oo E{Zt} > —oo. П
4 Quasimartingales
Let X be a cadlag, adapted process defined on [0,оо].х
Definition. A finite tuple of points r = (t0, ti,..., tn+i) such that 0 = t0 <
ti < • • • < tn+i = oo is a partition of [0, oo].
Definition. Suppose that т is a partition of [0, oo] and that Xti G L1, each
U e т. Define
i—0
The variation of X along r is defined to be
Varr(X) - E{C(X,t)}.
The variation of X is defined to be
Var(X) =supVarr(X),
r
where the supremum is taken over all such partitions.
Definition. An adapted, cadlag process X is a quasimartingale on [0, oo]
if J5{|Xt|} < oo, for each t, and if Var(X) < oo.
Before stating the next theorem we recall the following notational conven-
convention. If X is a random variable, then
X+ =(,),
X~ =-min(X,0).
Also recall that by convention if X is defined only on [0, oo), we set Xoo = 0.
Theorem 14. Let X be a process indexed by [0, oo). Then X is a quasi-
quasimartingale if and only if X has a decomposition X = Y — Z where Y and Z
are each positive right continuous supermartingales.
1 It is convenient when discussing quasimartingales to include oo in the index set,
thus making it homeomorphic to [0, t] for 0 < t < oo. If a process X is defined
only on [0, oo) we extend it to [0, oo] by setting Хж = 0.

4 Quasimartingales 117
Proof. For given s > 0, let XXs) denote the set of finite subdivisions of [s, oo].
For each r G XXs )> se*
У; = Д{С(Х,т)+|.Гв} and ZJ = E{C{X,t)-\Fs)
where C(X, r)+ denotes X^e-r^i^i ~ -^ti+il^i}") and analogously for
C(.X", r)~. Also let -< denote the ordering of set containment. Suppose а, т G
XXs) with a < т. We claim Y/ < YST a.s. To see this let a = (t0,.. .,tn). It
suffices to consider what happens upon adding a subdivision point t before
?o> after tn, or between ti and ?i+i. The first two situations being clear, let us
consider the third. Set
A = E{Xti - Xt\Fu}; В = E{Xt - Xti+1\Ft};
С = E{Xti - Xti+1\fti};
then С = A + E{B\Fti}, hence
C+ < A+ + E{B\ft.}+
by Jensen's inequality. Therefore
and we conclude Y° < YJ. Since E{Y^} is bounded by VarpsT), taking limits
in L1 along the directed ordered set XXs) we define
Ys = limYsT,
and we can define Zs analogously. Taking a subdivision with to = s and
tn+i = oo, we see YJ - ZTS = SjC" - C~\TS} = Xs, and we deduce that
Ys — Zs = Xs. Moreover if s < t it is easily checked that Ys > E{Yt\Ts} and
Zs > E{Zt\Ts}. Define the right continuous processes Yt = Yt+, Zt = Zt+,
with the right limits taken through the rationals. Then Y and Z are positive
supermartingales and Ys — Zs = Xs.
For the converse, suppose X = Y — Z, where Y and Z are each positive
supermartingales. Then for a partition r of [0, t]
UET
E{YU - Yti+1\fu}} + E{J2 E{Zt
i} - E{Yu+l} + E{ZU} -
= E{Y0} + E{Z0} - (E{Yt} + E{Zt}).
Thus X is a quasimartingale on [0,t], each t > 0. ?

118 III Semimartingales and Decomposable Processes
Theorem 15 (Rao's Theorem). A quasimartingale X has a unique decom-
decomposition X = M + A, where M is a local martingale and A is a predictable
process with paths of locally integrable variation and Aq = 0.
Proof. This theorem is a combination of Theorems 13 and 14. ?
5 Compensators
Let Л be a process of locally integrable variation, hence a fortiori an FV
process. A is then locally a quasimartingale, and hence by Rao's Theorem
(Theorem 15), there exists a unique decomposition
A = M + A
where Л is a predictable FV process. In other words, there exists a unique,
predictable FV process A such that A — A is a local martingale.
Definition. Let A be an FV process with Ao = 0, with locally integrable
total variation. The unique FV predictable process A such that A - A is a
local martingale is called the compensator of A.
The most common examples are when A is an increasing process locally
of integrable variation. When A is an increasing process it is of course a
submartingale, and thus by the Doob-Meyer Theorem we know that its com-
compensator A is also increasing. We also make the obvious observation that
E{At} = E{At} for gi\t,O<t< oo.
Theorem 16. Let A be an increasing process of integrable variation, and let
H eh be such that E{f* HsdAs} < oo. Then,
E{ HsdAs} = E{ HsdAs}.
Jo Jo
Proof. Since A - A is a martingale, so also is / Hsd(As — As), and hence it
has constant expectation equal to 0. ?
In Chap. IV we develop stochastic integration for integrands which are
predictable, and Theorem 16 extends with H G L replaced with H predictable.
One of the simplest examples to consider as an illustration is that of the
Poisson process N = (Nt)t>o with parameter A. Recall that Nt — \t is a
martingale. Since the process At = \t is continuous and obviously adapted,
it is predictable (natural). Therefore Nt — Xt, t > 0. A natural extension of
the Poisson process case is that of counting processes without explosions. We
begin however with a counting process that has only one jump. Let r\ be the
counting process щ = 1{*>t}j where r is a nonnegative random variable. Let
F be the minimal filtration making r a stopping time.

5 Compensators 119
Theorem 17. Let P{r = 0) = 0 and Р(т > t) > 0, each t > 0. Then the F
compensator A off], where r\t = 1{*>т}> *s given fey
where F is the cumulative distribution function of т. If т has a diffuse
distribution (that is, if F is continuous), then A is continuous and At =
-F(t At)).
Before proving Theorem 17 we formalize an elementary result as a lemma.
Lemma. Let (п, Т, P) be a complete probability space. In addition, suppose
r is a positive T measurable random variable and J^° — а{т A t}, where
J*> = J*>° v Af, N are the null sets of T, and Tt = П«>г^5- Then F s°
constructed is the smallest filtration making r a stopping time. Let
Y 6 LX{T). Then
E{Y\Tt) = E{Y\r}l{t>_T} + Mr>p^
Proof. By the hypotheses on r the cr-algebra J^ is equal to the Borel u-
algebra on [0,t] together with the indivisible atom (t, oo). Observe that T^ =
<т{т Аи; и < t}, and the result follows easily. ?
Proof of Theorem 17. Fix t0 > 0 and let ?rn be a sequence of partitions of
[0,t0] with limn-™meshGrn) = 0. Define A? = ^ E{r]u+1 - т]и\Ти} for
0 < t < t0. Then
it X -n 1 , E{1{r>ti}Vti+1}л
by the lemma preceding this proof. The first term on the right above
is Vu+11{u>t} = l{ti+i>i-}l{ti>T} = !{4i>T}- Furthermore, 1{т>ц}Г]и+1 =
1{*i+i>-r} = 1{*i<r<4i+i}- Hence,
Therefore,
1 -тЛЪЛ = 1{^>г} + 1+^ три- \ Ъ
1 r УН)
These simple calculations yield

120 III Semimartingales and Decomposable Processes
+1 - *j*j = E F{ti+-
which are Riemann sums converging to /0 T(l — F(u—))~1dF(u). The re-
remaining claims of the theorem are obvious. D
Corollary 1. If F in Theorem 17 is absolutely continuous, then so also is A.
Corollary 2. If r in Theorem 17 has an exponential distribution, then At =
т At.
Proof. Since in this case F is continuous and a distribution function, we can
represent it as F(x) = 1 - е~ф{-х). We have that At = -ln(l - F(t A t))
and equivalently we can write At = ф(т At). In the case of this corollary, the
function ф(х) = x, and the result follows. ?
Remark. Theorem 17 gives an explicit formula for the compensator of a
process rjt = l{t>T} when one uses the minimal filtration making r a stopping
time. A perhaps more interesting question is the following: suppose that one
has a non-trivial filtration F satisfying the usual hypotheses, and a strictly
positive random variable L that is not a stopping time. Can one expand the
filtration to make a larger filtration G that renders laG stopping time, and
then calculate the G compensator of /ij = l{t>L}? This has a positive answer
with the right hypotheses. To see how to do this, let Xt = 1[o,l)W> which is
not an F adapted process. Let Z denote the optional projection of X onto F.
We prove in Chap. VI that Z is a supermartingale. Let Z = M—A be its Doob-
Meyer decomposition with A predictable. (Note that Mt = E{A00\Jrt}-) Using
the techniques presented in Chap. VI, one can show that if L is "the end of an
F predictable set," then the G compensator of/it = 1{*>L} is A-u which is an F
predictable process! By the end of a predictable set, we mean that there exists
an F predictable set Лс[0,оо]хП such that L(u>) = sup{t < oo : (t,w) е Л}.
In Chap. VI we discuss times that are the ends of optional sets, known as
"honest times," and this is of course less restrictive than being the end of a
predictable set, since any predictable set is also optional.2 See [115] for this
result described here and related ones.
Theorem 17 treats the case of a point process with one jump. A special
case of interest is counting processes. We have the following.
Theorem 18. Let N be a counting process without explosions, adapted to a
filtration G satisfying the usual hypotheses. Then the compensator of N, call
it A, always exists.
Proof Since N has non-decreasing paths, to ensure the existence of a com-
compensator we need only to show that N is locally of integrable variation. Let
Tn be the time of the n-th jump of N. Since N has no explosions, the times
See Exercise 4.

5 Compensators 121
Tn increase to oo a.s. Moreover \Nt^xn\ — ^tATn < n and thus N is locally of
bounded variation, hence certainly locally of integrable variation. ?
Let Nt = Yli>\ l{*>Ti} be a counting process without explosions and let
Si = Ti — Tj_i be its interarrival times, where we take To = 0. Let J^ =
<?{NS; s < tj, and let Tt = J*> V Af, the filtration completed by the .F-null
sets, as usual. (Note that this filtration is already right continuous.) Let us
define the cumulative distribution functions for the interarrival times slightly
informally as follows (note that we are using the unusual format of P(S > x)
rather than the customary P(S < x)):
> t)
Fi(slt..., *_i; t) = P(Si > t\S! = slt..., S^ = st-i)
where the Sj are in [0, oo], j > 1. (If one of the Sj takes the value +oo, then
the corresponding cumulative distribution function is concentrated at +oo.)
Define
i-i; t) =
f* -1
/ — rdFi(si,...,Si-i; s).
Jo *i{si,---,Si-i; s-)
We now have the tools to describe the compensator of N for its minimal
completed filtration. We omit the proof, since it is simply a more notationally
cumbersome version of the proof of Theorem 17.
Theorem 19. Let N be a counting process without explosions and let F be
its minimal completed filtration. Then the compensator A of N is given by
At = ^i(Si) + 02(Si; S2) + ¦¦¦+ &_i(Si,..., 5,_2; Si-i)
+ 0i(Si,-- • ,Si-i; t -T^
on the event {Ti <t< Ti+1}.
Corollary. Let N be a counting process without explosions and independent
interarrival times. Then the functions <j>i defined in equation (*) have the
simplified form
&(si,..., Si_i; t) =
and the compensator A is given by
-1
+фм - Ti)}i{Ti<t<Ti+1}
Example (hazard rates and censored data). In applications, an intuitive
interpretation often used is that of hazard rates. Let Г be a positive random

122 III Semimartingales and Decomposable Processes
variable with a continuous density for its distribution function F, given by /.
The hazard rate Л is defined, when it exists, to be
X(t) = lim -P(t <T <t + h\T>t).
The intuition is that this is the probability the event will happen in the next
infinitesimal unit of time, given that it has not yet happened. Another way
of viewing this is that P(t < T < t + h\T > t) = Xh + o(h). By Theorem 17
we have that the compensator of Nt = l{t>T] is At = /q \(s)ds. Or in
language closer to stochastic integration,
Л/\Т
Nt —I X(s)ds is a local martingale.
Jo
In medical trials and other applications, one is faced with situations where
one is studying random arrivals, but some of the data cannot be seen (for
example when patients disappear unexpectedly from clinical trials). This is
known as arrivals with censored data. A simple example is as follows. Let T
be a random time with a density and let U be another time, with an arbitrary
distribution. T is considered to be the arrival time and U is the censoring
time. Let X = Г Л U, Nt = l{t>x}l{c/>T}, and JV/7 = l{t>x}l{T>c/} with
J?t = a{Nu,N^; и <t}. Let Л be the hazard rate function for Т. Л is known
as the net hazard rate. We define A#, known as the crude hazard rate, by
X* = lim \-P(t <T<t + h\T>t,U>t)
h—>0 h
when the limit exists. We then have that the compensator of N is given by
At = /0 l{X>u}^(u)du = /o X#{u)du = /0 X#(u)du, or equivalently
rt
Nt — I l{x>u}^(u)du is a local martingale.
Jo
If we impose the condition that Л = А#, which can be intuitively interpreted
as
P(s < T < s + ds\T >s) = P(s<T<s + ds\T >s,U> s),
then we have the satisfying result that
r*
Nt — I l{x>u}A(u)<iu is a local martingale,
./o
As we saw in Chap. II, processes of fundamental importance to the the-
theory of stochastic integration are the quadratic variation processes [X, X] =
([X,X]t)t>o, where X is a semimartingale.
Definition. Let X be a semimartingale such that its quadratic variation
process [X, X] is locally integrable. Then the conditional quadratic vari-
variation of X, denoted (X, X) = ({X,X)t)t>o, exists and it is defined to be the
compensator of [X,X]. That is (X,X) = \X^X).

5 Compensators 123
If X is a continuous semimartingale then [X, X] is also continuous and
hence already predictable; thus [X, X] = (X, X) when X is continuous. In
particular for a standard Brownian motion B, [B, B)t = {B,B)t = t, all
t > 0. The conditional quadratic variation is also known in the literature by
its notation. It is sometimes called the sharp bracket, the angle bracket, or the
oblique bracket. It has properties analogous to that of the quadratic variation
processes. For example, if X and Y are two semimartingales such that (X, X),
(Y,Y), and (X + Y,X + Y) all exist, then (X,Y) exists and can be defined
by polarization
(X,Y) = ±((X + Y,X + Y) - (X,X) - (
However {X, Y) can be defined independently as the compensator of [X, Y]
provided of course that [X, Y] is locally of integrable variation. In other
words, there exist stopping times (Tn)n>i increasing to oo a.s. such that
E{JQ \d[X,Y]s\} < oo for each n. Also, (X,X) is a non-decreasing pro-
process by the preceding discussion, since [X, X] is non-decreasing. The condi-
conditional quadratic variation is inconvenient since unlike the quadratic variation
it doesn't always exist. Moreover while [X,X], [X,Y], and [Y,Y] all remain
invariant with a change to an equivalent probability measure, the sharp brack-
brackets in general change with a change to an equivalent probability measure and
may even no longer exist. Although the angle bracket is ubiquitous in the
literature it is sometimes unnecessary as one can often use the quadratic vari-
variation instead, and indeed whenever possible we use the quadratic variation
rather than the conditional quadratic variation (X, X) of a semimartingale X
in this book. Nevertheless the process (X, X) occurs naturally in extensions
of Girsanov's theorem for example, and it has become indispensable in many
areas of advanced analysis in the theory of stochastic processes.
We end this section with several useful observations that we formalize as
theorems and a corollary. Note that the second theorem below is a refinement
of the first (and see also Exercise 25).
Theorem 20. Let A be an increasing process of locally integrable variation,
all of whose jumps occur at totally inaccessible stopping times. Then its com-
compensator A is continuous.
Theorem 21. Let A be an increasing process of locally integrable variation,
and let T be a jump time of A which is totally inaccessible. Then its compen-
compensator A is continuous at T.
Proof. Both theorems are simple consequences of Theorem 7. D
Corollary. Let A be an increasing predictable process of locally integrable
variation and let Г be a stopping time. If Р(А? ф Ат~) > 0 then Г is not
totally inaccessible.

124 III Semimartingales and Decomposable Processes
Proof. Suppose T were totally inaccessible. Let A be the compensator of A.
Then A is continuous at Г. But since A is already predictable, A — A, and
we have a contradiction by Theorem 21. D
Theorem 22. Let T be a totally inaccessible stopping time. There exists a
martingale M with paths of finite variation and with exactly one jump, of size
one, occurring at time T (that is, Mf ф Мт- on {T < oo}j.
Proof Define
Ut — l{t>T}-
Then U is an increasing, bounded process of integrable variation, and we
let A — U be the compensator of U. A is continuous by Theorem 20, and
M = U — A is the required martingale. ?
6 The Fundamental Theorem of Local Martingales
We begin with two preliminary results.
Theorem 23 (Le Jan's Theorem). Let T be a stopping time and let H be
an integrable random variable such that Е{Н\Тт-} = 0 on {T < oo}. Then
the right continuous martingale Ht = E{H\Tt) is zero on [0, T) = {(t,ui) :
0<t< T(lj)}.
Proof. Since the martingale (Ht)t>o is right continuous it suffices to show that
Я41{4<Т} = 0 almost surely for all t. Let Л G Tt. Then An{t <T} belongs
both to Tt- and also to Tt- Hence
0 = E{HlAn{t<T}} = E{HtlAn{t<T}}.
Since Л e Tt is arbitrary, this implies that Я41{4<Т} = 0 a.s. If H is Tt
measurable then H\{T<t} G Tt, and E{H\Tt} = H on {T < t} and the
theorem is proved. ?
Theorem 24. Let T be a predictable stopping time and let A be increasing,
predictable, and locally of integrable variation. Then At and ААт are each
Tt- measurable.
Proof. Let Sn be a sequence of stopping times announcing T. Then Asn G
Tsn С Тт- for each n. Since At- = Шип-юс Asn we have that At- G
Tt-. To show that At G Tt- it suffices to show E{ATH} — 0 for every
bounded random variable H such that Е{Н\Тт-} = 0. Let H be such a
random variable and let Ht = E{H\Tt}, the right continuous martingale with
terminal value H. Then Ht{w) = 0 for 0 < t < T{uj) by Le Jan's Theorem
(Theorem 23). This implies that the left continuous version of Я, Ht~, equals
0 on [0, Г]. Then since the hypotheses imply that A is natural, we have
/•OO
E{HAT} = E{ Hs_l{s<T}dAs} + E{[H,A}T}= 0 + 0 = 0. D
Jo

6 The Fundamental Theorem of Local Martingales 125
Theorem 25 (Fundamental Theorem of Local Martingales). Let M
be a local martingale and let /3 > 0. Then there exist local martingales N,
D such that D is an FV process, the jumps of N are bounded by 2/3, and
M = N + D.
Proof. We first set
ct= Yl \Ams\1{\am3\>/3},
0<s<t
and we want to show С is locally integrable. By stopping we can assume
without loss that M is a uniformly integrable martingale and that Mo = 0.
For each w, Ct{u>) is a finite sum. Define
Rn - in({t :Ct>n or \Mt\ > n).
Then \AMRn\ < \MRn\ + |МН„_| < \MRn\ + n. Moreover CRn < C7Hn_ +
|ДМдп| < |Мн„| + 2n, which is in L1. Since Rn increases to oo, we have С is
locally integrable. By stopping we further assume that E{Coo} < oo.
Next we define
0<s<4
and since С is assumed integrable we deduce that Л is a quasimartingale. By
Theorems 13 and 14 we know that A decomposes as follows:
where L is a local martingale and B1, B2 are non-decreasing, nonnegative,
predictable processes, such that ^{В^} < oo, ^{В^,} < oo. Let
At = B\ - B2t.
Then \At — At\ < Cqo +-Втс + -^то 6 L1, and hence the martingales A — A and
N = M-(A-A)
are uniformly integrable. As usual, we write A for the compensator of A. Then
D = A-A.
It remains only to show that thejjumps of N are bounded by 2/3. To this
end, let T be a stopping time. If AAT = 0 a.s., then
ANT = AMT - AAT
This implies |AiV^| < /3, and we are done by Theorem 7 of Chap. I.
We therefore assume if Л = {\AAT\ > 0}, then P(A) > 0. Let R = Тл.
By Theorem 3 we can decompose R as R — Ra Л Rb, where Ra is accessible
and Rb is totally inaccessible. Then P(Rb < oo) = 0 by Theorem 7, whence

126 III Semimartingales and Decomposable Processes
R = Ra- Let (Tfc)fc>i be a sequence of predictable times enveloping R. It
will suffice to show \ANxk\ < 2/3, for each Tk- Thus without loss of generality
we can take R — T to be predictable. By convention, we set ANT = 0 on
{T = oo}. Since we know that A Ax S Ft- by Theorem 24, we have
ANT = ANT - E{ANT\TT-}
= A(M - A)T + AAT - E{A(M - A)T\TT-} - E{AAT\TT-}
= A(M - A)T - E{A(M - A)T\TT-}.
Since |A(M - A)T\ < /3, the result follows. D
If the jumps of a local martingale M with Mo = 0 are bounded by a
constant /3, then M itself is locally bounded. Let Tn = in?{t : \Mt\ > n}.
Then \Mt/KTn | < n + C. Therefore M is a fortiori locally square integrable.
We thus have a corollary.
Corollary. A local martingale is decomposable.
Of course if all local martingales were locally square integrable, they would
then be trivially decomposable. The next example shows that there are mar-
martingales that are not locally square integrable (a more complex example is
published in Doleans-Dade [50]).
Example. Let (п, J7, P) be complete probability space and let X be a random
variable such that X e L1, but X $. L2. Define the filtration
t>l,
where Tt=J^vM', with N all the P-null sets of T. Let Mt = E{X\Tt}, the
right continuous version. Then M is not a locally square integrable martingale.
The next example shows another way in which local martingales differ from
martingales. A local martingale need not remain a local martingale under a
shrinkage of the filtration. They do, however, remain semimartingales and
thus they still have an interpretation as a differential.
Example. Let У be a symmetric random variable with a continuous distri-
distribution and such that .E{|Y|} = oo. Let Xt = У1{4>1}, and define
* \a{Y}, t>l,
where G = (Qt)t>o is the completed filtration. Define stopping times Tn by
rn = f 0, if \Y\ > n,
I oo, otherwise.

7 Classical Semimartingales 127
Then Tn reduce X and show that it is a local martingale. However X is not
a local martingale relative to its completed minimal filtration. Note that X is
still a semimartingale however.
The full power of Theorem 25 will become apparent in Sect. 7.
7 Classical Semimartingales
We have seen that a decomposable process is a semimartingale (Theorem 9
of Chap. II). We can now show that a classical semimartingale is indeed a
semimartingale as well.
Theorem 26. A classical semimartingale is a semimartingale.
Proof. Let X be a classical semimartingale. Then Xt = Mt + At where M is
a local martingale and A is an FV process. The process A is a semimartin-
semimartingale by Theorem 7 of Chap. II, and M is decomposable by the corollary of
Theorem 25, hence also a semimartingale (Theorem 9 of Chap. II). Since semi-
semimartingales form a vector space (Theorem 1 of Chap. II) we conclude X is a
semimartingale. D
Corollary. A cadlag local martingale is a semimartingale.
Proof. A local martingale is a classical semimartingale. ?
Theorem 27. A cadlag quasimartingale is a semimartingale.
Proof. By Theorem 15 a quasimartingale is a classical semimartingale. Hence
it is a semimartingale by Theorem 26. ?
Theorem 28. A cadlag supermartingale is a semimartingale.
Proof. Since a local semimartingale is a semimartingale (corollary to The-
Theorem 6 of Chap. II), it suffices to show that for a supermartingale X, the
stopped process X1 is a semimartingale. However for a partition т of [0,t],
= E{X0} - E{Xt}.
Therefore X1 is a quasimartingale, hence a semimartingale by Theorem 27. D
Corollary. A submartingale is a semimartingale.

128 III Semimartingales and Decomposable Processes
We saw in Chap. II that if X is a locally square integrable local martingale
and H ? L, then the stochastic integral H ¦ X is also a locally square inte-
integrable local martingale (Theorem 20 of Chap. II). Because of the corollary of
Theorem 25 we can now improve this result.
Theorem 29. Let M be a local martingale and let H G L. Then the stochastic
integral H ¦ M is again a local martingale.
Proof. A local martingale is a semimartingale by the corollary of Theorem 25
and Theorem 9 of Chap. II; thus H ¦ M is defined. By the Fundamental
Theorem of Local Martingales (Theorem 25) for /3 > 0 we can write M =
N + A where N, A are local martingales, the jumps of N are bounded by /3,
and A has paths of finite variation on compacts. Since N has bounded jumps,
by stopping we can assume N is bounded. Define T by
T = inf{t>0: / \dAs\ >m].
Jo
Then ?{/0*AT \dAs\} < m +/3 + E{\AMT\} < oo, and thus by stopping A can
0
be assumed to be of integrable variation. Also by replacing H by }
for an appropriate stopping time S we can assume without loss of generality
that H is bounded, since H is left continuous. We also assume without loss
that Mo — ЛГо = Ao = 0. We know H • N is a local martingale by Theorem 20
of Chap. II, thus we need show only that H ¦ A is a local martingale.
Let on be a sequence of random partitions of [0, t] tending to the identity.
Then Y2 Ht™ (At" - AT") tends to (H ¦ A)t in ucp, where on is the sequence
0 = Tq4 < Tf < • • • < T? < ¦ ¦ ¦. Let (nk) be a subsequence such that the
sums converge uniformly a.s. on [0,t]. Then
E { Г HudAu|Ts\ = E j lim VHTn
Uo J [ Пк ^
р1 - A^")\TS
= lim E \ У Нт«к (Лр1 - A?" )\TS
by Lebesgue's Dominated Convergence Theorem. Since the last limit above
equals (H ¦ A)s, we conclude that H ¦ A is indeed a local martingale. ?
A note of caution is in order here. Theorem 29 does not extend completely
to processes that are not in L but are only predictably measurable, as we will
see in Emery's example of a stochastic integral behaving badly on page 176
in Chap. IV.
Let X be a classical semimartingale, and let Xt = Xq + Mt + At be a
decomposition where Mq = Aq = 0, M is a local martingale, and A is an FV

7 Classical Semimartingales 129
process. Then if the space {?l,T, {Tt)t>o,P) supports a Poisson process N,
we can write
Xt=X0 + {Mt + Nt-t} + {At -Nt+t]
as another decomposition of X. In other words, the decomposition of a classi-
classical semimartingale need not be unique. This problem can often be solved by
choosing a certain canonical decomposition which is unique.
Definition. Let X be a semimartingale. If X has a decomposition Xt =
Xo + Mt + At with Mo = Ao = 0, M a local martingale, A an FV process,
and with A predictable, then X is said to be a special semimartingale.
To simplify notation we henceforth assume Xo = 0.
Theorem 30. If X is a special semimartingale, then its decomposition X =
M + A with A predictable is unique.
Proof. Let X = N + В be another such decomposition. Then M — N = B — A,
hence В — A is an FV process which is a local martingale. Moreover, В — A
is predictable, and hence constant by Theorem 12. Since Bo — Ao = 0, we
conclude В = A. О
Definition. If X is a special semimartingale, then the unique decomposition
X = M + A with Mo = Xo and Ao = 0 and A predictable is called the
canonical decomposition.
Theorem 15 shows that any quasimartingale is special. A useful sufficient
condition for a semimartingale X to be special is that X be a classical semi-
semimartingale, or equivalently decomposable, and also have bounded jumps.
Theorem 31. Let X be a classical semimartingale with bounded jumps. Then
X is a special semimartingale.
Proof. Let Xt = Xo + Mt + At be a decomposition of X with Mo = Aq = 0,
M a local martingale, and A an FV process. By Theorem 25 we can then also
write
Xt - Xo + Nt + Bt
where N is a local martingale with bounded jumps and В is an FV process.
Since X and TV each have bounded jumps, so also does B. Consequently, it is
locally a quasimartingale and therefore decomposes
B = L + B
where L is a local martingale and Б is a predictable FV process (Theorem 15).
Therefore
Xt = Xo + {Nt + Lt} + Bt
is the canonical decomposition of X and hence X is special. ?

130 III Semimaxtingales and Decomposable Processes
Corollary. Let X be a classical semimartingale with continuous paths. Then
X is special and in its canonical decomposition
the local martingale M and the FV process A have continuous paths.
Proof. X is continuous hence trivially has bounded jumps, so it is special by
Theorem 31. Since X is continuous we must have
AMt = -AAT
for any stopping time T (ААт — 0 by convention on {T = oo}). Suppose
A jumps at a stopping time T. By Theorem 3, Г = Гд Л Гв, where TA
is accessible and Tg is totally inaccessible. By Theorem 21 it follows that
P(\AAtb\ > 0) = 0. Hence without loss of generality we can assume T is
accessible. It then suffices to consider T predictable since countably many
predictable times cover the stopping time T. Let Sn be a sequence of stopping
times announcing T. Since A is predictable, we know by Theorem 24 that
ААт is Tt- measurable. Therefore AMt is also Тт- measurable. Stop M
so that it is a uniformly integrable martingale. Then
AMT = E{AMT\TT-}
= 0,
and M, and hence A, are continuous, using Theorem 7 of Chap. I. ?
Theorem 31 can be strengthened, as the next two theorems show. The
criteria given in these theorems are quite useful.
Theorem 32. Let X be a semimartingale. X is special if and only if the
process Jt = sups<t |AXe| is locally integrable.
Theorem 33. Let X be a semimartingale. X is special if and only if the
process X* = sups<t I-Xsl is locally integrable.
Before proving the theorems, we need a preliminary result, which is inter-
interesting in its own right.
Theorem 34. Let M be a local martingale and let Mt* = sups<t \MS\. Then
the increasing process M* is locally integrable.
Proof. Without loss of generality assume Mq = 0. Let Tn be a sequence
of stopping times increasing to oo such that MTn is a uniformly integrable
martingale for each n. Since we can replace Tn with TnAn if necessary, without
loss of generality we can assume that the stopping times Tn are each bounded.
Set Sn = TnAinft{|M4| > ra}. Since MTn is uniformly integrable, and Sn < Tn,
we have that Msn is integrable. But we also have that M?n < n V \Msn | which
is in L1. Since Sn increases to oo a.s., the proof is complete. ?

8 Girsanov's Theorem 131
Proof of Theorems 32 and 33. Since X is special the process A of its canonical
decomposition X = M + A is of locally integrable variation. Writing Xt* <
Mt* + /0 | dAs |, and since Mt* is locally integrable by Theorem 34 and since A is
of locally integrable variation, we have that X* is locally integrable. Further,
note that |AXt| = \Xt — Xt_ | < 2|Xt* |, hence Jt < X* and we have that Jt is
also locally integrable.
For the converse, it will suffice to show that there exists a decomposition
X = M + A where A is of locally integrable variation, since then we can take
the compensator of A and obtain a canonical decomposition. To this end note
that \AAS\ < \AMS\ + \AXS\ < 2M* + Js, which is locally integrable by the
hypotheses of Theorem 32 together with Theorem 34. Since Jt < Xf we have
that | A As | is locally integrable by the hypotheses of Theorem 33 together with
Theorem 34 as well. To complete the proof, let Tn = inf{f > 0 : Jo \dAs\ > n},
and note that /0 n |cL4s| = /0 n~ |cL4s| + |АЛТ„| < n + sups<Tn |A.AS|, which
is in L1 as soon as sups<Tn \AAS\ is. However this last term is locally in L1 as
we have seen, and the proof is complete with one more sequence of stopping
times Rn putting sups<fln | A^4S| in L1 and then constructing a new, single
sequence by taking minimums from the two sequences. ?
A class of examples of special semimartingales is Levy processes with
bounded jumps. By Theorem 31 we need only show that a Levy process Z
with bounded jumps is a classical semimartingale. This is merely Theorem 40
of Chap. I.
For examples of semimartingales which are not special, we need only to
construct semimartingales with finite variation terms that are not locally
integrable. For example, let X be a compound Poisson process with non-
integrable, nonnegative jumps. Or, more generally, let X be a Levy process
with Levy measure v such that /x xv(dx) = oo and/or /^ |a;|^(rfa;) = oo.
8 Girsanov's Theorem
Let X be a semimartingale on a space (il, J7, F, P) satisfying the usual hy-
hypotheses. We saw in Chap. II that if Q is another probability law on (?l,F),
and Q«P, then X is a Q semimartingale as well. If X is a classical semi-
semimartingale (or equivalently, if X is decomposable) and has a decomposition
X = M + A, M a local martingale and A an FV process, then it is often
useful to be able to calculate the analogous decomposition X — N + B, if it
exists, under Q. This is rather tricky unless we make a simplifying assumption
which usually holds in practice: that both Q < P and ?«Q.
Definition. Two probability laws P, Q on (fl, J7) are said to be equivalent
if P < Q and Q < P. (Recall that P«Q denotes that P is absolutely
continuous with respect to Q.) We write Q ~ P to denote equivalence.

132 III Semimartingales and Decomposable Processes
If Q -C P, then there exists a random variable Z in Lx(dP) such that
^ = Z and Ep{Z} = 1, where Ep denotes expectation with respect to the
law P. We let
be the right continuous version. Then Z is a uniformly integrable martingale
and hence a semimartingale (by the corollary of Theorem 26). Note that if Q
3p ) •
We begin with a simple lemma, the easy proof of which we leave as an
exercise for the reader.
Lemma. Let Q ~ P, and Zt = Ep < ^[p\J~t \- An adapted, cadlag process M
is a Q local martingale if and only if MZ is a P local martingale.
Proof. See Exercise 20. ?
Theorem 35 (Girsanov-Meyer Theorem). Let P and Q be equivalent.
Let X be a classical semimartingale under P with decomposition X — M + A.
Then X is also a classical semimartingale under Q and has a decomposition
X = L + C, where
/•* 1
Lt = Mt- / —d[Z,M]s
Jo Zs
is a Q local martingale, and С = X — L is a Q FV process.
Proof. Recall that by Theorem 2 of Chap. II it is trivial that X is a Q semi-
semimartingale. We need to show it is a classical semimartingale, with the above
decomposition being valid.
Since M and Z are P local martingales, they are semimartingales (corol-
(corollary of Theorem 26) and
Z-dM+ I M-dZ
is a local martingale as well (Theorem 29). Using integration by parts we have
ZM - [Z, M}= f Z_dM+ I M-dZ (*)
is also a P local martingale. Since Z is a version of Ep{^\Tt}, we have ;|-
is a cadlag version of Eq{^\Tt}\ therefore ^ is a Q semimartingale. Since
P <C Q, it is also a P semimartingale. Multiplying equation (*) by -| we have
(**)
Note that if we were to multiply the right side of (**) by Z we would obtain
a P local martingale. We conclude by the lemma preceding this theorem that

8 Girsanov's Theorem 133
M — (j^)[Z,M] is a Q local martingale. We next use integration by parts
(under Q):
j-d[z,M}
Let N — J[Z, M]-d (^). Since ^ is a Q local martingale, so also is N (The-
(Theorem 29). The above becomes:
+ Nt + [[Z,M], \\t
^j [Z,M]t = J* ~d[Z,
= f J-d[Z,
O As
Adding our two Q local martingales yields
N + M - ( \ ] [Z, M] = N + M - f -d[Z, M]-N
\ZJ J Z
= M-
which, being the sum of two Q local martingales, is itself one. This establishes
the theorem. ?
The Girsanov-Meyer Theorem has another version using the sharp bracket.
It transforms a P local martingale into a Q special semimartingale and gives its
canonical decomposition. However an extra integrability hypothesis is needed
for the theorem to be true. Note that in the predictable version the integrand
-%- is replaced by its left continuous version -^—.
Theorem 36 (Girsanov-Meyer Theorem: Predictable Version). Let X
be a P local martingale with Xq — 0. Let Q be another probability equivalent
to P and let Zt = E{-jp\J-t}. If (X,Z) exists for the P probability, then the
canonical Q decomposition of X is
Xt = (Xt- j ~d(X,Z)s)+ j
Jo "S- Jo
Proof. Using the same ideas as in the proof of Theorem 35, we want to find
a predictable finite variation term A such that Y = Z(X —A) is a P local
martingale. Using integration by parts yields
ZtXt = [ Zs_dXs + f Xs_dZs + [Z,X]
Jo Jo
= Nt + [Z,X}t-(X,Z)t + {X,Z)t
= Mt + (X,Z)t

134 III Semimartingales and Decomposable Processes
where both N = /„* Zs_dXs + /0*Xs-dZs and M = N + [Z,X] - {X, Z) are
P local martingales. Therefore ZX — {X, Z) is a P local martingale.
Let A = (X,Z), which of course is a predictable FV process. Moreover
we know that A does not jump at totally inaccessible times, and that we
can cover its jumps with predictable times. This implies that A is locally
bounded. Let Tn = ini{t > 0 : \At\ > n}, and let ATn~ be denned by
ATn~ = J l[o,Tn)(s)<?Aa. From now on we assume A = ATn~. Then A is still
an FV process and is still predictable, |Л| < n, and the stopping times Tn
increase to oo a.s. Integration by parts gives us also
1 /"* 1 /"*
-At = / ~-dAa+ /
At J0 ^s- JO
i]..
¦ If we show that [A, ^j is also a Q local martingale, we will be done. It suffices
to show that Eq{[A, ^]t} = 0 for every stopping time T. Since A is FV and
predictable, by stopping at T— if necessary, we can assume the jumps of A are
bounded. And since ^ is a Q local martingale, its supremum process is locally
integrable (Theorem 34). This is enough to give us that [A, j^] is Q locally
of integrable variation. By stopping we assume it is integrable. We then have
that A is Q natural, so if T is a stopping time, we replace A with AT and we
still have a natural process, and thus we have Eq{[A, ^}t} = 0. Since T was
arbitrary, this implies that [A, \\ is a martingale. Of course, the process is
implicitly stopped at several stopping times, so what we have actually shown
is that [A,^] is a Q local martingale.
We now have that
1 /"* 1
—-At = Q-local martingale + / dAs
Zt Jo Zs-
which in turn gives us, recalling that A = {X, Z),
t t
1 /¦* 1
— — Mt + Q-local martingale + / ——d(X, Z)s.
Zt Jo As-
Since Z(jjM) = M = P local martingale, we have that j^M is also a Q local
martingale. Thus finally Xt — /0* -^d(X, Z)s is a Q local martingale, and the
proof is done. ?
Let us next consider the interesting case where Q is absolutely continuous
with respect to P, but the two probability measures are not equivalent. (That
is, we no longer assume that P is also absolutely continuous with respect to
Q.) We begin with a simple result, and we will assume J-o = {0,^} a.s.
Theorem 37. Let X be a P local martingale with Xo = 0. Let Q be another
probability absolutely continuous with respect to P, and let Zt = ^

8 Girsanov's Theorem 135
Assume that (X,Z) exists for P. Then At = JQ -^d(X,Z)s exists a.s. for
the probability Q, and Xt — Jo -^^d(X, Z)s is a Q local martingale.
Proof. Zo = E{Z} = 1, so if we let Rn - ini{t > 0 : Zt < 1/n}, then
Rn increase to oo, Q-a.s., and the process -?— is bounded on [0, Rn]. By
Theorem 36, XRn - f* ^-d{X, Z)fn is a Q local martingale, each n. Since a
local, local martingale is a local martingale, we axe done. ?
We now turn to the general case, where we no longer assume the existence
of (X, Z), calculated with P. (As before, we take X to be a P local martingale.)
We begin by denning a key stopping time: R = inf{f > 0 : Zt = 0, Zt_ > 0}.
Note that Q(R < oo) = 0, but it is entirely possible that P{R < oo) > 0. We
further define Ut — Д^д1{*>д}- Then U is an FV process, and moreover U
is locally integrable (dP). Let Tn increase to oo and be such that XTn is a
uniformly integrable martingale. Then
|Д^я = 1^Т„ЛД - X(TnAR)-\ < |^Т„ЛД| + \X(TnAR)-\ ? L ¦
Thus U has a compensator U, and of course U is predictable and U — U is a
P local martingale.
Theorem 38 (Lenglart-Girsanov Theorem). Let X be a P local martin-
martingale with Xq = 0. Let Q be a probability absolutely continuous with respect
to P, and let Zt = EP{^\Tt}, R = ini{t > 0 : Zt = 0, Zt_ > 0}, and
Ut = kXRl{t>R}. Then
is a Q local martingale.
Proof. Let Rn = inf{t > 0 : Zt < A}. (Recall that Zo = 1, and also note
that it is possible that Rn = R.) Then both XRn and ZRn are P local
martingales. Also note that ARn = /0* -ф^1{2нп>0^[Хк«,ZR«)S, URn, and
уд„ = XRn _ ^Д„ + [/я. аге ац p-well-defined. We can define
/* 1
At = / -^1{Z,:
JO ^s
on [0,R), since d[XRn,ZRn]s does not charge (R,oo), and —h:^szRn>o] = ®
at R. Thus we need only to show YRn is a Q local martingale for each fixed
n, which is the same as showing that ZRnYRn is a P local martingale. Let us
assume all these processes are stopped at Rn to simplify notation. We have
ZY = ZX -ZA + ZU.

136 III Semimartingales and Decomposable Processes
Hence,
d{ZX) =Z^dX + X-dZ + d[Z, X]
= local martingale + d[Z, X]
At Л У\ Л АУ _l 7 А Л
U\s\?j I — s\ LiZj -}- ZjUSI.
= local martingale + ZdA
= local martingale + l^z>Oyd[X, Z]
d{ZU) = Z-dU + UdZ
= local martingale + Z^dU
= local martingale + Z^dU
where the last equality uses that U—U is a local martingale (dP). Summarizing
we have
ZY = ZX -ZA + ZU
= local martingale + \Z, X] — local martingale
+ / ~L{z>o}d[X,Z\) + local martingale + / Z_dU
which we want to be a local martingale under dP. This will certainly be the
case if
d[Z,X] - l{Z>oyd[X,Z] + Z_dU = 0. (*)
However (*) equals
But AZr = Zr — Zr- = 0 — Zr- = —Zr-, and this implies that equation (*)
is indeed zero, and thus the Lenglart-Girsanov Theorem holds. ?
Corollary. Let X be a continuous local martingale under P. Let Q be abso-
absolutely continuous with respect to P. Then (X,Z) = [Z,X] = \Z,X]C exists,
and
Xt - f -l-d[Z,X}cs = Xt - I asd[X,X]s
JO ^s- JO
which is a Q local martingale.
Proof. By the Kunita-Watanabe inequality we have
which shows that it is absolutely continuous with respect to d[X, X]s a.s.,
whence the result. ?

8 Girsanov's Theorem 137
We remark that if Z is the solution of a stochastic exponential equation
of the form dZs = Zs-HsdXs (which it often is), then as = Hs.
Example. A problem that arises often in mathematical finance theory is
that one has a semimartingale S = M + A denned on a filtered probability
space (O,.F,F, P) satisfying the usual hypotheses, and one wants to find an
equivalent probability measure Q such that under Q the semimartingale X is
a local martingale, or better, a martingale. In essence this amounts to finding
a probability measure that "removes the drift." To be concrete, let us suppose
S is the solution of a stochastic differential equation3
dSs = h(s, Ss)dBs + b(s; Sr;r < s)ds,
where В is a standard Wiener process (Brownian motion) under P. Let us
postulate the existence of a Q and let Z = ^ and Zt = E{Z\Tt}, which is
clearly a cadlag martingale. By Girsanov's Theorem
f h(s,Ss)dBs- f ^rd[Z, f h(r,Sr)dBr]s
J0 Jo "s Jo
is a Q local martingale. We want to find the martingale Z. In Chap. IV
we will study martingale representation and show in particular that every
local martingale on a Brownian space is a stochastic integral with respect to
Brownian motion. Thus we can write Zt = 1 + JQ JsdBs for some predictable
process J. If we assume Z is well behaved enough to define Hs = -^f-, then
we have Zt = 1 + /0 HsZsdBs, which gives us a linear stochastic differential
equation to solve for Z. Thus if we let Nt = jQ HsdBs, we get that Zt =
?(N)t.4 It remains to determine H. We do this by observing from our previous
Girsanov calculation that
f h(s,Ss)dBs- f ^rZsHsh(s,Ss)ds= f h(s,Ss)dBs- f Hsh(s,Ss)ds
Jo JO 6S Jq Jq
is a Q local martingale. We then choose Hs = —¦ ' r'—r=—-, which yields
h(s,Ss)
St= I h(s,Ss)dBs+ I b{s;Sr;r<s)ds
Jo Jo
[*" b(s' S ¦ r < s)
is a local martingale under Q. Letting Mt = Bt+ I ' /' ~—-ds denote
Jo 4S> ss)
this Q local martingale, we get that [M, M]t — [B, B]t = t, and by Levy's
Theorem M is a Q-Brownian motion. Finally, under Q we have that S satisfies
the stochastic differential equation
dSt = h(t,St)dMt.
3 Stochastic differential equations are introduced and studied in some detail in
Chap. V.
4 The stochastic exponential E is defined on page 85.

138 III Semimartingales and Decomposable Processes
There is one problem with the preceding example: we do not know a priori
whether our solution Z is the Radon-Nikodym density for simply a measure
Q, or whether Q is an actual bona fide probability measure. This is a constant
problem. Put more formally, we wish to address this problem:
Let M be a local martingale. When is ? (M) a martingale?
The only known general conditions that solve this problem are Kazamaki's
criterion and Novikov's criterion.5 Moreover, these criteria apply only to lo-
local martingales with continuous paths. Novikov's is a little less powerful than
Kazamaki's, but it is much easier to check in practice. Since Novikov's crite-
criterion follows easily from Kazamaki's, we present both criteria here. Note that
if M is a continuous local martingale, then of course ?(M) is also a continuous
local martingale. Even if, however, it is a uniformly integrable local martin-
martingale, it still need not be a martingale; we need a stronger condition. As an
example, one can take u(x) = \\x\\~1 and Nt = u(Bt) where В is standard
three dimensional Brownian motion. Then N is a uniformly integrable local
martingale but not a martingale.
Nevertheless, whenever M is a continuous local martingale, then ?{M) is a
positive supermartingale, as is the case with any nonnegative local martingale.
Since ?(M)q = 1, and since ?(M) is a positive supermartingale, we have
E{?(M)t} < 1, all t. (See the lemma below.) It is easy to see that Z is a
true martingale if one also has E{?(M)t} = 1, for all t. We begin with some
preliminary results.
Lemma. Let M be a continuous local martingale with Mo = 0. Then
E{?(M)t} < 1 for all t > 0.
Proof. Recall that ?(M)o = 1. Since M is a local martingale, ?(M) is a
nonnegative local martingale. Let Tn be a sequence of stopping times reducing
?{M). Then E{?(M)tATn} = 1, and using Fatou's Lemma,
E{?(M)t} = Е{Итт{?(М)глТп} < liminiE{?(M)tATn} = 1. ?
Theorem 39. Let M be a continuous local martingale. Then
Proof.
which implies that
5 There also exist partial results when the local martingale is no longer continuous
but only has cadlag paths. See Exercise 14 of Chap. V for an example of these
results.

8 Girsanov's Theorem 139
and this together with the Cauchy-Schwarz inequality and the fact that
E{?(M)t} < 1 gives the result. D
Lemma. Let M be a continuous local martingale. Let 1 < p < oo, ^ +1 = 1.
Taking the supremum below over all bounded stopping times, assume that
s\rpE\eK2^-l!MT\ < oo.
т
Then ?(M) is an Lq bounded martingale.
Proof. Let 1 < p < oo and r = y-_1 ¦ Then s = ^~— and ^ + -j — 1. Also we
note that (g — y/lj:)s — 2i /-'-n which we use in the last equality of the proof.
We have
?{M) =
We now apply Holder's inequality for a stopping time S:
E{?(M)%} = E
Recalling that E{?(.y/qrM)s} < 1, we have the result. D
Theorem 40 (Kazamaki's Criterion). Let M be a continuous local mar-
martingale. Suppose supT Е{е^Мт)у < oo, where the supremum is taken over all
bounded stopping times. Then ?{M) is a uniformly integrable martingale.
Proof. Let 0 < a < 1, and p > 1 be such that tjp_u < ?• Our hypothesis
combined with the preceding lemma imply that ?(aM) is an LQ bounded
martingale, where - + - = 1, which in turn implies it is a uniformly integrable
martingale. However
?(aM) = e°-M
= ?(M)a2 ea{1~a)M,
and using Holder's inequality with a~2 and A - a2) yields (where the 1 on
the left side comes from the uniform integrability):

140 III Semimartingales and Decomposable Processes
Now let a increase to 1 and the second term on the right side of the last
inequality above converges to 1 since 2a(l — a) —> 0. Thus 1 < E{?(M)OO},
and since we know that it is always true that 1 > ?{?(M)OO}, we are done. ?
As a corollary we get the very useful Novikov's criterion. Because of its
importance, we call it a theorem.
Theorem 41 (Novikov's Criterion). Let M be a continuous local martin-
martingale, and suppose that
Then ?{M) is a uniformly integrable martingale.
Proof. By Theorem 39 we have Е{е?Мт} < Е{е^м'м^т}ъ, and we need only
to apply Kazamaki's criterion (Theorem 40). ?
We remark that it can be shown that 1/2 is the best possible constant
in Novikov's criterion, even though Kazamaki's criterion is slightly stronger.
Note that in the case of the example treated earlier, we have [N, N]t = JQ H^ds
where H = ^'Лгт''. Clearly whether or not \ [N, N]t has a finite exponential
moment will depend on our choice of b and of h. A standard simple solution
is to choose b and h so that b/h is bounded.
Partial results exist that give an analogue of Novikov's criterion in the
general (cadlag) case. See for example Exercise 14 of Chap. V, and the
sources [156] and [157].
Example. This example is taken from statistical communication theory. Sup-
Suppose we are receiving a signal corrupted by noise, and we wish to determine
if there is indeed a signal, or if we are just receiving noise (e.g., we could be
searching for signs of intelligent life in our galaxy with a radio telescope). Let
x(t) be the received signal, ?(i) the noise, and s(t) the actual (transmitted)
signal. Then
A frequent assumption is that the noise is "white." A white noise is usually
described as a second order wide sense stationary process with a constant
spectral density function (that is, E{?t} = 0, and V(r) = E{XtXt+T} = 5TS0,
where ST is the Dirac delta function at r, and So is the constant spectral
density; one then has S(v) = /^ el2nuTVг(т)йт). Such a process does not exist
in a rigorous mathematical sense. Indeed it can be interpreted as the derivative
of the Wiener process, in a generalized function sense. (See Arnold [2, page
53], for example.) This suggests that we consider the integrated version of our
signal-noise equation:
[
Jo
Xt= [ s(u)du + Wt
J

8 Girsanov's Theorem 141
where IF is a standard Wiener process and where St is thought of as
"/0 x(s)ds," the cumulative received signal.
The key step in our analysis is the following consequence of the Girsanov-
Meyer Theorem.
Theorem 42. Let W be a standard Brownian motion on (fi,.F,F, P), and
let H G L be bounded. Let
I*
Hsds + Wt
and define Q by ^ = exp{/0 -HsdWs — \j0 H2ds}, for some T > 0. Then
under Q, X is a standard Brownian motion for 0 < t < T.
Proof. Let ZT = exp{/0T -HsdWs - \ JQT H2ds}. Then if Zt = E{ZT\Tt} we
know by Theorem 37 of Chap. II that Z satisfies the equation
Zt = l- f Zs-HsdWs.
Jo
By the Girsanov-Meyer Theorem (Theorem 35), we know that
,t 1
Nt = Wt- / — d[Z,W]s
Jo ^s
is a Q local martingale. However
[Z, W]t = \-Z_H ¦ W, W\t = f -ZsHsd[W, W]s
Jo
f*
= — I ZsHsds,
Jo
since [W, W]t = t for Brownian motion. Therefore
Г —
Jo %s
pt
= Wt+ Hsds
Jo
hence X is a Q local martingale. Since (Jo Hsds)t>o is a continuous FV pro-
process we have that [X, X]t = [W, W]t = t, and by Levy's Theorem (Theorem 39
of Chap. II) we conclude that X is a standard Brownian motion. D
Corollary. Let W^bea standard Brownian motion and H G L be bounded.
Then the law of
Xt= / Hsds + Wt,
Jo
0 < t < T < oo, is equivalent to Wiener measure.

142 III Semimartingales and Decomposable Processes
Proof. Let C[0,T] be the space of continuous functions on [0, T\ with values
in M (such a space is called a path space). If W = (Wt)o<t<T is a standard
Brownian motion, it induces a measure fiy/ on C[0, T):
fiw(A) = P{w : t » Wt{w) e A}.
Let fix be the analogous measure induced by X. Then by Theorem 42 we
have fix ~ f*w and further we have
= exp ( Г -HsdWs - \ [H>ds\ . D
[J?2 J J
Remark. We have not tried for maximum generality here. For example the
hypothesis that H be bounded can be weakened. It is also desirable to weaken
the restriction that H G L. Indeed we only needed that hypothesis to be able
to form the integral /0 HsdWs. This is one example to indicate why we need
a space of integrands more general than L.
We are now in a position to consider the problem posed earlier: is there
a signal corrupted by noise, or is there just noise (that is, does s(t) = 0 a.e.,
a.s.)? In terms of hypothesis testing, let Ho denote the null hypothesis, Hx
the alternative. We have:
Ho : XT = WT
HX:XT= f Hsds + WT-
Jo
We then have
= exp I / -HsdWs -- H*ds ,
'o z Jo J
by the preceding corollary. This leads to a likelihood ratio test:
if -r-^-(w) < A, reject Ho,
dfix
if -t^-(oj) > A, fail to reject Ho,
dfix
where the threshold level A is chosen so that the fixed Type I error is achieved.
To indicate another use of the Girsanov-Meyer Theorem let us consider
stochastic differential equations. Since stochastic differential equations6 are
treated systematically in Chap. V we are free here to restrict our attention to
a simple but illustrative situation. Let W be a standard Brownian motion on a
space {?l,J-,F,P) satisfying the usual hypotheses. Let fi(w,s,x) be functions
satisfying (г = 1,2):
6 Stochastic "differential" equations have meaning only if they are interpreted as
stochastic integral equations.

9 The Bichteler-Dellacherie Theorem 143
(i) \fi(w,s,x) - fi(u,s,y)\ < K\x-y\ for fixed (w,s);
(ii) /;(-,s,:r)e.Fs for fixed (s,:r);
(iii) fi(w,-,x) is left continuous with right limits for fixed (w,x).
By a Picard-type iteration procedure one can show there exists a unique so-
solution (with continuous paths) of
Xt = X0+ f f1(-,s,Xs)dWs + f f2{.,s,Xs)ds. (***)
Jo Jo
The Girsanov-Meyer Theorem allows us to establish the existence of solutions
of analogous equations where the Lipschitz hypothesis on the "drift coeffi-
coefficient" /2 is removed. Indeed if X is the solution of (***), let 7 be any
bounded, measurable function such that -y(w,s,Xs) e L. Define
g(w,s,x) = f2(u>,s,x) +f1(w,s,x)'y(s,u,x).
We will see that we can find a solution of
= Y0+ [ M-,a,Ya)dBa+ f g(-,s,Ys
Jo Jo
)ds
provided we choose a new Brownian motion В appropriately.
We define a new probability law Q by
dQ ( fT 1 fT \
-^=exp / ~f{s,Xa)dWa-- / -y(s,XsJds .
«^ \Jo г Jo )
By Theorem 42 we have that
Bt = Wt- [ i(s,Xs)ds
Jo
is a standard Brownian motion under Q. We then have that the solution X
of (* * *) also satisfies
Xt=X0+ f h(;S,Xs)dBs+ f (h + fl<y)(;8,Xa)d8
Jo Jo
= X0+ [ fi(;8,Xa)dBa+ f g(;8,Xa)d8,
Jo Jo
which is a solution of a stochastic differential equation driven by a Brownian
motion, under the law Q.
9 The Bichteler-Dellacherie Theorem
In Sect. 7 we saw that a classical semimartingale is a semimartingale. In this
section we will show the converse.

144 III Semimartingales and Decomposable Processes
Theorem 43 (Bichteler-Dellacherie Theorem). An adapted, cadlag pro-
process X is a semimartingale if and only if it is a classical semimartingale. That
is, X is a semimartingale if and only if it can be written X = M + A, where
M is a local martingale and A is an FV process.
Proof. The sufficiency is exactly Theorem 26. We therefore establish here only
the necessity.
Since X is cadlag, the process Jt = Ylo<s<t AXsl{\AXa\>i} has paths of
finite variation on compacts; hence it is an FV process. Let Y — X — J.
Then Y has bounded jumps. If we show that X is a classical semimartingale
on [O,io], some io, then Y = X — J is one as well; moreover У is special
by Theorem 31. Let Y = Yo + M + A be the canonical decomposition of Y.
Suppose ii > t0 and X is also shown to be a classical semimartingale on [0, ti].
Then let Y = Yo + N + В be the canonical decomposition of Y on [0, ti]. We
have that, for t < to,
M -N = B~~A~
by subtraction. Thus В — A is a predictable, FV process which is a local
martingale; hence by Theorem 12 we have В — A. Thus if we take tn increasing
to oo and show that X is a classical semimartingale on [0, tn], each n, we have
that X is a classical semimartingale on [0, oo).
We now choose u0 > 0, and by the above it suffices to show X is a classical
semimartingale on [0, щ]. Thus it is no loss to assume X is a total semimartin-
semimartingale on [0, uo] ¦ We will show that X is a quasimartingale, under an equivalent
probability Q. Rao's Theorem (Theorem 15) shows that X is a classical semi-
semimartingale under Q, and the Girsanov-Meyer Theorem (Theorem 35) then
shows that X is a classical semimartingale under P.
Let us take H e S of the special form:
n-l
^t = 2j^»1№,Ti+i] (*)
i=0
where 0 = To < T\ < ¦ ¦ ¦ < Tn-\ < Tn = u0. In this case the mapping Ix is
given by
IX{H) = (H ¦ X)uo = H0(XTl -Xo) + --- + tfn_!(XU0 -Xr,.,).
The mapping Ix : Su —> L° is continuous, where L° is endowed with the
topology of convergence in probability, by the hypothesis that X is a total
semimartingale. Let
В = {H e S : H has a representation (*) and \H\ <1}.
Let /3 = Ix{B), the image of В under Ix- It will now suffice to find a
probability Q equivalent to P such that Xt e ^(dQ), 0 < t < u0 and
such that supuep Eq(U) = с < oo. The reason this suffices is that if we
take, for a given 0 — to < t\ < ¦ ¦ ¦ < tn = uq, the random variables

9 The Bichteler-Dellacherie Theorem 145
Ho = sign(EQ{Xtl - X0\F0}), #! = sign(?Q{Xt2 - Xu |.Ftl}),..., we have
that for this H G B,
T? ( т 1 TT\\ Z?flZ?fV v If II IP fv V 'XT Л\Л
^Q^xy^i)) = &Q\\&Q\Ato - Atl|y-Oj| + • • • + \?iQ\Atnl — Лгп А„_1/|/-
Since this partition r was arbitrary, we have Var(X) — suprVarr(X) <
suPuep ^q(^) = c < °o> and so X is a Q quasimartingale.
Lemma 1. lim sup P(\Y\ > c) = 0.
c—ooye/3
Proof of Lemma 1. Suppose linic^ooSupyg^Pdyi > c) > 0. Then there ex-
exists a sequence с„ tending to oo, Yn G C, and a > 0 such that Р(|У^| > d) > a,
all n. This is equivalent to
Since Yn G C, there exists tf" G В such that /х(Яп) = Yn. Then Ix{±-Hn) =
-1-1х{Нп) = -^У„ 6 /3, if с„ > 1. But ^Я" tends to 0 uniformly a.s. which
implies that Ix{^-Hn) = — Yn tends to 0 in probability. This contradicts
(*)• " " ?
Lemma 2. There exists a law Q equivalent to P such that Xt G Lx{dQ),
0<t<u0.
Proof of Lemma 2. Let Y = sup0<t<uo\Xt\. Since X has cadlag paths,
Y < oo a.s. Moreover if D is a countable dense subset of [0,щ], then
У = supteZ) |Xt|. Hence У is a random variable. Let Am — {m < У < m+1},
and set Z = Em=o^mW' Then Z is bounded, strictly positive, and
YZ 6 L^dP). Define Q by setting $ = ^^yZ. Then ?д{|Х4|} <
= Ep{YZ}/EP{Z} < oo. Hence EQ{|Xt|} < oo, 0 < t < u0. П
Observe that C С L1(dQ) for the law Q constructed in Lemma 2. Lemma 3
below implies that X is an R quasimartingale for R ~ Q ~ P. Hence by
Rao's Theorem (Theorem 15) it is a classical R semimartingale, and by the
Girsanov-Meyer Theorem (Theorem 35) it is a classical semimartingale for
the equivalent law P as well. Thus Lemma 3 below will complete the proof of
Theorem 43. We follow Yan [233].
Lemma 3. Let C be a convex subset of L1(dQ), 0 € /3, that is bounded in
probability. That is, for any e > 0 there exists а с > 0 such that <3(? > c) < e,
for any С G C. Then there exists a probability R equivalent to Q, with a
bounded density, such that supue/3 Eu(U) < oo.
Proof of Lemma 3 and end of proof of Theorem 43. To begin, note that the
hypotheses imply that C С L1(dR). What we must show is that there ex-
exists a bounded random variable Z, such that P(Z > 0) = 1, and such that
suPce/3 eq(z0 <oo.

146 III Semimartingales and Decomposable Processes
Let ief such that Q(A) > 0. Then there exists a constant d such that
Q(( > d) < Q(A)/2, for all ( e /3, by assumption. Using this constant d,
let с = 2d, and we have that 0 < c\A ф /3, and moreover if B+ denotes all
bounded, positive r.v., then clA is not in the Lx{dQ) closure oif3—B+, denoted
C-B+. That is, clA ? /3 - B+. Since the dual of L1 is L°°, and /3 - B+ is
convex, by a version of the Hahn-Banach Theorem (see, e.g., Treves [223, page
190]) there exists a bounded random variable Y such that
sup EQ{Y(C-v)}<cEQ{YlA}. (*)
CB
Replacing ry by al{y<o} and letting a tend to oo shows that Y > 0 a.s., since
otherwise the expectation on the left side above would get arbitrarily large.
Next suppose r\ = 0. Then the inequality above gives
supEQ{Y(} < cEq{YIa} < +oo.
Now set H = {Y G B+ : supCe/3 Eq{YQ < oo}. Since 0 G B+, we know H
is not empty. Let Л = {all sets of the form {Z = 0},Z 6 H}. We wish to
show that there exists a Z € 7i such that Q(Z — 0) — inf^e^ Q(A). Suppose,
then, that Zn is a sequence of elements of 7i. Let Cn — sup^e/3 E{Zn?} and
dn = ||ZnlU°°- (Since 0 G /3, we have с„ > 0). Choose Ь„ such that ХЖС« <
oo and J2bndn < oo, and set Z = ^2bnZn. Then clearly Z G W. Moreover,
{Z = 0} = f]n{Zn = 0}. Thus Л is stable under countable intersections, and
so there exists a Z such that Q(Z = 0) = inf^e^ Q(A).
We now wish to show Z > 0 a.s. Suppose not. That is, suppose Q(Z =
0) > 0. Let У satisfy (*) (we have seen that there exists such a Y and that
it hence is in Ti). Further we take for our set A in (*) the set A = {Z = 0},
for which we are assuming Q(A) > 0. Since 0 G /3 and 0 G B+, we have from
Lemma 2 that
} = E{Yl{z=0}}.
Since each of Y and Z are in 7i, their sum is in 7i as well. But then the above
implies
Q{Y + Z = 0} = Q{Z = 0} - Q({Z = 0} П {Y > 0}) < Q(Z = 0).
This, then, is a contradiction, since Q(Z = 0) is minimal for Z G 7i. Therefore
we conclude Z > 0 a.s., and since Z G B+, it is bounded as well, and Lemma 3
is proved; thus also, Theorem 43 is proved. D
We state again, for emphasis, that Theorems 26 and 43 together allow
us to conclude that semimartingales (as we have defined them) and classical
semimartingales are the same.

Exercises for Chapter III 147
Bibliographic Notes
The material of Chap. Ill comprises a large part of the core of the "general
theory of processes" as presented, for example in Dellacherie [41], or alterna-
alternatively Dellacherie and Meyer [45, 46, 44]. We have tried once again to keep the
proofs non-technical, but instead of relying on the concept of a natural process
as we did in the first edition, we have used the approach of R. Bass [11], which
uses the P. A. Meyer classification of stopping times and Doob's quadratic in-
inequality to prove the Doob-Meyer Theorem (Theorem 13), the key result of
the whole theory.
The Doob decomposition is from Doob [55], and the Doob-Meyer decom-
decomposition (Theorem 13) is originally due to Meyer [163, 164]. The theory of
quasimartingales was developed by Fisk [73], Orey [187], К. М. Rao [207],
Strieker [218], and Metivier-Pellaumail [159].
The treatment of compensators is new to this edition. The simple example
of the compensator of a process with one jump of size one (Theorem 17), dates
back to 1970 with the now classic paper of Dellacherie [40]. The case of many
jumps (Theorem 19) is due to C. S. Chou and P. A. Meyer [30], and can be
found in many texts on point processes, such as [24] or [139]. The example of
hazard rates comes from Fleming and Harrington [74].
The Fundamental Theorem of Local Martingales is due to J. A. Yan and
appears in an article of Meyer [172]; it was also proved independently by
Doleans-Dade [51]. Le Jan's Theorem is from [112].
The notion of special semimartingales and canonical decompositions is due
to Meyer [171]; see also Yoeurp [234]. The Girsanov-Meyer theorems (The-
(Theorems 35 and 36) trace their origin to the 1954 work of Maruyama [152],
followed by Girsanov [83], who considered the Brownian case only. The two
versions presented here are due to Meyer [171], and the cases where the two
measures are not equivalent (Theorems 37 and 38) are due to Lenglart [142].
The example from finance theory (starting on page 137) is inspired by [203].
Kazamaki's criterion was published originally in 1977 [124]; see also [125],
whereas Novikov's condition dates to 1972 [184].
The Bichteler-Dellacherie Theorem (Theorem 43) is due independently to
Bichteler [13, 14] and Dellacherie [42]. It was proved in the late 1970's, but
the first time it appeared in print was when J. Jacod included it in his 1979
tome [103]. Many people have made contributions to this theorem, which had
at least some of its origins in the works of Metivier-Pellaumail, Mokobodzki,
Nikishin, Letta, and Lenglart. Our treatment was inspired by Meyer [176] and
by Yan [233].
Exercises for Chapter III
Exercise 1. Show that the maximum and the minimum of a finite number
of predictable stopping times is still a predictable stopping time.

148 Exercises for Chapter III
Exercise 2. Let S be a totally inaccessible stopping time, and let T = S + 1.
Show that Г is a predictable stopping time.
Exercise 3. Let S, T be predictable stopping times. Let Л = {S = T}. Show
that Sa. is a predictable stopping time.
Exercise 4. Let (Q,T, F, P) be a filtered probability space satisfying the
usual hypotheses. Show that the predictable cr-algebra (on K+ x Q) is con-
contained in the optional cr-algebra. (Hint: Show that a cadlag, adapted process
can be approximated by processes in L.)
Exercise 5. Let S, T be stopping times with S < T. Show that (S, T] =
{{t,u) : S(u) < t < T(u)} is a predictable set. Show further that [S,T)
and (S,T) are optional sets. Last, show that if T is predictable, then (S, T)
is a predictable set, and if both S and T are predictable, then \S,T) is a
predictable set.
Exercise 6. Let Г be a predictable stopping time, and let (Sln)n>i be a
sequence of stopping times announcing T. Show that Тт- — Vn Fsn-
Exercise 7. A filtration F is called quasi left continuous if for every pre-
predictable stopping time T one has Тт = -Яг--7 Show that if F is a quasi left
continuous filtration, then whenever one has a non-decreasing sequence of
stopping times (Sn)n>\, with ИгПп^оо Sn = T, it follows that Тт = Vn ^~sn-
(Note: One can have a quasi left continuous filtration such that there exists
a stopping time T with Тт- ф J~t-)
Exercise 8. Show that the natural completed filtration of a Levy process is
a quasi left continuous filtration.
*Exercise 9. Let X be a semimartingale with E{[X, X]^} < oo and suppose
X has a decomposition X = Xq -f M + A, with A predictable. Show that
E{[A, A]oo} < ?{[X,X]oo}. (Hint: Recall that if T is a predictable stopping
time, then
ДЛт1{т<оо} =?{ДХг1{т<оо}|Яг-})
and also that the set {(t,u>) : AAt(u>) Ф 0} is a countable disjoint union of
sets of the form {(t,uj) : Т»(ш) = <}, where the stopping times (Tj)j>j are
predictable.)
Exercise 10. Let X and Y be semimartingales with the processes [X, X] and
[Y, Y] locally integrable. Prove the following "sharp bracket" version of the
Kunita-Watanabe inequality, for jointly measurable processes H, K:
f \HsKs\\d(X,Y)s\ < ( I H2sd(X,X)s)^( f K2sd(Y,Y)s)i
Jo Jo Jo
a.s.
(Warning: Recall that (X,Y) can exist as the compensator of [X,Y] even
when (X,X) and (Y,Y) do not exist.)
7 A quasi left continuous filtration is formally defined in Chap. IV on page 189.

Exercises for Chapter III 149
Exercise 11. Show that if A is a predictable finite variation process, then it
is of locally integrable variation. That is, show that there exists a sequence of
stopping times (Tn)n>i such that E{J0 " |cL4s|} < oo.
Exercise 12. Let TV be a counting process with its minimal completed fil-
filtration, with independent interarrival times. Show that the compensator A of
TV has absolutely continuous paths if and only if the cumulative distribution
functions of the interarrival times are absolutely continuous.
Exercise 13. Let T be an exponential random variable and Nt = l{t>T}-
Let To = cr{T} and suppose the filtration F is constant: Ft — Ро for all
t > 0. Show that the compensator A of TV is At = 1{<>т}- (This illustrates
the importance of the filtration when calculating the compensator.)
Exercise 14. Let TV be a compound Poisson process, Nt = ^Zj>i Uil{t>Ti}
where the times Tj are the arrival times of a standard Poisson process with
parameter A and the ?/, are i.i.d. and independent of the arrival times. Suppose
JS{|?/j|} < °° and E{Ui} — ц. Show that the compensator A of TV for the
natural filtration is given by At = A//i.
Exercise 15. Show that if T is exponential with parameter A, its hazard rate
is constant and equal to A. Show also that if R is Weibull with parameters
{a, 13), then its hazard rate is \(t) = aft"?*-1.
Exercise 16. Let T be exponential with parameter A and have joint distri-
distribution with U given by P(T > t,U > s) = exp{-At - /xs - 6ts} for t > 0,
s > 0, where A, //, and в are all positive constants and also в < A//. Show that
the crude hazard rate of (T, U) is given by A* (t) = A + 0t.
*Exercise 17. Let M be a martingale on a filtered space where the filtra-
filtration is quasi left continuous. Show that (M,M) is continuous. (Hint: See the
discussion on quasi left continuous filtrations on page 189 of Chap. IV.)
Exercise 18. Let X be a semimartingale such that the process Dt =
sups<t|AX,| is locally integrable. Show that every decomposition of X,
X = Xo + M + C, has that the process С is of locally integrable variation.
Exercise 19. Suppose that X is locally special, that is, there exists a
sequence (Tn)n>i of stopping times increasing to oo a.s. such that XTn =
Mn + An, with An a predictable finite variation process, for each n. Show
that X is special. (That is, о semimartingale which is locally special is also
special.)
Exercise 20. Prove the Lemma on page 132. Let Q ~ P, and Zt =
Ep I jp \Ft \ ¦ Show that an adapted, cadlag process M is a Q local martingale
if and only if MZ is a P local martingale.
*Exercise 21. Let F с G be filtrations satisfying the usual hypotheses, and
let X be a G quasimartingale. Suppose that Yj = E{Xt\J-t}, and that Y can
be taken cadlag (this can be proved to be true, although it is a little hard to

150 Exercises for Chapter III
prove). Show that Y is also a quasimartingale for the filtration F. (Hint: Use
Rao's Theorem.)
*Exercise 22. Suppose that A is a predictable finite variation process with
?{[A, A\1} < oo, and that M is a bounded martingale. Show that [A, M] is
a uniformly integrable martingale.
Exercise 23. Let Г be a strictly positive random variable, and let Ft =
a{T Л s; s < t}. Show that (J-t)t>o is the smallest filtration making T a
stopping time.
Exercise 24. Let Z be a cadlag supermartingale of Class D with Zq = 0 and
suppose for all predictable stopping times T one has ?^{А^т|Яг-} = О, a.s.
Show that if Z = M — A is the unique Doob-Meyer decomposition of Z, then
A has continuous paths almost surely.
Exercise 25. Let A be an increasing process of integrable variation, and let
Г be a predictable jump time of A such that Е{ААт\Тт-} = О. Then its
compensator A is continuous at T. (This exercise complements Theorem 21.)
*Exercise 26. A supermartingale Z is said to be regular if whenever a
sequence of stopping times (Tn)n>i increases to T, then limn_,00.E{iy7'Ti} =
E{Zt}- Let Z be a cadlag supermartingale of Class D with Doob-Meyer
decomposition Z = M — A. Show that A is continuous if and only if Z is
regular.
*Exercise 27 (approximation of the compensator by Laplacians). Let
Z be a cadlag positive supermartingale of Class D with \hat^ooE{Zt} — 0.
(Such a supermartingale is called a potential.) Let Z — M — A be its Doob-
Meyer decomposition and assume further that A is continuous. Define
At=\j {Zs - E{Zs+h\fs})ds
Show that for any stopping time T, lim/j_»o AT = At with convergence in L1.
*Exercise 28. Let Z be a cadlag positive supermartingale of Class D with
lim^oo E{Zt} = 0. Let Z = M — A be its Doob-Meyer decomposition. Let Ah
be as given in Exercise 27. Show that for any stopping time T, lim/i_,o AT —
At, but in this case the convergence is weak for L1; that is, the convergence
is in the topology cr(Ll,L°°).8
Exercise 29 (discrete Laplacian approximations). Let Z be a cadlag
positive supermartingale of Class D with lim^oo E{Zt} = 0. Let Z = M - A
be its Doob-Meyer decomposition and assume further that A is continuous.
Define
Xn converges to X in ^(Z/1,!/00) if Xn, X are in Ll and for any a.s. bounded
random variable Y, E(XnY) ->• E(XY).

Exercises for Chapter III 151
k=0
Show that limn^oo A1^ = Аж with convergence in L1.
Exercise 30. Use Meyer's Theorem (Theorem 4) to show that if X is a strong
(Markov) Feller process for its natural completed filtration FM, and if X has
continuous paths, then the filtration F^ has no totally inaccessible stopping
times. (This implies that the natural filtration of Brownian motion does not
have any totally inaccessible stopping times.)
*Exercise 31. Let (Q,^, F, P) be the standard Brownian space. Show that
the optional cr-algebra and the predictable cr-algebra coincide. (Hint: Use
Meyer's Theorem (Theorem 4) and Exercise 30.)
* Exercise 32. Let (fl,.F,F, P) be a filtered probability space satisfying
the usual hypotheses. Let X be a (not necessarily adapted) cadlag stochas-
stochastic process such that for A > 0, E{fQ°° e~xt\Xt\dt} < oo. Let R\(Xt) =
E{J0°° e~AsXt+sds|^rt}, the right continuous version. Show that
f*
Mx(t) = Rx{Xt) - RX(XO) + X,- XRx(Xs)ds
Jo
is an F martingale.
*Exercise 33 (Knight's compensator calculation method). Let X be
a cadlag semimartingale. In the framework of Exercise 32 suppose the limits
below exist both pathwise a.s. and are in L1, and are of finite variation in
finite time intervals:
At = lim A / (Xu - XRxXu)du, 0 < t.
A-*°° Jo
Show that X is a special semimartingale, and A is the predictable term in its
semimartingale decomposition.

IV
General Stochastic Integration and Local
Times
1 Introduction
We defined a semimartingale as a "good integrator" in Chap. II, and this led
naturally to defining the stochastic integral as a limit of sums. To express an
integral as a limit of sums requires some path smoothness of the integrands and
we limited our attention to processes in L, the space of adapted processes with
paths that are left continuous and have right limits. The space L is sufficient
to prove Ito's formula, the Girsanov-Meyer Theorem, and it also suffices in
some applications such as stochastic differential equations. But other uses,
such as martingale representation theory or local times, require a larger space
of integrands.
In this chapter we define stochastic integration for predictable processes.
Our extension from Chap. II is very roughly analogous to how the Lebesgue
integral extends the Riemann integral. We first define stochastic integration
for bounded, predictable processes and a subclass of semimartingales known
as H2. We then extend the definition to arbitrary semimartingales and to
locally bounded predictable integrands.
We also treat the issue of when a stochastic integral with respect to a
martingale or a local martingale is still a local martingale, which is not always
the case. In this respect we treat the subject of sigma martingales, which has
recently been shown to be important for the theory of mathematical finance.
2 Stochastic Integration for Predictable Integrands
In this section, we will weaken the restriction that an integrand H must be
in L. We will show our definition of stochastic integrals can be extended to a
class of predictably measurable integrands.
Throughout this section X will denote a semimartingale such that Xq = 0.
This is a convenience involving no loss of generality. If Y is any semimartin-
semimartingale we can set Yt = Yt — Yq, and if we have defined stochastic integrals for

154 IV General Stochastic Integration and Local Times
semimartingales that are zero at 0, we can next define
HsdYs + H0Y0.
f HsdYs= f\
Jo Jo
When Yo ф 0, recall that we write Jo HsdYs to denote integration on @, t],
and JQ HsdYs denotes integration on the closed interval [0,t].
We recall for convenience the definition of the predictable cr-algebra, al-
already defined in Chap. III.
Definition. The predictable cr-algebra V on K+ x Q is the smallest cr-
algebra making all processes in L measurable. That is, V — a{H : H G L}.
We let hV denote bounded processes that are V measurable.
Let X = M + A be a decomposition of a semimartingale X, with Xo —
Mo = Ao = 0. Here M is a local martingale and A is an FV process (such
a decomposition exists by the Bichteler-Dellacherie Theorem (Theorem 43
of Chap. Ill))- We will first consider special semimartingales. Recall that a
semimartingale X is called special if it has a decomposition
X = N + A
where TV is a local martingale and Л is a predictable FV process. This decom-
decomposition is unique by Theorem 30 in Chap. Ill and it is called the canonical
decomposition.
Definition. Let X be a special semimartingale with canonical decomposition
X = TV + A~. The 7i2 norm of X is defined to be
Jo
\dAs
The space of semimartingales 7i2 consists of all special semimartingales
with finite H2 norm.
In Chap. V we define an equivalent norm which we denote || • ||Я2.
Theorem 1. The space ofH2 semimartingales is a Banach space.
Proof. The space is clearly a normed linear space and it is easy to check
that || • ||W2 is a norm (recall that E{NX\ = E{[N,N]00}, and therefore
2
\\X||^2 = 0 implies that E{NOO} = 0 which implies, since N is a martingale,
that TV = 0). _
To show completeness we treat the terms TV and A separately. Consider
first TV. Since ^{TV^} = ||[TV,TV]to2|||2, it suffices to show that the space of
L2 martingales is complete. However an L2 martingale M can be identified
with Moo G L2, and thus the space is complete since L2 is complete.
Next suppose (An) is a Cauchy sequence of predictable FV processes in
|| • ||2 where \\A\\P = \\ /0°° |cL4s|||LP, p > 1. To show (An) converges it suffices

2 Stochastic Integration for Predictable Integrands 155
to show a subsequence converges. Therefore without loss of generality we can
assume ^2n ||A™||2 < oo.
Then Y An converges in || • ||j to a limit A. Moreover
lim У / \dA™\ =0
n>m
in L1 and is dominated in L2 by 5Zn Jo°° l^"l- Therefore J] An converges
to the limit A in || • Ц2 as well, and there is a subsequence converging almost
surely. To see that the limit A is predictable, note that since each term in
the sequence {An)n>\ is predictable, the limit A is the limit of predictably
measurable processes and hence also predictable. ?
For convenience we recall here the definition of L.
Definition. L (resp. bL) denotes the space of adapted processes with caglad1
(resp. bounded, caglad) paths.
We first establish a useful technical lemma.
Lemma. Let A be a predictable FV process, and let H be in L such that
E{jo°° \Hs\\dAs\} < 00. Then the FV process (/„* HadAs)t>0 is also pre-
predictable.
Proof. We need only to write the integral JQ HsdAs as the limit of Riemann
sums, each one of which is predictable, and which converge in ucp to Jo HsdAs,
showing that it too is predictable. ?
The results that follow will enable us to extend the class of stochastic
integrands from bL to bP, with X G 7i2 (and Xq — 0). First we observe that
if Я jE_bL_and X e H2, then the stochastic integral Я • X_ e W2._Also if
X = N + A is the canonical decomposition of X, then Я - N + H ¦ A is the
canonical decomposition of Я • X by the preceding lemma. Moreover,
\\H-X\\H2 = ||( ГH2d{N,N]s)i\\L2 + || Г \Hs\\dAs
Jo Jo
№¦
The key idea in extending our integral is to notice that [N,N] and A are
FV processes, and therefore oj-by-a; the integrals /0 H2(u>)d[N, N]s(u>) and
/0* |#s||cL4e| make sense for any Я G hV and not just Я G L.
Definition. Let X G H2 with X = N + A its canonical decomposition, and
let H, J e hV. We define dx(H, J) by
dx(H,J) = ||( Г(н8 - JS)MWW1/2IU* + II Г \н8 - J8\\dA8\\\La.
Jo Jo
1 "caglad" is the French acronym for left continuous with right limits.

156 IV General Stochastic Integration and Local Times
Theorem 2. For X G H2 the space bL is dense in bP under dx(-, ¦)•
Proof. We use the Monotone Class Theorem. Define
Л = {H G bP : for any e > 0, there exists J G bL such that dx(#, •/) <
Trivially Л contains bL. If Я™ G Л and Я™ increases to Я with H bounded,
then Я G bP, and by the Dominated Convergence Theorem if S > 0 then for
some N(S), n > iV(<J) implies dx(H,Hn) < S. Since each Я™ G Л, we choose
no > JV(($) and there exists J G bL such that dx(J, Hn°) < 8. Therefore given
e > 0, by taking S = e/2 we can find J G bL such that dx (J, H) < e, and
therefore ЯбЛ An application of the Monotone Class Theorem yields the
result. D
Theorem 3. Let X G H2 and Hn G bL such that Hn is Cauchy under dX-
Then Hn -X is Cauchy in H2.
Proof. Since \\Hn ¦ X -Hm ¦ X\\H2 = dx(Hn,Hm), the theorem is immediate.
?
Theorem 4. Let X G П2 and H G bP. Suppose Hn G bL and Jm G bL
are two sequences such that limn dx(Hn,H) = limm dx(Jm, H) = 0. Then
Hn ¦ X and Jm ¦ X tend to the same limit in 7i2.
Proof. Let Y = limn^oo Hn ¦ X and Z = limm-xx, Jm ¦ X, where the limits
are taken in Ti2. For e > 0, by taking n and m large enough we have
\\Y - Z\\H2 < \\Y - Hn ¦ X\\H2 + \\Hn -X-Jm- X\\H2 + \\Jm ¦ X - Z\\H2
<2e+\\Hn-X-Jm-X\\H2
<2e + dx(Hn,Jm)
<2e + dx(Hn,H) + dx(H,Jm)
<4?,
and the result follows. ?
We are now in a position to define the stochastic integral for H G bV (and
X G 7i2).
Definition. Let X be a semimartingale in H2 and let Я G bP. Let Hn G bL
be such that limn^oo dx(Hn, H) = 0. The stochastic integral H ¦ X is the
(unique) semimartingale Y G H2 such that Нт^^Я™ • X = Y in 7i2. We
write H-X = {J* HsdXs)t>0.
We have defined our stochastic integral for predictable integrands and
semimartingales in H2 as limits of our (previously defined) stochastic integrals.
In order to investigate the properties of this more general integral, we need to
have approximations converging uniformly. The next theorem and its corollary
give us this.

2 Stochastic Integration for Predictable Integrands 157
Theorem 5. Let X be a semimartingale in 7i2. Then
E{(Snp\Xt\J}<S\\X\\2n2.
t
Proof. For a process H, let H* = supt \Ht\. Let X = N + Abe the canonical
decomposition of X. Then
/»oo
X* <N*+ / \dls\.
Jo
Doob's maximal quadratic inequality (Theorem 20 of Chap. I) yields
and using (a + bJ < 2a2 + 2b2 we have
E{(X*J} < 2E{(N*J} + 2E{( / \dAs\J}
Jo
POO
<8E{[N,N]oo} + 2\\ \dAs\\\h
Jo
<8\\Х\\2Ю. П
Corollairy. Let (Xn) be a sequence of semimartingales converging to X in
7i2. Then there exists a subsequence (n^) such that limnk^oo(Xnk — X)* = 0
a.s.
Proof. By Theorem 5 we know that (Xn - X)* = supt \X™ — Xt\ converges
to 0 in L2. Therefore there exists a subsequence converging a.s. ?
We next investigate some of the properties of this generalized stochastic
integral. Almost all of the properties established in Chap. II (Sect. 5) still
hold.2
Theorem 6. Let X,Y etf and H, К е hV. Then
and
Proof. One need only check that it is possible to take a sequence Hn e bL
that approximates H in both dx and dy. ?
Theorem 7. Let T be a stopping time. Then (H ¦ X)T — .fflrox] " X =
Т
2 Indeed, it is an open question whether or not Theorem 16 of Chap. II extends to
integrands in bV. See the discussion at the end of this section.

158 IV General Stochastic Integration and Local Times
Proof. Note that 1[0,т] e bL, so #1[о,т] e bP. Also, XT is clearly still in W2.
Since we know this result is true for H e bL (Theorem 12 of Chap. II), the
result follows by uniform approximation, using the corollary of Theorem 5. ?
Theorem 8. The jump process (A(H ¦ X)s)s>o is indistinguishable from
(Hs(AXs))s>0-
Proof. Recall that for a process J, AJt — Jt — Jt-, the jump of J at time
t. (Note that H ¦ X and X are cadlag semimartingales, so Theorem 8 makes
sense.) By Theorem 13 of Chap. II we know the result is true for Я е bL.
Let H e hV, and let Hn e bL such that lim^oo d* (#n, #) = 0. By the
corollary of Theorem 5, there exists a subsequence (nk) such that
lim (Hnk -X-H-Xy =0 a.s.
This implies that, considered as processes,
lim А(НПк -Х)=А(Н-Х),
outside of an evanescent set.3 Since each Hnk e bL, we have A(H"k ¦ X) =
H"k(AX), outside of another evanescent set. Combining these, we have
lim Hn"(AX)l{AX^0} = lim A(Hn" ¦ X)l{Axm
= A(H ¦
and therefore
In particular, the above implies that lim^^oo H"k (ш) exists for all (t,w) in
{AX ф 0}, a.s. We next form
А = {ш: there exists t>0 such that lim Н"к{ш)фНг{ш) and АХг(ш)ф0\.
Bl-tOO
Suppose P(A) > 0. Then
dx(Hnk,H)>\\lA{f {H™k-HsJd{ ]T (ANUJ)}\\L2
^° 0<u<s . .
,00 (*)
+ ||1л{/ \H?*-He\d( Yl
0<u<
and if AXS ф 0, then |ДЖ,| + \AAS\ > 0. The left side of (*) tends to 0
as nk -» со, and the right side of (*) does not. Therefore P(A) — 0, and we
conclude A(H ¦ X) = limnfc^oo HnkAX = HAX. D
3 A set Л С R+ x П is evanescent if 1л is a process that is indistinguishable from
the zero process.

2 Stochastic Integration for Predictable Integrands 159
Corollary. Let X e 7i2, H e bV, and T a finite stopping time. Then
Proof. By Theorem 8, (H ¦ X)T~ = (H ¦ X)T - НтАХт1{г>т}- On the other
hand, XT~ = XT - AXrl{t>T}- Let At = AXTl{t>T}- By the bilinearity
(Theorem 6), H ¦ (XT~) = H ¦ (XT) -HA. Since Я • (XT) = (H ¦ X)T by
Theorem 7, and НА — НтАХт1{г>т], the result follows. G
The next three theorems all involve the same simple proofs. The result is
known to be true for processes in bL; let (Hn) € bL approximate H € hV in
dx(•,•)> and by the corollary of Theorem 5 let n^ be a subsequence such that
lim (НПк -Х-Н-ХУ=О a.s.
Then use the uniform convergence to obtain the desired result. We state these
theorems, therefore, without proofs.
Theorem 9. Let X e 7i2 have paths of finite variation on compacts, and
H e hV. Then H ¦ X agrees with a path-by-path Lebesgue-Stieltjes integral.
Theorem 10 (Associativity). Let X e H2 and H,K ? bV. Then К ¦ X 6
H2 and H-{K-X) = (HK) ¦ X.
Theorem 11. Let X € 7i2 be a (square integrable) martingale, and H € hV.
Then H ¦ X is a square integrable martingale.
Theorem 12. Let X,Y etf and H,K e bV. Then
[H-X,K-Y]t= j HsKsd{X,Y}s, (t > 0),
and in particular
[H ¦ X,H ¦ X]t = [ H2d[X,X}3, (t > 0).
Jo
Proof. As in the proof of Theorem 29 of Chap. II, it suffices to show
[H-X,Y]t= fHsd[X,Y]s.
Jo
Let (Яп) G bL such that lim^oo dx(Hn, H) = 0. Let Tm = inf{? > 0 : \Yt\ >
m}. Then (T) are stopping times increasing to со a.s. and |yj"* | < m.4 Since
it suffices to show the result holds on [0, Tm), each m, we can assume without
loss of generality that У_ is in bL. Moreover, the Dominated Convergence
Theorem gives limn^oodx(HnY-,HY-) = 0. By Theorem 29 of Chap. II, we
have
Recall that Y_ denotes the left continuous version of Y.

160 IV General Stochastic Integration and Local Times
[Hn-X,Y]t = f H?d[X,Y]a (t>0),
Jo
and again by dominated convergence
lim [Hn -X,Y} = [ Hsd[X,Y}s (t > 0).
It remains only to show lim^ooftf" • X,Y] = [H ¦ X,Y]. Let Zn = Hn ¦ X,
and let nk be a subsequence such that limnk^oo(Z"k — Z)* = 0 a.s., where
Z — H ¦ X (by the corollary to Theorem 5). Integration by parts yields
[Zn\Y] = ZnkY - (У_) • Znk - [Zn_k)-Y
= ZnkY - (Y_Hnk) ¦ X - {Z"Lk) ¦ У,
where we have used associativity (Theorem 10). We take limits so that
lim [Znk,Y] = ZY -Y- ¦ (H ¦ X) - Z- -Y
= ZY - У_ ¦ (Z) - Z_ • У
= {Z,Y} = [H-X,Y]. а
At this point the reader may wonder how to calculate in practice a canon-
canonical decomposition of a semimartingale X in order to verify that X € H2.
Fortunately Theorem 13 will show that H2 is merely a mathematical conve-
convenience.
Lemma. Let A be an FV process with Ao — 0 and /0°° |cL4s| el2. Then
A e П2. Moreover \\A\\H2 < 6|| /0°° |di4,|||L2.
Proof. If we can prove the result for A increasing then the general result will
follow by decomposing A = A+ — A~. Therefore we assume without loss of
generality that A is increasing. Hence as we noted in Sect. 5 of Chap. Ill, the
compensator Л of Л is also increasing and E{Aoo} = ^{Лоо} < oo.
Let M be a martingale bounded by a constant k. Since A - A is a local
martingale, Corollary 2 to Theorem 27 of Chap. II shows that
L = M{A -A)-[M,A- A]
is a local martingale. Moreover
sup \Lt\ < к(Аоо + Лоо) +2к^ \&(А ~ А)*\
S
< ЗЛКД» + 4») eL1.
Therefore L is a uniformly integrable martingale (Theorem 51 of Chap. I) and
Е{ЬЖ} = E{L0} = 0. Hence

2 Stochastic Integration for Predictable Integrands 161
A - Л)оо} = E{[M,A- i]oc}
= E{[M,A]0O}-E{[M,A]0O}
= E{[M,A}00],
because A is natural. By the Kunita-Watanabe inequality (the corollary to
Theorem 25 of Chap. II)
E{\[M,AU\} <
к
where the second inequality uses 2ab < a2 +b2. However
E^M^U) = E{Ml)
(Corollary 4 of Theorem 27 of Chap. II) and also [А, А}^ < А2^ a.s. Therefore
Since M is an arbitrary bounded martingale we are free to choose
Moo = (A - А)о
and we obtain
and using the Monotone Convergence Theorem we conclude
??{(Л-Л&}<Я{^}.
Consequently
E{Al} < 2E{Al} + 2E{(A - 1I} < ЩА^} < oo,
and A — A is a square integrable martingale, and
\\A\\w = \\(A- lUlU» + \\Aooh' < ЗЦЛооЦ^
for A increasing. ?
Remarks. The constant 6 can be improved to l+\/8 < 4 by not decomposing
A into A+ and A~. This lemma can also be proved using the Burkholder-
Gundy inequalities (see Meyer [171, page 347]).

162 IV General Stochastic Integration and Local Times
In Chap. V we use an alternative norm for semimartingales which we
denote || • ||я_р, 1 < p < со. The preceding lemma shows that the norms
|| ¦ ||И2 and || • \\gi are equivalent.
The restrictions of integrands to ЪТ> and semimartingales to H2 are
mathematically convenient but not necessary. A standard method of re-
relaxing such hypothesis is to consider cases where they hold locally. Recall
from Sect. 6 of Chap. I that a property тг is said to hold locally for
a process X if there exists a sequence of stopping times (Tn)n>o such that
0 = T° < T1 < T2 < ¦ ¦ ¦ < Tn < ¦ ¦ ¦ and lim^oo Tn = со a7s., and such
that Хг1{Г»>0} has property тг for each n. Since we are assuming our semi-
semimartingales X satisfy Xo = 0, we could as well require only that XT" has
property тг for each n. A related condition is that a property hold prelocally.
Definition. A property тг is said to hold prelocally for a process X with
Xo = 0 if there exists a sequence of stopping times (Tn)n>i increasing to со
a.s. such that XT"~ has property тг for each n > 1.
Recall that XT~ = Xtl{o<t<T} +^T-l{t>T}- The next theorem shows
that the restriction of semimartingales to H2 is not really a restriction at all.
Theorem 13. Let X be a semimartingale, Xq — 0. Then X is prelocally in
7i2. That is, there exists a non-decreasing sequence of stopping times {Tn),
ooT™ = со a.s., such that XT"~ e H2 for each n.
Proof. Recall that XT"~ = Xtl[OtTn) + ^т"-1[Т",оо)- By the Bichteler-
Dellacherie Theorem (Theorem 43 of Chap. Ill) we can write X = M + A,
where M is a local martingale and A is an FV process. By Theorem 25 of
Chap. Ill we can further take M to have bounded jumps. Let C be the bound
for the jumps of M. We define
Tn = inf{t > 0 : [M,M]t > n or / \dA3\ > n}
Jo
and let Y = XT"~. Then Y has bounded jumps and hence it is a special
semimartingale (Theorem 31 of Chap. III). Moreover
Y = XTn~ = MT" + ATn~ - (ДМГI[Т,]00),
C,
where L = MT" and С = Ат"~ - (ДМт»I[т«)Оо)- Then [L, L] < n + 01, so
L is a martingale in Ti2 (Corollary 4 to Theorem 27 of Chap. II), and also
/•OO
/ \dCs\ <n+ |ДМт«| < 2n + C,
Jo
hence С e H2 by the lemma. Therefore XT"~ = L + С € V2. ?

2 Stochastic Integration for Predictable Integrands 163
We are now in a position to define the stochastic integral for an arbi-
arbitrary semimartingale, as well as for predictable processes which need not be
bounded.
Let X be a semimartingale in H2. To define a stochastic integral for pre-
predictable processes H which are not necessarily bounded (written Я е ?), we
approximate them with Hn e bV.
Definition. Let X e H2 with canonical decomposition X = N + A. We say
Я € V is (H2,X) integrable if
/*oo /*
E{ H2d[N,N}s} + E{(
Jo Jo
< CO.
Theorem 14. Let X be a semimartingale and let H e V be (Ti2,X) inte-
integrable. Let H" = Я1{|я|<п} € ЬР. Then H" ¦ X is a Cauchy sequence in
H2.
Proof. Since H" e bV, each n, the stochastic integrals H" ¦ X are defined.
Note also that итп^юЯ" = H and that \Hn\ < \H\, each n. Then
\\Hn-X-Hm-X\\H2 =dx(Hn,Hm)
/•OO /*OO
= ||( / {Щ - H™Jd\N,-N\sYi2\\L. +1| / |#; - H?\\dAe\\\L,,
Jo Jo
and the result follows by two applications of the Dominated Convergence
Theorem. ?
Definition. Let X be a semimartingale in H2, and let H e V be (H2,X)
integrable. The stochastic integral Я • X is defined to be limn^oo Hn ¦ X,
with convergence in H2, where Я™ = #
Note that Я • X in the preceding definition exists by Theorem 14. We
can "localize" the above theorem by allowing both more general Я е V and
arbitrary semimartingales with the next definition.
Definition. Let X be a semimartingale and И е V. The stochastic integral
И X is said to exist if there exists a sequence of stopping times T" increasing
to со a.s. such that XT"~ e H2, each n > 1, and such that H is (W2,XT"~)
integrable for each n. In this case we say H is X integrable, written Я е
L(X), and we define the stochastic integral by
H ¦ X = H ¦ (Хтп~), on[0,Tn),
each n.
Note that if m > n then

164 IV General Stochastic Integration and Local Times
where Hk = Hl{\H\<k}, by the corollary of Theorem 8. Hence taking limits we
have H ¦(XT"l^)T"~ = H-(XT"~), and the stochastic integral is well-defined
for Я € L{X). Moreover let R* be another sequence of stopping times such
that XRl~ e H2 and such that Я is (H2,XRe~) integrable, for each I. Again
using the corollary of Theorem 8 combined with taking limits we see that
on [0, Rl Л T"), each ? > 1 and n > 1. Thus in this sense the definition of
the stochastic integral does not depend on the particular sequence of stopping
times.
If Я e hV (i.e., Я is bounded), then Я € L(X) for all semimartingales
X, since every semimartingale is prelocally in Ti2 by Theorem 13.
Definition. A process Я is said to be locally bounded if there exists a
sequence of stopping times (Sm)m>i increasing to со a.s. such that for each
m > 1, (HtAS™l{sm>o})t>o is bounded.
Note that any process in L is locally bounded. The next example is suffi-
sufficiently important that we state it as a theorem.
Theorem 15. Let X be a semimartingale and let H &V be locally bounded.
Then H e L(X). That is, the stochastic integral H ¦ X exists.
Proof. Let (Sm)m>i, (Tn)n>i be two sequences of stopping times, each in-
increasing to со a.s., such that Hsml{sm>o} is bounded for each m, and
XT"- E П2 for each n. Define Rn = minE",T"). Then Я = Ял{Лп>0}
on @,Rn) and hence it is bounded there. Since XR"~ charges only @,Rn),
we have that H is (H2,XRn~) integrable for each n > 1. Therefore using the
sequence Rn which increases to oo a.s., we are done. ?
We now turn our attention to the properties of this more general integral.
Many of the properties are simple extensions of earlier theorems and we omit
their proofs. Note that trivially the stochastic integral Я • X, for H e L(X),
is also a semimartingale.
Theorem 16. Let X be a semimartingale and let H,j? L{X). Then aH +
PJ E L{X) and {aH + pj) ¦ X = aH ¦ X + pj ¦ X. That is, L{X) is a linear
space.
Proof. Let (Rm) and (Tn) be sequences of stopping times such that Я is
(H2,XRm~) integrable, each m, and J is (H2,XT"~) integrable, each n. Tak-
Taking Sn = Rn ЛГ, it is easy to check that aH + pj is (H2,XS"~) integrable
for each n. ?
Theorem 17. Let X,Y be semimartingales and suppose H e L{X) and
H E L(Y). Then H e L(X + Y) and H ¦ (X + Y) = H ¦ X + H Y.

2 Stochastic Integration for Predictable Integrands 165
Theorem 18. Let X be a semimartingale and Я 6 L{X). The jump process
(Д(Я ¦ X)s)s>o is indistinguishable from (Hs(AXs))s>o.
Theorem 19. LetT be a stopping time, X a semimartingale, and H € L(X).
Then
(Я ¦ Xf = Я1[0>Т] • X = Я ¦ (XT).
Moreover, letting со— equal со, we have moreover
(Н-Х)Т-=Н-(ХТ~).
Theorem 20. . Let X be a semimartingale with paths of finite variation on
compacts. Let Я € L(X) be such that the Stieltjes integral Jo \Hs\\dXs\ exists
a.s., each t > 0. Then the stochastic integral H ¦ X agrees with a path-by-path
Stieltjes integral.
Theorem 21 (Associativity). Let X be a semimartingale with К б L(X).
Then H e L(K ¦ X) if and only if HK e L(X), in which case H ¦ (K ¦ X) =
{HK) ¦ X.
Theorem 22. Let X,Y be semimartingales and let H e L(X), К е L(Y).
Then
[H-X,K-Y}t= f HsKsd[X, Y]s (t > 0).
Jo
Note that in Theorem 22 since Я • X and Я • Y are semimartingales, the
quadratic covariation exists and the content of the theorem is the formula.
Indeed, Theorem 22 gives a necessary condition for Я to be in L(X), namely
that /0* H^d[X, X]s exists and is finite for all t > 0. The next theorem (The-
(Theorem 23) is a special case of Theorem 25, but we include it because of the
simplicity of its proof.
Theorem 23. Let X be a semimartingale, let H € L(X), and suppose Q is
another probability with Q -C P. If Hq ¦ X exists, it is Q indistinguishable
from Hp ¦ X.
Proof. Hq ¦ X denotes the stochastic integral computed under Q. By Theo-
Theorem 14 of Chap. II, we know that Hq ¦ X = Hp ¦ X for Я е L, and therefore
if X e H2 for both P and Q, they are equal for Я G hV by the corollary
of Theorem 5. Let (Re)e>i, (Tn)n>i be two sequences of stopping times in-
increasing to со a.s. such that Я is {7i2,XR'~) integrable under Q, and Я is
(П2,ХТ"~) integrable under P, each ? and n. Let Sm = RmATm, so that Я
is (H2, Xsm~) integrable under both P and Q. Then Я ¦ X = lim^oo Я™ • X
on [0,Sm) in both dx{P) and dx{Q), where Hn = Я1{|Я|<п} е hV. Since
H'p ¦ X — Hq ¦ X, each n, the limits are also equal. D
Much more than Theorem 23 is indeed true, as we will see in Theorem 25,
which contains Theorem 23 as a special case. We need several preliminary
results.

166 IV General Stochastic Integration and Local Times
Lemma. Let X e Ti2 and X = N + A be its canonical decomposition. Then
E{[N,NU} <E{{X,XU}.
Proof. First observe that
[X,X} = [N,N} + 2{N,A] + iA,A\.
It suffices to show E{[N, А}^} = 0, since then
?{[ЛГ,ЛПоо} = E{[X,XU - p, Л]»},
and the result follows since [A, A}oo > 0. Note that
??{|[ЛГ,Л]оо|} < (^[Ж^]оо}I/2(Я{[ЗДоо}I/2 < oo,
by the Kunita-Watanabe inequalities. Also -Е{[М,Л]оо} = О for all bounded
martingales because A is natural. Since bounded martingales are dense in
the space of L2 martingales, there exists a sequence (Mn)n>i of bounded
martingales such that lim^oo E{[Mn -77, Mn-Щ^} = О. Again using the
Kunita-Watanabe inequalities we have
E{\[N - Mn Ди} < (E{{N - Mn,N - МП}ОО}I'2{Е{\А,А~}ОО}I'2
and therefore lim^oo ^{[M™^]^} = .Е{[77Д]оо}. Since E{[Mn, A}^} = 0,
each n, it follows that ?{[iV, Л]^} = 0. П
Note that in the preceding proof we established the useful equality
E{[X,X]t} = E{[~N,TV}t} + E_\\A^A\t} for a semimartingale X e H2 with
canonical decomposition X = N + A.
Theorem 24. For X a semimartingale in Ti2,
\\X\\n*< sup \\(H ¦ Xy^L* +2\\[X,X}V2\\L* <5\\X\\n2.
|Я|1
Proof. By Theorem 5 for |tf | < 1
\\(H ¦ X)*J\Li < V8\\H ¦ X\\H* < V8\\X\\Hi.
Since
2\\[X,X]1J12\\li <2||[M,M]V2||l2+2|| Г \dAe\\\L,
Jo
where X — M + A is the canonical decomposition of X, we have the right
inequality.
For the left inequality we have \\{М,М}^2\\^ < ||pf,X]^2||L2 by the
lemma preceding this theorem. Moreover if \H\ < 1, then

2 Stochastic Integration for Predictable Integrands 167
\\(H-A)t\\La<\\(H-X)t\\L2
<\\{h ¦ xy^Wv
Next take Я — щ; this exists as a predictable process since A is predictable,
and therefore A predictable implies that |A|, the total variation process, is also
predictable. Consequently we have
/¦O
| /
Jo
. a
We present as a corollary to Theorem 24, the equivalence of the two
pseudonorms sup^^ \\(H ¦ -X")Soll-c2 an<^ ll^llw2- We will not have need of
this corollary in this book,5 but it is a pretty and useful result nevertheless.
It is originally due to Yor [241], and it is actually true for all p, 1 < p < oo.
(See also Dellacherie-Meyer [46, pages 303-305].)
Corollary. For a semimartingale X (with Xo = 0),
and in particular sup^^j ||(# • X)Io\\l2 < oo if and only if ||X||W2 < oo.
Proof. By Theorem 5 if \\X\\ns < oo and |#| < 1 we have
• X\\H2 < VS\\X\\H2.
Thus we need to show only the left inequality. By Theorem 24 it will suffice
to show that
\\[X,X}H2\\L2< sup
||
for a semimartingale X with Xo = 0.
To this end fix a * > 0 and let 0 = To < Тг < ¦ ¦ ¦ < Tn = t be a random
partition of [0,t]. Choose ei,... ,en non-random and each equal to 1 or —1
and let H = ^"=1 ?»1(Т;_1,т<]- Then Я is a simple predictable process and
Let a — sup^^j ||(Я • X)^o\\l2. We then have
5 However the corollary does give some insight into the relationship between The-
Theorems 12 and 14 in Chap. V.

168 IV General Stochastic Integration and Local Times
a2 > E{(H
2 > E{(H ¦ X)l} = V
If we next average over all sequences E\,... ,en taking values in the space
{±1}", we deduce
a2
Next let am = {27™} be a sequence of random partitions of [0,t] tending to
the identity. Then Х^(-^т™ —XTr^ J converges in probability to [X,X]t. Let
{mk} be a subsequence so that 5Zi(^Tmfc ~ XT™kJ converges to [X, X]t a.s.
Finally by Fatou's Lemma we have
E{[X,X]t} < lirnmf E{J2(XT^k - Хт^J} < а2.
г
Letting t tend to oo we conclude that
E{[X,XU}<a2= sup
||
It remains to show that if sup^j^j \\(H ¦ -X')?o||/,2 < oo, then X s H2. We
will show the contrapositive. If X ? H2, then sup^^ \\(H ¦ ^)^o||l2 = oo.
Indeed, let Tn be stopping times increasing to oo such that XT"~ is in Ti2
for each n (c?, Theorem 13). Then
\\Хтп-\\П2<3 sup
Letting n tend to oo gives the result. ?
Before proving Theorem 25 we need two technical lemmas.
Lemma 1. Let A be a non-negative increasing FV process and let Z be a
positive uniformly integrable martingale. Let T be a stopping time such that
A = AT~ (that is, A^ = At-) and let к be a constant such that Z < к on
[0,T). Then
Z^} <
Proof. Since A0- — Zo- = 0, by integration by parts
AtZt = I As_dZs + I Zs-dAs + [A,Z]t
Jo Jo
f As_dZs + f
Jo Jo
= I As_dZs+ f ZsdAs
J jU/Ag

2 Stochastic Integration for Predictable Integrands 169
where the second integral in the preceding is a path-by-path Stieltjes integral.
Let Rn be stopping times increasing to oo a.s. that reduce the local martingale
(Jo As-dZs)t>o- Since dAs charges only [О, Г) we have
ft rt
kdA,
/ ZsdAs <
Jo Jo
for every t > 0. Therefore
E{(AZ)Rn} = E{ / As.dZs} + E{ / ZsdAs}
Jo Jo
The result follows by Fatou's Lemma. D
Lemma 2. Let X be a semimartingale with Xo — 0, let Q be another prob-
probability with Q -C P, and let Zt = Ep{^\!Ft\- If Г is a stopping time such
that Zt < к on [0, Г) for a constant k, then
\\XT-\\H2{Q) <5Vk\\XT-\\H2{P).
Proof. By Theorem 24 we have
\\XT-\\n2(Q) < sup EQ{((H ¦ XT-)*J}i +2EQ{[XT-,XT-\i}
< sup EP I -j^-((H ¦ XT~)*
sup EP{((H ¦ XT-)*)-
where we have used Lemma 1 on both terms to obtain the last inequality
above. The result follows by the right inequality of Theorem 24. ?
Note in particular that an important consequence of Lemma 2 is that if
Q < P with ^ bounded, then X <s H2(P) implies that X <s H2(Q) as well,
with the estimate ||X||^2(qj < 5-\/fc||X||ft2(p), where к is the bound for -jp.
Note further that this result (without the estimate) is obvious if one uses the
equivalent pseudonorm given by the corollary to Theorem 24, since
sup EQ{((H ¦ X)^J} = smp EP{^(
<k sup ЕрЩН-Х)^J},
where again к is the bound for ^.

170 IV General Stochastic Integration and Local Times
Theorem 25. Let X be a semimartingale and H s L(X). If Q -С P, then
H e L(X) under Q as well, and Hq ¦ X = HP ¦ X, Q-a.s.
Proof. Let Tn be a sequence of stopping times increasing to oo a.s. such that
H is (Хтп-,П2) integrable under P, each n > 1. Let Zt = EP{^jj;\Ft}, the
cadlag version. Define Sn = M{t > 0 : \Zt\ > n} and set Rn = Sn A Tn.
Then XRn~ € H2(P)nH2(Q) by Lemma 2, and Я is (H2,XRn-) integrable
under P. We need to show Я is (W2,Xfi"~) integrable under Q, which will
in turn imply that Я e -Ь(^) under Q. Let Хл"~ = N + С be the canonical
decomposition under Q. Let Hm = Я1{|я|<т}- Then
| |tf Л
= ||tfm • XR"-\\H2{Q) <
and then by monotone convergence we see that H is (H2,XRn~) integrable
under Q. Thus H <s L(X) under Q, and it follows that HQ ¦ X = HP ¦ X,
<2-a.s. О
Theorem 25 can be used to extend Theorem 20 in a way analogous to the
extension of Theorem 17 by Theorem 18 in Chap. II.
Theorem 26. Let X, X be two semimartingales, and let H 6 L(X), H e
L(X). Let A = {w : Н.(ш) = tf.(w) and Х.(ш) = X(w)}, and let В = {w :
11—> Xt(oj) is of finite variation on compacts}. Then H¦ X = H-X on A, and
H ¦ X is equal to a path-by-path Lebesgue-Stieltjes integral on B.
Proof. Without loss of generality assume P(A) > 0. Define Q by Q(A) =
P(A\A). Then Q -C P and therefore H e L(X), H 6 L(X) under_Q as well as
under P by Theorem 25. However under Q the processes H and H as well as
X and X are indistinguishable. Thus Hq X = HqX and hence ЙХ = HX
P-a.s. on A by Theorem 25, since Q «P.
The second assertion has an analogous proof (see the proof of Theorem 18
of Chap. II). ?
Note that one can use stopping times to localize the result of Theorem 26.
The proof of the following corollary is analogous to the proof of the corollary
of Theorem 18 of Chap. II.
Corollary. With the notation of Theorem 26, let S, T be two stopping times
with S < T. Define
С = {a; : Щш) = Ht{w); Xt(uj) = Xt(u>); S(u) < t < Т(ш)},
D = {a> : t н-> Xt(tjS) is of finite variation on S(u>) < t < Т(ш)}.
Then Я • XT - H ¦ Xs = H ¦ X~T - H ¦ ~XS on С and Я • XT - H ¦ Xs equals
a path-by-path Lebesgue-Stieltjes integral on D.

2 Stochastic Integration for Predictable Integrands 171
Theorem 27. Let Pk be a sequence of probabilities such that X is a Pk
semimartingale for each k. Let R = Y^k=i -^fc-ffc where Afc > 0, each k, and
J2kLiXk = 1. Let H e L(X) under R. Then Я e L(X) under Pk and
Hr • X = Hpk ¦ X, Pk-a.s., for all к such that Xk > 0.
Proof If Afc > 0 then Pk < R. Moreover since Рк(Л) < -^R(A), it follows
that Я <s L(X) under Pk. The result then follows by Theorem 25. D
We now turn to the relationship of stochastic integration to martingales
and local martingales. In Theorem 11 we saw that if M is a square integrable
martingale and Я e bV, then Я ¦ M is also a square integrable martingale.
When M is locally square integrable we have a simple sufficient condition for
H to be in L(M).
Lemma. Let M be a square integrable martingale and let Я e V be such
that E{f?° H2d[M, M]s} < oo. Then Я ¦ M is a square integrable martingale.
Proof. If Яk 6 bV, then Hk ¦ M is a square integrable martingale by The-
Theorem 11. Taking Hk = Hl{\H\<ky, and since Я is (H2,M) integrable, by
Theorem 14 Hk ¦ M converges in H2 to H ¦ M which is hence a square inte-
integrable martingale. ?
Theorem 28. Let M be a locally square integrable local martingale, and let
H & V. The stochastic integral H ¦ M exists (i.e., H e L(M)) and is a locally
square integrable local martingale if there exists a sequence of stopping times
(Tn)n>i increasing to oo a.s. such that E{JQ H2d[M,M]s} < oo.
Proof. We assume that M is a square integrable martingale stopped at the
time Tn. The result follows by applying the lemma. D
Theorem 29. Let M be a local martingale, and let H sV be locally bounded.
Then the stochastic integral H ¦ M is a local martingale.
Proof. By stopping we may, as in the proof of Theorem 29 of Chap. Ill, assume
that H is bounded, M is uniformly integrable, and that M = N + A where
N is a bounded martingale and A is of integrable variation. We know that
there exists Rk increasing to oo a.s. such that MR sH2, and since H 6 bV
there exist processes H? s bL such that \\He ¦ MRk~ - H ¦ MRk~\\H2 tends
to zero. In particular, He ¦ MR ~ tends to H ¦ MR ~ in ucp. Therefore we
can take Hl such that Hl ¦ MRk~ tends to Я • M in ucp, with He <s bL.
Finally without loss of generality we assume He ¦ M converges to Я • M in
ucp. Since He ¦ M = He ¦ N + He ¦ A and He ¦ N converges to H ¦ N in ucp,
we deduce He ¦ A converges to Я • A in ucp as well. Let 0 < s < t and assume
Y 6 bFs. Therefore, since A is of integrable total variation, and since He ¦ A
is a martingale for H^ e bL (see Theorem 29 of Chap. Ill), we have

172 IV General Stochastic Integration and Local Times
E{Y f HudAu} = E{Y f lim H*udAu} = E{Y lim f HidAu}
J5+ J S-\- УО° УО° J S-\-
rt
= lim E{Y / HeudAu}
e-*°° Js+
= 0,
where we have used Lebesgue's Dominated Convergence Theorem both for
the Stieltjes integral w-by-w (taking a subsequence if necessary to have a.s.
convergence) and for the expectation. We conclude that (f* HsdAs)t>0 is a
martingale, hence H ¦ M = H • N + H ¦ A is also a martingale under the
assumptions made; therefore it is a local martingale. D
In the proof of Theorem 29 the hypothesis that H ? V was locally bounded
was used to imply that if M — N + A with N having locally bounded jumps
and A an FV local martingale, then the two processes
л
\ H2d[N,N}s and
Jo
are locally integrable. Thus one could weaken the hypothesis that H is locally
bounded, but it would lead to an awkward statement. The general result, that
M a local martingale and H s L(M) implies that H ¦ M is a local martingale,
is not truel
See Emery's example, which precedes Theorem 34 for a stochastic integral
with respect to an Ti2 martingale which is not even a local martingale! Emery's
counterexample has lead to the development of what are now known as sigma
martingales, a class of processes which are not local martingales, but can be
thought to be "morally local martingales." This is treated in Section 9 of this
chapter.
Corollary. Let M be a local martingale, Mo = 0, and let Г be a predictable
stopping time. Then MT~ is a local martingale.
Proof. The notation MT~ means
M?~ = Mtl{t<Ty + MT-l{t>T},
where Mo_ = Mo = 0. Let (Sn)n>i be a sequence of stopping times announc-
announcing T. On {T > 0},
and since l[o,s»] is a left continuous, adapted process it is predictable; hence
1[о,т) is predictable. But /0 l{0,T)(s)dMs = MtT~ by Theorem 18, and hence
it is a local martingale by Theorem 29. D

2 Stochastic Integration for Predictable Integrands 173
Note that if M is a local martingale and T is an arbitrary stopping time,
it is not true in general that MT~ is still a local martingale.
Since a continuous local martingale is locally square integrable, the theory
is particularly nice.
Theorem 30. Let M be a continuous local martingale and let H 6 V be such
that /0 H2d[M, M]s < oo a.s., each t>0. Then the stochastic integral H ¦ M
exists (i.e., Я e L(M)) and it is a continuous local martingale.
Proof. Since M is continuous, we can take
Rk = mi{t>0: \Mt\ >k}.
Then \MtARk\ < к and therefore M is locally bounded, hence locally square
integrable. Also M continuous implies [M, M] is continuous, whence if
= inf{t > 0 : / H2d[M,M]s > к},
Jo
we see that (/0* H2d[M, M]s)t>0 is also locally bounded. Then H-M is a locally
square integrable local martingale by Theorem 28. The stochastic integral H ¦
M is continuous because Д(Я-М) = H(AM) and AM = 0 by hypothesis. D
In the classical case where the continuous local martingale M equals В, а
standard Brownian motion, Theorem 30 yields that if H S V and /0 H^ds <
oo a.s., each t > 0, then the stochastic integral (Я • Bt)t>0 = (Jo* HsdBs)t>0
exists, since [B,B]t = t.
Corollary. Let X be a continuous semimartingale with (unique) decomposi-
decomposition X = M + A. Let Я е V be such that
/ H2sd[M,M]s+ \Hs\\dAs\ <oo a.s.
Jo Jo
each t > 0. Then the stochastic integral (Я • X)t = /0 HsdXs exists and it is
continuous.
Proof. By the corollary of Theorem 31 of Chap. Ill, we know that M and A
have continuous paths. The integral Я • M exists by Theorem 30. Since Я • A
exists as a Stieltjes integral, it is easy to check that Я s L(A), since A is
continuous, and the result follows from Theorem 20. ?
In the preceding corollary the semimartingale X is continuous, hence
[X, X] = [M, M] and the hypothesis can be written equivalently as
/ H?d[X,X].+ I \Hs\\dAs
o Jo
< oo a.s.
Jo ~ Jo
each t > 0.
We end our treatment of martingales with a special case that yields a
particularly simple condition for Я to be in L(M).

174 IV General Stochastic Integration and Local Times
Theorem 31. Let M be a local martingale with jumps bounded by a constant
P. Let H eV be such that f* H2d{M,M]s < oo a.s., t > 0, and E{H%} < oo
for any bounded stopping time T. Then the stochastic integral (Jo HsdMs)t>o
exists and it is a local martingale.
Proof Let Rn = inf{t > 0 : fQH2sd[M,M]s > n}, and let Tn = mm(Rn,n).
Then T™ are bounded stopping times increasing to oo a.s. Note that
E{ f
./о
2
H2d[M,M]s}<n-
< n + C2Е{Щп} < oo,
and the result follows from Theorem 28. ?
The next theorem is, of course, an especially important theorem, the Dom-
Dominated Convergence Theorem for stochastic integrals.
Theorem 32 (Dominated Convergence Theorem). Let X be a semi-
martingale, and let Hm e V be a sequence converging a.s. to a limit H. If
there exists a process G 6 L(X) such that \Hm\ < G, all m, then Hm, H are
in L(X) and Hm ¦ X converges to H ¦ X in ucp.
Proof. First note that if \J\ < G with J e V, then J e L(X). Indeed, let
(T")n>i increase to oo a.s. such that G is (H2,XTn~) integrable for each n.
Then clearly
E{ / J2d[N,N]s} + E{( / |Js||cL4s|J}
Jo Jo
/•OO POO
<E{ G2d[N,N]s} + E{( |Gs||rfls|J}<oo,
Jo Jo
and thus J is (H2,XT"~) integrable for each n. (Here N + A is the canonical
decomposition of XT"~.)
To show convergence in ucp, it suffices to show uniform convergence in
probability on intervals of the form [0, to] for to fixed. Let e > 0 be given, and
choose n such that P(Tn < t0) < e, where Хт"~ E H2 and G is (W2,XT"~)
integrable. Let XT"~ = N + A, the canonical decomposition. Then
E{snp\Hm ¦ X1- - H ¦ XTn~\2}
t<t0
< 2EISUP \(Hm - H) ¦ N\2} + 2E{( / \H™ - Hs\\dAs\J}.
t<t0 J0
The second term tends to zero by Lebesgue's Dominated Convergence Theo-
Theorem. Since \Hm -H\< 2G, the integral (Hm - H) ¦ ~N is a square integrable
martingale (by the lemma preceding Theorem 28). Therefore using Doob's
maximal quadratic inequality, we have

2 Stochastic Integration for Predictable Integrands 175
?{sup \{Hm - H) ¦ АГ|2} < AE{\{Hm - H) ¦ Nto\2}
t<t0
rto
= 4E{[°(H™-HsJd{N,N}s},
Jo
and again this tends to zero by the Dominated Convergence Theorem. Since
convergence in L2 implies convergence in probability, we conclude for 8 > 0,
limsupP{sup \Hm Xt-H -Xt\>6}
m—юо t<t0
< limsupP{sup \((Hm - H) ¦ XTn~)t\ > 6} + P(Tn < t0)
m->oo t<t0
and since e is arbitrary, the limit is zero. ?
We use the Dominated Convergence Theorem to prove a seemingly in-
innocuous result. Generalizations, however, are delicate as we indicate following
the proof.
Theorem 33. Let F = (^)«>о and G — (Gt)t>o be two filtrations satisfying
the usual hypotheses and suppose Tt С Qt, each t > 0, and that X is a
semimartingale for both F and G. Let H be locally bounded and predictable for
F. Then the stochastic integrals Hr ¦ X and HG ¦ X both exist, and they are
equal.6
Proof. It is trivial that H is locally bounded and predictable for (Gt)t>o as
well. By stopping, we can assume without loss of generality that H is bounded.
Let
H = {all bounded, T predictable Я such that H? ¦ X = HG ¦ X}.
Then H is clearly a monotone vector space, and H contains the multiplicative
class bL by Theorem 16 of Chap. II. Thus using Theorem 32 and the Monotone
Class Theorem we are done. D
It is surprising that the assumption that H be locally bounded is impor-
important. Indeed, Jeulin [114, pages 46, 47] has exhibited an example which shows
that Theorem 33 is false in general.
Theorem 33 is not an exact generalization of Theorem 16 of Chap. II.
Indeed, suppose F and G are two arbitrary filtrations such that X is a semi-
semimartingale for both F and G, and H is bounded and predictable for both of
them. If It = Tt П Qt, then X is still an (Xt)t>o semimartingale by Strieker's
Theorem, but it is not true in general that H is (lt)t>o predictable. It is an
open question as to whether or not Hf ¦ X = Hg ¦ X in this situation. For a
partial result, see Zheng [248].
¦ X and Hg ¦ X denote the stochastic integrals computed with the filtrations
)г>о and (Gt)t>o, respectively.

176 IV General Stochastic Integration and Local Times
Example (Emery's example of a stochastic integral behaving badly).
The following simple example is due to M. Emery, and it has given rise to the
study of sigma martingales, whose need in mathematical finance has become
apparent. Let X = (Xt)t>o be a stochastic process given by the following
description. Let T be an exponential random variable with parameter Л = 1,
let U be an independent random variable such that P{U = 1} = P{U =
-1} = 1/2, and set X = Ul{t>Ty. Then X together with its minimal filtration
satisfying the usual hypotheses is a martingale in Ti2. That is, X is a stopped
compound Poisson process with mean zero and is an L2 martingale. Let Ht =
f l{t>o}- Therefore H is a deterministic integrand, continuous on @,oo), and
hence predictable. Consequently the path-by-path Lebesgue-Stieltjes integral
Zt = /o HsdXs exists a.s. However H ¦ X is not locally in W for any p > 1.
(However since it is still a semimartingale7, it is prelocally in H2.) Moreover,
even though X is an L2 martingale and Я is a predictable integrand, the
stochastic integral H ¦ X is not a local martingale because -EJIZsl} = оо for
every stopping time S such that P(S > 0) > 0.
The next theorem is useful, since it allows one to work in the convenient
space Ti2 through a change of measure. It is due originally to Bichteler and
Dellacherie, and the proof here is due to Lenglart. We remark that the use of
the exponent 2 is not important, and that the theorem is true for H? for any
p> 1.
Theorem 34. Let X be a semimartingale on a filtered complete probability
space (fi, T, F, P) satisfying the usual hypotheses. Then there exists a proba-
probability Q which is equivalent to P such that under Q, X is a semimartingale
in Ti2. Moreover, ^ can be taken to be bounded.
Before we begin the proof we establish a useful lemma.
Lemma. Let (fi, J7, P) be a complete probability space and let Xn be a se-
sequence of a.s. finite valued random variables. There exists another probability
Q, equivalent to P and with a bounded density, such that every Xn is in
L2(dQ).
Proof. Assume without loss that each of the Xn is positive. For a single ran-
random variable X we take Ak = {k < X < к +1} and Y = J2k>i 2~к1лк- Then
Y is bounded and
For the general case, choose constants an such that P(Xn > an) < 2~". The
Borel-Cantelli Lemma implies that a.s. Xn < an for all n sufficiently large.
Next choose constants cn such that ^n>i c«an < °°- Let Yn be the bounded
7 H ¦ X is still a semimartingale since it is of finite variation a.s. on compact time
sets.

2 Stochastic Integration for Predictable Integrands 177
density chosen for Xn individually as done in the first part of the proof, and
take Y = J2n>i спУп so that we have the result. ?
Proof of Theorem 34- Using the lemma we make a first change of measure
making all of the random variables [X, X]n integrable. Recall that [X, X] is
invariant under a change to an equivalent probability measure. By abusing
notation, we will still denote this new measure by P. This implies that if
Jt = sups<t |ДХЯ|, then J2 < [X, X]t and thus J is locally in I?. Hence it
is also a fortiori locally in L1, and thus X is special. We write its canonical
decomposition X = M + A, with Xo = Ao. We now make a second change
to an equivalent law, this time called Q, again using the lemma, such that
Jq \dAs\ 6 L2(dQ) for each n. Then X is special under Q as well, since each
[X, X]n remains in L1 for Q. Let X = N + C denote its Q decomposition. We
apply Girsanov's Theorem to get
I
-z—d(M, Z)a
0 ^s-
where Zt = Ep{Z\Tt\ with Z = ^p. For convenience we write this as Ct =
At + Vt. We now need to establish that ?q{(/0* \dv*\?} = EQ{\v\t} < °°-
Note that we have Eq{V?} < oo, since Vt = Mt - At, and At ? L2(dQ). It
is easy to check that Ep{[M,M]n} < 4Ep{[X,X]n} < oo by an argument
similar to the proof of the lemma preceding Theorem 24. Finally L2(dP) С
L2(dQ). Since V is predictable, there exists a predictable process H taking
values in { — 1,1} such that Jo \dVs\ = JQ HsdVs. Setting 5 = H ¦ X, we have
[S,S] = [X, X], and the random variables [S,S]n are then integrable (dP). S
has a decomposition 5 = L + D where L = H -M and D = H -A, hence under
the law Q we have L = H-N+H-V, always using that the stochastic integral is
invariant under a change to an equivalent probability measure. We deduce that
Eq{(!o HsdVsJ} < oo. Since EQ{(J* HsdVsJ} = EQ{(J* \dVs\J}, coupled
with \H ¦ A\t = \A\t, the result follows. ?
Corollary (Lenglart's Inclusion Theorem). Let (?l,J-,W,P) satisfy the
usual hypotheses, and suppose Q is an equivalent probability measure with a
bounded density with respect to P. Then H2(P) С H2(Q). That is, U2 with
respect to P is contained in the space of H2 semimartingales with respect to
Q.
Proof. Let X be an H2 semimartingale with respect to P. As we saw in
the proof of Theorem 34, since Q has a bounded density, we obtain that
[X, X] is in Ll{dQ), and hence X is special for Q. We then obtain a canonical
decomposition of X and show both terms are in 7i2 analogously to how we
did it in the proof of Theorem 34. D

178 IV General Stochastic Integration and Local Times
3 Martingale Representation
In this section we will be concerned with martingales, rather than semimartin-
gales. The question of martingale representation is the following. Given a col-
collection Л of martingales (or local martingales), when can all martingales (or
all local martingales) be represented as stochastic integrals with respect to
processes in A? This question is surprisingly important in applications, and
it is particularly interesting (in finance theory, for example) when Л consists
of just one element.
Throughout this section we assume as given an underlying complete, fil-
filtered probability space (fi, J-, (J-t)t>o, P) satisfying the usual hypotheses. We
begin by considering only I? martingales. Later we indicate how to extend
these results to locally square integrable local martingales.
Definition. The space M2 of I? martingales is all martingales M such
that supt E{M?} < oo, and Mo = 0 a.s. Notice that if M e M2, then
lim^oo^jM2} = E{M^} < oo, and Mt = E{Moo\J:t}. Thus each MeM2
can be identified with its terminal value M^. We can endow M2 with a norm
and also with an inner product
(M,N)=E{MOONOO},
for M,N ? M2. It is evident that M2 is a Hilbert space and that its dual
space is also M2.
If E{M2} < oo for each t, we call M a square integrable martingale.
If in addition supt E{M2} = E{M%O) < oo, then we call M an L2 martingale.
The next definition is a key idea in the theory of martingale representation.
It differs slightly from the customary definition because we are assuming all
martingales are zero at time t = 0. If we did not have this hypothesis, we
would have to add the condition that for any event Л 6 To, any martingale
M in the subspace F, then M\a ? F.
Definition. A closed subspace F of M2 is called a stable subspace if it is
stable under stopping (that is, if M e F and if T is a stopping time, then
MT e F).8
Theorem 35. Let F be a closed subspace o/M2. Then the following are
equivalent.
(a) F is closed under the following operation. For M ? F, (M - МгIл ? F
for Л ? Tt, any t > 0.
(b) F is a stable subspace.
Recall that Mj = MtAT, and Mo = 0.

3 Martingale Representation 179
(c) If M 6 F and H is bounded, predictable, then (f* HsdMs)t>o = H ¦ M e
F.
(d) If M e F and H is predictable with ?{/0°° H2d[M,M]s} < oo, then
H -M ? F.
Proof Property (d) implies (c), and it is simple that (c) implies (b). To get
(b) implies (a), let T = tA, where
Then T — tA is a stopping time when A 6 Tt, and (M - МгIа = M — MT;
since F is assumed stable, both M and MT are in F. It remains to show only
that (a) implies (d). Note that if H is simple predictable of the special form
г=\
with Ai ? Tti, 0 < i < n, 0 = t0 < tx < ¦ ¦ ¦ < tn+1 < oo, then H ¦ M e F
whenever M ? F. Linear combinations of such processes are dense in bL which
in turn is dense in bV under <^м(-, •) by Theorem 2. But then hV is dense in
the space of predictable processes Ti such that E{f^° H^d[M, M]s} < oo, as
is easily seen (cf., Theorem 14). Therefore (a) implies (d) and the theorem is
proved. ?
Since the arbitrary intersection of closed, stable subspaces is still closed
and stable, we can make the following definition.
Definition. Let Л be a subset of M2. The stable subspace generated by
Л, denoted <S(.4), is the intersection of all closed, stable subspaces containing
Л.
As already noted on page 178, we can identify a martingale M e M2
with its terminal value M^ e L2. Therefore another martingale ./V € M2 is
(weakly) orthogonal to M if ^{Л^ооМоо} = О. There is however another,
stronger notion of orthogonality for martingales in M2.
Definition. Two martingales N, M e M2 are said to be strongly orthog-
orthogonal if their product L = NM is a (uniformly integrable) martingale.
Note that if JV, M € M2 are strongly orthogonal, then NM being a
(uniformly integrable) martingale implies that [N, M] is also a local martin-
martingale by Corollary 2 of Theorem 27 of Chap. II. It is a uniformly integrable
martingale by the Kunita-Watanabe inequality (Theorem 25 of Chap. II).
Thus M,N ? M2 are strongly orthogonal if and only if [M, N] is a uni-
uniformly integrable martingale. If N and M are strongly orthogonal then
^{Л^ооМоо} = S{Loo} = E{L0} = 0, so strong orthogonality implies or-
orthogonality. The converse is not true however. For example let M e M2, and

180 IV General Stochastic Integration and Local Times
let Y e To, independent of M, with P(Y = 1) = P(Y = -1) = \. Let
Nt = YMU t > 0. Then TV G M2 and
EiN^M^} = E{YMl) = E{Y}E{Ml} = 0,
so M and TV are orthogonal. However MN = YM2 is not a martingale (unless
M = 0) because ?{yM2|JF0} = y?{Mt2|.Fo} ^ о = M02.
Definition. For a subset A of M2 we let Ax (resp. Ax) denote the set of all
elements of M2 orthogonal (resp. strongly orthogonal) to each element of A.
Lemma 1. If A is any subset of M2, then Ax is (closed and) stable.
Proof. Let Mn be a sequence of elements of Ax converging to M, and let
TV 6 A. Then MnN is a martingale for each n and Ax will be shown to be
closed if MN is also one, or equivalently that [M, TV] is a martingale. However
E{\[Mn,N] - [M,N}t\} = E{\[Mn-M,N]t\}
< (E{[Mn -M,Mn- \t]t})^2(E{[N, Nit}I/2
by the Kunita-Watanabe inequalities. It follows that [Mn,N]t converges to
[M, N]t in L1, and therefore [M, N] is a martingale, and Ax is closed. Also Ax
is stable because M e Ax, N 6 A implies [MT, N] = [M, N]T is a martingale
and thus MT is strongly orthogonal to N. D
Lemma 2. Let N, M be in M2. Then the following are equivalent.
(a) M and N are strongly orthogonal.
(b) S(M) and N are strongly orthogonal.
(c) S(M) and S(N) are strongly orthogonal.
(d) S(M) and N are weakly orthogonal.
(e) S(M) and S(N) are weakly orthogonal.
Proof. If M and N are strongly orthogonal, let A = {./V} and then M e Лх.
Since Ax is a closed stable subspace by Lemma 1, S(M) с {Аг}х- Therefore
(b) holds and hence (a) implies (b). The same argument yields that (b) implies
(c). That (c) implies (e) which implies (d) is obvious. It remains to show that
(d) implies (a).
Suppose N is weakly orthogonal to S(M). It suffices to show that [N, M]
is a martingale. By Theorem 21 of Chap. I it suffices to show E{[N, М]т} = О
for any stopping T. However E{[N,M]T} = E{[N, M7}^ = 0, since TV is
orthogonal to MT which is in S(M). О
Theorem 36. Let M1,..., Mn 6 M2, and suppose Мг, iVP are strongly or-
orthogonal for i ф j. Then S(M1,... ,M") consists of the set of stochastic
integrals
n
H1 ¦ M1 + ¦ ¦ ¦ + Hn ¦ Mn = ]T #* • M\

3 Martingale Representation 181
where Hl is predictable and
Е{П{Н\)Ч{М\М%}
Jo
poo
< oo, 1 < i < n.
Proof. Let I denote the space of processes Yl7=i ^% ' Мг, where Нг satisfy
the hypotheses of the theorem. By Theorem 35 any closed, stable subspace
must contain X. It is simple to check that X is stable, so we need to show only
that X is closed. Let
/•OO
= {HeV:E{ H*d[M, M}s} < oo}.
Jo
Then the mapping (ff^ff2,... ,#") —> J2"=i H% ' Ml is an isometry from
L2Mi © • • • © L2Mn into M2. Since it is a Hilbert space isometry its image X is
complete, and therefore closed. ?
Theorem 37. Let Abe a subset of M2 which is stable. Then Ax is a stable
subspace, and if M 6 A then M is strongly orthogonal to A. That is, A-1 =
Ax, andS(A) = A^ =AxX =AXX.
Proof. We first show that Ax - Ax. Let M ? A and N ? Ax. Since TV is or-
orthogonal to S(M), by Lemma 2, TV and M are strongly orthogonal. Therefore
Ax С Ax. However clearly Ax С Ах, whence Ax = Ах, and thus Ах = Ax
is a stable subspace by Lemma 1.
By the above applied to Ax, we have that (Ах)х = (Ax)x.
It remains to show that S(A) — Ахх. Since Axx = A, the closure of A
in M2, it suffices to show that A is stable. However it is simple to check that
condition (a) of Theorem 35 is satisfied for A, since it already is satisfied for
A, and we conclude A is a stable subspace. ?
Corollary 1. Let A be a stable subspace of M2. Then each M e M2 has a
unique decomposition M = A + B, with A e A and В G Ax.
Proof. A is a closed subspace of M2, so each M e M2 has a unique decom-
decomposition into M = A + В with A ? A and В e Ax. However Ax = Ax by
Theorem 37. ?
Corollary 2. Let M, TV e M2, and let L be the projection of TV onto S(M),
the stable subspace generated by M. Then there exists a predictable process
H such that L = H -M.
Proof. We know that such an L exists by Corollary 1. Since {M} consists of
just one element we can apply Theorem 36 to obtain the result. D

182 IV General Stochastic Integration and Local Times
Definition. Let A be finite set of martingales in M2. We say that A has the
(predictable) representation property if T = M2, where
n
1={X : X = ]?я* • M*, Af* e A},
each Hг predictable such that
poo
E{ I {HiJd[M\M%} < oo, 1 < i < n.
Jo
Corollary 3. Let A = {M1,...,Mn} С М2, and suppose M\M^> are
strongly orthogonal for i ф j. Suppose further that if N 6 M2, N ± A in the
strong sense implies that N = 0. Then A has the predictable representation
property.
Proof. By Theorem 36 we have S(A) = I. The hypotheses imply that
= {0}, hence S(A) = M2. ?
Stable subspaces and predictable representation can be considered from
an alternative perspective. Up to this point we have assumed as given and
fixed an underlying space (Q, F, (^)г>о, Р), and a set of martingales A in
M2. We will see that the property that S(A) = M2, intimately related to
predictable representation (cf., Theorem 36), is actually a property of the
probability measure P, considered as one element among the collection of
probability measures that make L2 (ft)t>o martingales of all the elements of
A.
Our first observation is that since the filtration F = (J-t)t>o is assumed to
be P complete, it is reasonable to consider only probability measures that are
absolutely continuous with respect to P.
Definition. Let А С M2. The set of M2 martingale measures for A,
denoted Ai2(A), is the set of all probability measures Q defined on Vo<t<<x> -^t
such that
(i) Q « P,
(ii) д = Роп^0)
(Hi) if X G A then X e M2(Q),
where M2(<3) denotes all (Q,?) L2 martingales.9
Lemma. The set M.2 is a convex set.
9 Note that while F satisfies the usual hypotheses under P, Jo does not contain all
of the Q-null sets, and thus F does not completely satisfy the usual hypotheses
under Q. This does not create any problems, however.

3 Martingale Representation 183
Proof. Let Q and R ? M2(A), and let S = \Q + A - \)R, 0 < A < 1. Then
for X e M2(A),
supEs{M2} = sup[A?Q{M2} + A - \)ER{M2}} < oo,
t t
since Q, R e M2(A). Also if Я ? hTs, s <t, then
Es{MtH} = \EQ{MtH} + A - А)?Я{М4Я}
= A?q{Ms#} + A - \)ER{MSH}
= ES{MSH},
and 5 ? Л<2(Л). П
Definition. A measure Q e M2(A) is an extremal point of M2(A) if
whenever Q = \R + A - AM with R, S ? Л<2(Л), i? ^ 5, 0 < A < 1, then
A = 0or 1.
Theorem 38. ?e? А С M2. If S(A) = M2 Йеп Р is an extremal point of
M2(A).
Proof. Suppose P is not extremal. We will show that S(A) Ф M2. Since P is
not extremal, there exist Q,R? M2(A), Q Ф R, such that P = A<3 + A-A)i?,
0 < A < 1. Let
Loo = dP'
and let Lt = E{^p\J-t}- Then 1 = ^ = Aioo + A~A)^ > A-Lqo a.s., so that
-^oo < \ a.s. Therefore L is a bounded martingale with Lo = 1 (since Q = P
on .Fo), and thus L - Lo is a nonconstant martingale in M2(P). However, if
X ? A and Я € hTv,, then X is a Q martingale and for s < t,
EP{XtLtH} = EpiXtL^H} = EQ{XtH} = EQ{XSH}
= Ep{XsLooH}
= EP{XSLSH},
and XL is a P martingale. Therefore X(L - Lo) is a P martingale, and
L - Lo 6 M2 and it is strongly orthogonal to A. By Theorem 37 we cannot
have S(A) = M2. ?
Theorem 39. Let А С M2. If P is an extremal point of ЛЛ2{А), then every
bounded P martingale strongly orthogonal to A is null.
Proof. Let L be a bounded nonconstant martingale strongly orthogonal to A.
Let с be a bound for \L\, and set
dQ = (l -)dP and dR=(l + —)dP.
2c 2c
We have Q, R 6 M2(A), and P — \Q + ^R is a decomposition that shows
that P is not extremal which is a contradiction. D

184 IV General Stochastic Integration and Local Times
Theorem 40. Let Л = {M1,..., M"} С М2, with Мг continuous and M\
Mi strongly orthogonal for i ф j. Suppose P is an extremal point of Л42(А).
Then
(a) every stopping time is accessible;
(b) every bounded martingale is continuous;
(c) every uniformly integrable martingale is continuous; and
(d) Л has the predictable representation property.
Proof, (a) Suppose T is a totally inaccessible stopping time and P(T < oo) >
0. By Theorem 22 of Chap. Ill, there exists a martingale M with АМт =
1{Т<оо} and M continuous elsewhere. Moreover, the martingale M can be
taken of finite variation with Mo — 0. Therefore, since each M% is continuous,
[M, Мг] = О. Moreover taking (for example)
Rn = inf{t > 0 : \Mt
R
\MR | < n + 1 and thus M is locally bounded. Indeed, we then have M
are bounded martingales strongly orthogonal to Ml. By Theorem 39 we have
MR = 0 for each n. Since limn^oo-R" = o° &-s., we conclude M = 0, a
contradiction.
(b) Let M be a bounded martingale which is not continuous, and assume
Mq = 0. Let TE = inf{? > 0 : |AMt| > e}. Then there exists e > 0 such
that for T = Te, P{\AMT\ > 0} > 0. By part (a) the stopping time T is
accessible, hence without loss we may assume that T is predictable. Therefore
MT~ is a bounded martingale by the corollary to Theorem 29, whence N =
MT — MT~ = AMj1{ojj is also a bounded martingale. However N is a
finite variation bounded martingale, hence [N, Мг] = 0 each i. That is, N is
a bounded martingale strongly orthogonal to Л. Hence N = MT — MT~ = 0
by Theorem 39, and we conclude that M is continuous.
(c) Let M be a uniformly integrable martingale closed by M^. Define
Then Mn are bounded martingales and therefore continuous by part (b).
However
1
Pjsup \МГ - Mt\ > e} < - E{\M" - Moo }
t ?
by an inequality of Doob10, and the right side tends to 0 as n tends to сю.
Therefore there exists a subsequence (n^) such that limfc^oo M"k = Mt a.s.,
uniformly in t. Thus M is continuous.
(d) By Corollary 3 of Theorem 37 it suffices to show that if N ? Ax then
N = 0. Suppose N ? Ax. Then N is continuous by (c). Therefore N is locally
bounded, and hence by stopping, N must be 0 by Theorem 39. ?
See, for example, Breiman [23, page 88].

3 Martingale Representation 185
The next theorem allows us to consider subspaces generated by countably
infinite collections of martingales.
Theorem 41. Let M e M2, Y" € M2, n > 1, and suppose Y^ converges
to Уоо in L2, and that there exists a sequence Hn e L(M) such that Y™ =
j0 H™dMs, n > 1. Then there exists a predictable process H ? L(M) such
thatYt = f*HsdMs.
Proof. If Y? converges to Y"oo in L2, then Y" converges to Y in M2. By
Theorem 36 we have that S{M) = X(M), the stochastic integrals with respect
to M. Moreover Y" e S(M), each n. Therefore Y is in the closure of <S(M);
but S(M) is closed, so У € S(M) = I(M), and the theorem is proved. ?
Theorem 42. Let Л = {Ml,M2,..., M",...}, with M' ? M2, and suppose
there exist disjoint predictable sets A1 such that lAid[Ml,Ml] = d[Mi,Ml],
i>l.LetAt = YZLi /о иЛ»Нм^м%- Suppose that
(a) ElAoo} < oo; and
(b) for J* С J^oo such that for any Xi € ЪР, we have Xlt = Е{Х*\Тг} =
/0 H\dM\, t > 0, for some predictable process Hl.
Then M = Yl'iLi Мг exists and is in M2, and for any Yeb\JiJri,ifYt =
E{Y\Ft), we have that Yt = f0HsdMs, for the martingale M = ?Si Mi
and for some H € L(M).
Proof Let Nn = Y?i=iM\ Then [Nn,Nn}t = EtiS^MW]
hence E{(N^J} = E{{Nn, N"}^} < ?{Ax,}, and Nn is Cauchy in M2 with
limit equal to M. By hypothesis we have that if Xх 6 ЪТг then
= Х\= I H\dM\ = f \Ai
Jo Jo
= f iAiHt
Jo
Therefore if i ф j we have that
= f lAiHilAJ
Jo
- / 1л*пл>ЩЩ<1[М,М\.
Jo
= 0,
since Лг П A> =0, by hypothesis. However using integration by parts we have

186 IV General Stochastic Integration and Local Times
xixi = f x\_dxi + [ xi_dx\ + [x\
Jo Jo
= / X\_dX{ + j Xi_dX\
Jo Jo
/
o
i*dMa,
f
Jo
= f Hi*
Jo
where H1^ is defined in the obvious way. By iteration we have predictable
representation for all finite products \\i<n Xх. The Monotone Class Theorem
together with Theorem 41 then yields the result. D
We conclude this section by applying these results to a very important
special case, namely n-dimensional Brownian motion. This example shows
how these results can be easily extended to locally square integrable local
martingales.
Theorem 43. Let X = (X1,..., Xn) be an n-dimensional Brownian motion
and let F — (ft)o<t<oo denote its completed natural filtration. Then every
locally square integrable local martingale M for F has a representation
where Hl is (predictable, and) in L(Xl).
Proof. Fix to, 0 < t < to, and assume X is stopped at to- Then letting
Л = {Xx,...,Xn}, we have that Л С M2. Let М2(Л) be all probability
measures Q such that Q «; P, and under Q all of Л С M2(<2).n Since
[X1, Xi] is the same process under Q as under P we have by Levy's Theorem
(Theorem 40 of Chap. II) that X = {X1,...,Xn) is a Q-Brownian motion
as well. Therefore, for bounded Borel functions /i,..., /n, we have that for
t\ < t2 • ¦ • < tn < to, fi(Xtl) ПГ=1 fi{Xti+i —Хъ) have the same expectation
for Q as they do for P. Thus P = Q on Fto, and we conclude that M2(A)
is the singleton {P}. The probability law P is then trivially extremal for
Л42(Л), and therefore by Theorem 40 we know that Л has the predictable
representation property.
Suppose M is locally square integrable and Mt = MtM0 ¦ Let (Т*%>о be
a sequence of stopping times increasing to oo a.s., To = 0, and such that
MT € M2, each к > 0. Then by the preceding there exist processes H''k
such that
мтк = Тн*>к-.
11 Recall that M2(Q) denotes all L2 Q martingales.

3 Martingale Representation 187
By defining Hl = YX=\ нг'к\тк~\т^ we have that W e Ь{Хг) and also
м =
Next let (te)e>o be fixed times increasing to oo with to = 0. For each tt we
know there exist Нг'е such that for a given locally square integrable local
martingale N
*=i
By defining W = YT=iHi'ei(U-uU] one easily checks that W € ЦХг) and
that
П
Nt = ^Hl -X1. D
Corollary 1. As in Theorem 43, let F be the completed natural filtration of
an n-dimensional Brownian motion. Then every local martingale M for F is
continuous.
Proof. In the proof of Theorem 43 we saw that the underlying probability
law P is extremal for Л = {X1,... ,Xn}. Therefore by Theorem 40(c), ev-
every uniformly integrable martingale is continuous. The corollary follows by
stopping. D
Corollary 2. Let X — (X1,..., Xn) be an n-dimensional Brownian motion
and let F be its completed natural filtration. Then every local martingale M
for F has a representation
Mt = Mo + J2 I H\
where Hl are predictable.
Proof. By Corollary 1 any local martingale M is continuous, hence it is locally
square integrable. It remains only to apply Theorem 43. D
Corollary 3. Let X = {X1,..., Xn) be an n-dimensional Brownian motion
and let F be its completed natural filtration. Let Z e Т^ be in L1. Then
there exist H% predictable in L(X%) with /0°°(i/*Jds < oo a.s. such that
Htdxt.

188 IV General Stochastic Integration and Local Times
Proof. Let Zt = E{Z\Jr{\, taking the cadlag (and hence continuous) version.
By Corollary 2 we have
By Theorem 42 of Chap. II we have that Zt = B\ztz]t for some Brownian
motion B, 0 < t < oo. Letting t tend to oo shows that Z^ = B\\mt^oo^z,z\t^
which shows that \Z,Z\OO < oo a.s. However,
< oo a.s.
" /-OO " l-t
V / {Hlfds = lim V / {Hlfds = lim [Z,Z]t = [Z.Z]
Finally, take t = oo, observe that Z^ = Z and that Tq is a.s. trivial. Hence
Zq is constant and therefore Zq = E{Z}. О
Corollary 4. Let X = (X1,... ,Xn) be an n-dimensional Brownian motion
and let F be its completed natural filtration. Let Z e I/1(^roo) and Z > 0 a.s.
Then there exist Jl predictable with J^°(JlJds < oo a.s. such that
Proof. By Corollary 3 there exist predictable H% such that if Z( = E{Z\Tt\,
then
Zt = ?{*} ? Г
Therefore
+ E Г
(Note that since (^t)t>o is continuous and never 0, -^ is locally bounded.)
The proof is completed by setting J* = -%-Нг8 and taking exponentials of both
sides. G
Corollary 5. Let F be the completed natural filtration of an n-dimensional
Brownian motion. If T is a totally inaccessible stopping time, then T = oo
a.s.
Proof. This is merely Theorem 40 (a). D
We end this section with a quite general martingale representation result,
which while often true, nevertheless requires a countable number of martin-
martingales to have the representation. We then apply it to give a condition for
compensators to have absolutely continuous paths. But first we state a defi-
definition from measure theory that is not widely known.

3 Martingale Representation 189
Definition. A measurable space (f2, T) is said to be separable if there exists
a countable family of functions on (or subsets of) Q which generate the a-
algebra T.
Theorem 44. Let (fi, J7, F, P) be a filtered complete probability space satisfy-
satisfying the usual hypotheses. Assume also that the o-algebra T — T^ and that it is
separable. Then there exists a countable sequence of martingales (M\,M2, ¦ • •)
in L2 such that they are orthogonal, J2i>i E{[M\ M*]oo} < oo, and such that
if N is any L2 martingale, then there exists a sequence of predictable pro-
processes (HX,H2,...) such that Nt = J2i>i /о Щ^^а- That is, the martingales
{Мг,г > 1} have martingale representation.
Proof. Since J- is separable, there exists a countable basis of L2, call it
(M°,M1, M2,...), where M° is constant and Е{М*} = 0 for г ф 0. One can
take such a basis to be orthonormal in the usual way. That is, Е{МгМ:>} = О
for i ф j, and E{(M1J} = 1 for all i. If one then multiplies Mi by 2~г and
defines the martingales Ml = E{M%\Tt}, we have the construction. The space
of stochastic integrals with respect to all of the Ml where the sum converges
in L2 is easily seen to be a stable subspace. Next let N be any martingale in
L2 with No = 0. Then we have
rt
I + Lt
where L is orthogonal to the stable subspace generated by the countable
collection of martingales (Ml)i>\. This implies that L^ is orthogonal to all
of the random variables M^ — Ml. Since these random variables form a basis
of L2, this implies that L must be constant. Since Lo = 0 we have that L is
identically zero. ?
Before we apply Theorem 44 to compensators, we need to recall and for-
formalize two definitions, and prove Lebesgue's change of time formula. The first
definition we have already seen in Exercise 7 of Chap. Ill, and the second one
we saw in Theorem 42 of Chap. II as well as Exercises 31 through 36 of that
chapter.
Definition. A (right continuous) filtration F on a filtered probability space
satisfying the usual hypotheses is called quasi left continuous if for every
predictable stopping time T one has Тт = Ft- ¦
Note that if T is a predictable stopping time, and if Tn is an announcing
sequence of stopping times for T, and if X e L1, then limn^oo E{X\J-j-n} =
E{X\J-t-}. If M is a uniformly integrable martingale, then
Е{Мт\^т-} = lim E{MT\TTn} = lim MTn = MT--
n—>oo n—>oo
Therefore E{&Mt\Ft-} = 0 a.s. and if АМт is Тт- measurable, it must
be 0 almost surely. So if F is quasi left continuous, no martingales can jump

190 IV General Stochastic Integration and Local Times
at predictable times. Since the accessible part of any stopping time can be
covered by a countable sequence of predictable times, we conclude that if
F is quasi left continuous then martingales jump only at totally inaccessible
stopping times.
Definition. Let A — {At)t>o be an adapted, right continuous increasing
process, which need not always be finite-valued. The change of time (also
known as a time change) associated to A is the process
rt = inf{s > 0 : As > t}.
Some observations are in order. We have that t н-> rt is non-decreasing
and hence rt_ exists. Also, since {At > s} = U?>o{^< > s + ?}> we nave
that t н-> rt is right continuous. It is continuous if A has strictly increasing
paths. Moreover ATt > ATt_ > t, and A(Tt_)_ < -A(rt)_ < t. (Here we use the
convention tq- = 0.) Finally note that {rs_ < i) = {s < At}, which implies
that T(_ is a stopping time, and since Tj = lim?_>o T(t+?)- we conclude that r^
is also a stopping time.
Theorem 45 (Lebesgue's Change of Time Formula). Let a be a positive,
finite, right continuous, increasing function on [0, oo). Let с denote its right
continuous inverse (change of time). Let f be a positive Borel function on
[0, oo). If G is any positive, finite, right continuous function on [0, oo) with
G(O-) = 0, then
/•OO /*OO
/ f(s)dG(a(s)) = / f(c(s-))l{c(s_)<oo]dG(s),
Jo Jo
and in particular
лОО ЛОО
/ f(s)da(s) = / f(c(s))l{c(s)<oo}ds.
Jo Jo
Here the integrals are taken over the set [0, oo).
Proof. First consider / of the form f(s) = l[0)U](s), for 0 < и < со. The
left side of the equation then reduces to G(a(u)). For the right side note that
/(c(s-))l{c(s_)<oo} = l{c(s_)<u} = l{s<a(«)} and it follows that the right side
is also equal to G(a(u)). By subtraction and linearity we also have the result
for / of the form 1 («,„]. The Monotone Class Theorem gives the result for any
/ positive with compact support, and approximations then gives the general
case. If we take G(s) = s we get /0°° f(s)da(s) - /0°c/(c(s-))l{c(s_)<oo}ds,
and since с only jumps at most countably often because it is increasing, and
since ds does not charge countable sets, we have /0°° f(c(s—))l{c(s-)<oo}^s =
Jo° /(c(s))l{c(s)<oo}^s and the second statement is proved, whence the theo-
theorem. D

3 Martingale Representation 191
Corollary. Let a be a positive, finite, continuous, strictly increasing function
on [0,oo). Let с denote its continuous inverse (change of time). Let / be a
positive Borel function on [0,00). Then
fC(t) ft
/ f(s)da(s) = / f(c(s))ds.
Jo Jo
Proof. It suffices to rewrite the left side of the equation as
l[o,c(t)](s)f(s)da(s)
r°
/
Jo
/
Jo
and observe that с is also continuous and strictly increasing, which implies
VcWiW5)) = 1[o,*](s)- a
Our goal, stated informally in words, is if a filtration is quasi left contin-
continuous, then modulo a change of time, all compensators of adapted counting
processes with totally inaccessible jumps have paths which are absolutely con-
continuous. This is achieved in Theorem 47. We begin with two definitions.
Definition. Let (Q,J-, F, P) be a filtered probability space satisfying the
usual hypotheses, and let <S denote the space of all square integrable martin-
martingales with continuous paths a.s. It is easy to see that <S is a stable subspace.12
For an arbitrary square integrable martingale M, let Mc denote the orthogo-
orthogonal projection of M onto <S. Then Mc is called the continuous martingale
part of M. If we write M = Mc + (M — Mc) as its orthogonal decomposition,
then we have Md = (M-Mc) where Md is called the purely discontinuous
part of the martingale M.
Note that if a local martingale is locally square integrable, we can extend
the definition of continuous part and purely discontinuous part trivially, by
stopping. Also note that the term "purely discontinuous" is misleading: it
is not a description of a path property of a martingale, but rather simply
refers to the property of being orthogonal to the stable subspace of continuous
martingales. See Exercises 6 and 7 in this regard.
Definition. Let (fi, ,F,F, P) be a filtered probability space satisfying the
usual hypotheses. We call it an absolutely continuous space if for any
purely discontinuous locally square integrable martingale M, d{M, M)t <C dt.
Theorem 46. Let (fi, T, F, P) be an absolutely continuous space. Then the
compensators for all adapted counting processes with totally inaccessible jump
times and without explosions, are absolutely continuous.
Proof. Let N be an adapted counting process and let N be its compensator,
so that X = N — N is a locally square integrable local martingale. Since N is
continuous we have [X,X]t = Y,s<t(^NsJ = Nt, since (&NSJ = ANS = 1
12 See Exercise 6.

192 IV General Stochastic Integration and Local Times
when N jumps at s. Therefore N = (X,X), and since (X, X) has absolutely
continuous paths by hypothesis, we are done. D
Remark (Yan Zeng). With a little more work, one can show the following:
Let (П, J-, F, P) satisfy the usual hypotheses. If it is an absolutely continuous
space, then all jump times of square integrable martingales are totally inac-
inaccessible. Moreover it is an absolutely continuous space if and only if for every
totally inaccessible stopping time T, if Nt = l{t>r}> then dNt <c dt a.s.
If we take the uninteresting counting process Nt = J2i>i !{*>»} which
jumps at constant (and hence predictable) times, then its compensator is
simply Nt = Nt, and the martingale X = N - N is identically zero. To avoid
these trivialities we assume that the process N is continuous. Note however
that N is not a priori absolutely continuous. This is to explain our hypotheses
in the next theorem:
Theorem 47 (Absolutely Continuous Compensators). Let (?l,F,W,P)
be a filtered probability space satisfying the usual hypotheses. Assume that the
a-algebra J- = J-oq and that it is separable}3 Assume further that F is quasi
left continuous. To prevent trivialities assume also that it is not an absolutely
continuous space. Then there exists a change of time Tt and a new filtration G
given by Qt = TTt, such that if N is a G adapted counting process with totally
inaccessible jump times and without explosions, then dNt <C dt.
Proof. Let (MJ)i>i be the countable sequence constructed in Theorem 44.
Define the strictly increasing process At = t + ^2i>i[M%, M%]t, and note that
E{At\ < oo for 0 < t < oo. Let С denote the compensator of A. One can
check that Ct = t + '^2li>i{M%,M%)t. Then С is both strictly increasing and
continuous, since F is quasi left continuous. Let
Tt = inf{s > 0 : Cs > t} and then Ct = M{s > 0 : rs > t},
where r is the time change of the theorem statement.
Let N be a G adapted counting process and let X = N — N. As-
Assume, by stopping if necessary, that X^ e L2. We write (X, X)G to de-
denote the sharp bracket taken using the G filtration. Note that Lt = Nct
is also a counting process, and since r continuous implies that Qct С Tt
(see Exercise 38 of Chap. II), we conclude L is adapted to F. We then have
[L,L]t = Es<t(ALsJ = Lt, hence Lt = (L,L)*t. But also [X,X]t = Nt, and
we have
[X,X]t-{X,X)f = Nt-{X,X)f.
Hence
NCt-(X,X)%=Lt-(X,X)%,
A separable сг-algebra is defined on page 189.

4 Martingale Duality and Jacod-Yor Theorem 193
and by the uniqueness of the compensator of L we have (X, X)^t = (L, L)\.
On the other hand, by Theorem 44 we know that Lt - Lt = J2i>i /о
for some predictable processes (Нг), and thus
ST f i 2 i i?
I
[
o
= / J.dC
for some (predictable) nonnegative processes cl whose existence is assured
because A{М\М% « dCt a.s. The process Js = ?4>1(Я]Jс', ЬУ Fubini's
Theorem. Next, note that since (L-L, L-Lft = (X, X)%t, we have (X, X)f =
(L — L,L — L)?Tt = /on JsdCs, which by the corollary of Lebesgue's change of
time formula, implies {X,X)f = JQ JTsds. D
4 Martingale Duality and the Jacod-Yor Theorem on
Martingale Representation
We have already defined the space H2 for semimartingales; we use the same
definition for martingales, which in fact historically came before the definition
for semimartingales. We make the definition for all p, p > 1.
Definition. Let M be a local martingale with cadlag paths. We define the
W norm of M to be ||M||P = ?{[M,M]^2}1/p- If \\Щн* < oo, we say that
M is in HP. Finally we call Jiv the space of all cadlag martingales with finite
W norm.
It is simple to verify that || • ||WP is a norm if we identify martingales
that are almost surely equal. Indeed, the only property that is not obvi-
obvious is the subadditivity. For the case p = 1 for example, we have that
[M + N,M + N}1/2 < [M,M]^2 + [N,N]V2 since by the Kunita-Watanabe
inequality [M,N] < [M, M]1/2^,^]1/2, which gives the subadditivity. It is
also simple to verify that 7ip is a linear space. We state without proof the
important martingale inequalities known as the Burkholder-Davis-Gundy in-
inequalities, and often referred to by the acronym BDG. (For a proof one can
consult many different sources, including [146] and [44].)
Theorem 48 (Burkholder-Davis-Gundy Inequalities). Let M be a mar-
martingale with cadlag paths and let p > 1 be fixed. Let Mt* = sups<t |MS|. Then
there exist constants cp and Cp such that for any such M
E{[M,M]}}$ <cpE{{m;Y}^ <CpE{[M,M}f}p
for all t, 0 < t < oo. The constants are universal: they do not depend on the
choice of M.

194 IV General Stochastic Integration and Local Times
We have the following simple results concerning Ti1.
Theorem 49. Ti2 С Ti1 and local martingales of integrable variation are a
subset ofTt1.
Proof. First note that if M is a cadlag martingale, then E{^[M,M\OO\ <
(E{[M, Mjoo}I/2 which gives the first statement. For the second, if M has
integrable variation then [M,M]oo = ^8(ДМ8J < (%2S |AMS|J, whence
\/[M, M]oo < /0°° |<iMs|, and taking expectations, we have the result. D
Theorem 50. Ti2 is dense in Ti1 and bounded martingales are dense in Ti1.
Proof. Let M be a local martingale in Ti1. By the Fundamental Theorem of
Local Martingales (Theorem 25 of Chap. Ill) we have M = N + U where
N is a local martingale with jumps bounded by 1 and U has paths of locally
integrable variation. If (Tn)n>i is a sequence of stopping times increasing to oo
a.s., then ||MTn||-fti < ||M||^i, and also MTn converges to M in Ti1. Choose
Tn to be the first time [U, U]t has total variation larger than n, and also
I[N,N}T"| < n-1; that is, Tn = inf{t > 0 : /0*[17,U}t > n, or [N,N]t>n-1}
with Tn increasing to oo. Then [N, N]Tn < n and thus iVTn is in Ti1 for each n.
Therefore we turn out attention to UTn, and since [U, U]Tn < n + \А11тп\, we
need consider only the one jump of U at time Tn : AUrn ¦ Let ? denote
Letting ?k = ?l{|?|<fc}) since ^ is bounded we can compensate ?а
and call the resulting martingale V?. The compensator of ^1{(>т„} is m W2
because ?k is bounded. We have then that V? is in H2 and converges to Vn
as к —> oo in total variation norm, and hence also in Ti.1. This proves that Ti.2
is dense in Ti1, and since bounded martingales are dense in Ti.2, they too are
dense in Ti1. ?
Theorem 51. Let M be a local martingale. Then M is locally in Ti1.
Proof. By the Fundamental Theorem of Local Martingales we know that M =
N + U, where N has jumps bounded by a constant C and U is locally of
integrable variation. By stopping, we thus assume that N is bounded and
U has paths of integrable variation. The result then follows by the previous
theorem (Theorem 49). D
One can further show, by identifying Ttp with Lp through the terminal
value of the martingale, that Tip is a Banach space (in particular it is complete)
for each p > 1. One can further show that the dual space of continuous linear
functional on Tip is TiQ, where - + - = l,l<p<oo. This continues nicely
the analogy with IP except for the case p = 1. It can be shown (see [47])
that the dual of Ti1 is not Ti°°. It turns out that a better analogy than IP
is that of Hardy spaces in complex analysis, where C. Fefferman showed that
the dual of Ti1 can be identified with the space of functions of bounded mean
oscillation, known by its acronym as BMO. With this in mind we define the
space of BMO martingales.

4 Martingale Duality and Jacod-Yor Theorem 195
Definition. Let M be a local martingale. M is said to be in BMO if M is
in H2 and if there exists a constant с such that for any stopping time T we
have
?{(Моо - МТ-?\ТТ} < c2 a.s,
where Mo- = 0 by convention. The smallest such с is defined to be the BMO
norm of M, and it is written ЦМЦвмо- If the constant с does not exist, or if
M is not in H2, then we set ЦМЦвмо = со.
Note that in the above definition E{M^c\Jro\ < c2 (with the convention
that Mo_ = 0) and therefore ||M||W2 < ||М||вмо- Note in particular that
\\M\\bmo = 0 implies that M = 0. •
Let T be a stopping time and A ? J~t- Replacing Г with Тд shows that
the above definition is equivalent to the statement
?{(Moo - MT_J} < c2P(T < oo)
for every stopping time T. This in turn gives us an equivalent description of
the BMO norm:
where the supremum is taken over all stopping times T. Note that this second
characterization gives that || • Цвмо is a semi-norm, since it is the supremum
of quadratic semi-norms. An elementary property of BMO martingales is that
M € BMO if and only if all of the jumps of M are uniformly bounded. Thus
trivially, continuous I? martingales and bounded martingales are in BMO.
The next inequality is quite powerful.
Theorem 52 (Fefferman's Inequality). Let M and N be two local mar-
martingales. Then there exists a constant с such that
/•o
E{
./o
BMO-
Fefferman's inequality is a special case of the following more general result.
Theorem 53 (Strengthened Fefferman Inequality). There exists a con-
constant с such that for all local martingales M and N, and U an optional process,
E{ Г \Us\\d[M,N]s\} < cE{{ Г C
Jo Jo
Proof. Let Ct = /0* U2d[M, M]s and define H and К by
Ht = /7T- ,—7^=j=l{c!>o}, Kt =
y/Ct + /Gtl{t0}
BMO-

196 IV General Stochastic Integration and Local Times
Using integration by parts yields
Hfd[M,M}t = i{Ct>^J
From the definitions of H and К we have
н?к? > \u2i{Cl>0}.
The Kunita-Watanabe inequality implies
/•OO />OO
/ \Us\l{Cs=0}\d[M,N]s\<{ U2l{
Jo Jo
/•OO
= ( / l{cs
Jo
and since \d[M,N]s\ is absolutely continuous with respect to d[M,M]s as a
consequence of the Kunita-Watanabe inequality, we have
1 /-OO
}s\} = -^E{J \l{c3>o}Us\\d[M,N}s\}
= 0 a.s.,
/•OO
<
/•OO
E{ \HsKs\\d[M,N}s\}
Jo
< \\E{ I H^d[M,M]s} \\E{ I K2sd[N,N]s}.
But
/•OO /*O
E{ / fl?d[M, M]s} < E{ /
Jo Jo
and
/•OO />OO
/ K2d[N,N}s} = E{ ([N,N]00-[N,N}e-)dK?}
o Jo
= E{ f
Jo
But EiEUN^oolF.} - [N,N]S-} is bounded by \\N\\%MO on @, oo), hence
we have that
E{ Г K2d[N,N}s} = E{ r
Jo Jo
and the result follows. О

4 Martingale Duality and Jacod-Yor Theorem 197
Remark. The constant с in Theorems 52 and 53 can be taken to be y/2, as
can be seen from an analysis of the preceding proof.
Theorem 54. Let N e H2 ¦ Then N is in BMO if and only if there is a
constant с > 0 such that for all M € 7i2,
\E{[M,NU}\<c\\M\\nl.
Moreover \\N\\bmo < л/бс.
Proof. If N is in BMO, then we can take с = y/2 ^N\\bmo from Fefferman's
inequality (Theorem 52) and the remark following the proof of Theorem 53.
Now suppose that M is in H2 and that |.E{[M, JVjoo}! < c||M||^i; we want
to show that N is in BMO. We do this by first showing that |ЛГ0| < с a.s.,
and then showing that N has bounded jumps.
Let Л = {|JV0| > c}. Suppose P(A) > 0. Let $ = 5®^ilyl. Then ?;{|^|} =
1, and if we define the trivial martingale Mt = ? for all t > 0, then M ? 7i2
and ||M||wi = E{\?\} = 1, whence
J> С ——
This of course is a contradiction and we conclude |JV0| < с a.s.
We next show |AJV| < 2c. Since every stopping time T can be decomposed
into its accessible and totally inaccessible parts, and since each accessible time
can be covered by a countable collection of predictable times with disjoint
graphs, we can assume without loss of generality that T is either totally inac-
inaccessible or predictable. We further assume P(T > 0) = 1. Suppose then that
P(|AJVT| > 2c) > 0, and set
sign(AJVT)
Let M be the martingale consisting of ?l{t>r} minus its compensator. Then
M is in 7i2 and has at most one jump, which occurs at T. The jump is given
by
.}, T is predictable,
AMT = { s
I ? T is totally inaccessible.
Note that we also have ||M||^i is less than the expected total variation of M,
which in turn is less than 2.E{|?|} = 2. If T is totally inaccessible then
U} = E{AMTANT} = E{$ANT} =
P{\ANT\ > 2c) J
>2c>c\\M\\ni,
which is a contradiction. On the other hand, if T is predictable, then we know
that E{ANt\J~t-} = 0 and thus we are reduced to the same calculation and

198 IV General Stochastic Integration and Local Times
the same contradiction. We conclude that P(\ANt\ > 2c) = 0, and thus JV
has jumps bounded by 2c.
Last let T be any stopping time. Let M = N — NT and 77 = [N, JVjoo —
[iV,iV]T. Then M is in H2 and [M^M]^ = [M.iVjoo = rj. By our hypotheses
it now follows that
E{v] = ЕЦМ^и} <c\\M\\w =
= сЕ{л/п1{Т«х,}}
< сЕ{г,}г/2Р{Т < o
which in turn implies
E{[N,NU - [N,N}T} = E{V} < c2P{T < 00).
This then implies
Eil^NU^r} - [N,N]T < c2 a.s.,
and since
E^Noo - JVT_J|^T} = E{[N,NU\FT} - [N,N]T + iVo2l{T=o} + (AiVTJ
almost surely, we obtain
E{Nl~N2_}<6c2
which yields the result. ?
During the proof of Theorem 54 we proved, inter alia, that the jumps of a
local martingale in BMO are bounded. We formalize this result as a corollary,
which in turn has its own corollary.
Corollary 1. Let N be a local martingale in BMO. Then N has bounded
jumps.
Corollary 2. Let N be a local martingale in BMO. Then N is locally
bounded.
Proof. Let с be a bound for the jumps of N which we know exists by the
previous corollary. Next let Tn = inf{i > 0 : |iVt| > n]. Then |JVTn| < n + c,
and N is locally bounded. ?
The key result concerning Ti1 and BMO is the Duality Theorem which is
Theorem 55 that follows. First let us lay the foundation. For N chosen and
fixed in BMO we define the operator Ln from Ti1 to К by
LN(M) = E{[M,N}OO}
for all M in Ti1. Then one can easily check to see that Ln is linear, and Feffer-
man's inequality proves that it is bounded as well, and therefore continuous.
If BMO is the Banach space dual of H1 then it is also complete, a fact that
is apparently not easy to verify directly.

4 Martingale Duality and Jacod-Yor Theorem 199
Theorem 55 (The Dual of W} is BMO). The Banach space dual of all
(bounded) linear functionals on H1 can be identified with BMO. Moreover if
Ln is such a functional then the norms \\Lpf\\ and \\N\\bmo are equivalent.
Proof. Let N be in BMO. By Fefferman's inequality we have
\LN(M)\ = \E{[M,NU}\ <
for all M in Л1. This shows that Ln is in the dual of H1 and also that
\\Ln\\ < c\\N\\bmo- Note further that Ln cannot be trivial since Ln(N) =
E{[N, iVjoo} > 0 unless N is identically 0. Therefore the mapping F(N) = LN
is an injective linear mapping from BMO into Ti1*, the dual of Ti1.
Let L be an arbitrary linear functional in the dual of Ti1. We have
This means that L is also a bounded linear functional on Ti2. Since Ti2 is
isomorphic as a Hilbert space to the L2 space of the terminal random variables
of the martingales in Ti2, we have that there must exist a unique martingale
N in Ti2 such that for any M in Ti1 we have:
\LN{M)\ = ^{MooAUl = \E{[M,NU}\ <
Clearly Ln = L on Ti2, and since Ti2 is dense in Ti1 by Theorem 50, we
have that L and Ln are the same functional on Ц1. This shows that BMO
equipped with the norm \\LN\\ is isomorphic to H1* and thus, being the dual
of a Banach space, it is itself a Banach space and in particular it is complete.
Combining equation (*) with Theorem 54 we have that \\Lpf\\ and ЦЛ^Цдмо
are equivalent norms. This completes the proof. ?
While Fefferman's inequality, the space of BMO martingales, and the du-
duality of 7i°° and BMO are all of interest in their own right, we were motivated
to present the material in order to prove the important Jacod-Yor Theorem on
martingale representation, which we now present, after we recall the version
of the Hahn-Banach Theorem we will use.
Theorem 56 (Hahn-Banach Theorem). Let X be a Banach space and let
Y be a closed linear subspace. Then Y = X if and only if the only bounded
linear functional L which has the property that L(Y) = 0 is the functional
which is identically zero.
Theorem 57 (Jacod-Yor Theorem on Martingale Representation).
Let Л be a subset of H2 containing constant martingales. Then S(A), the
stable subspace of stochastic integrals generated by A, equals H2 if and only
if the probability measure P is an extremal point of Л42(А), the space of
probability measures making all elements of A square integrable martingales.

Then Q and R are both in M2(A), and P = \Q + \R shows that P
200 IV General Stochastic Integration and Local Times
Proof. The necessity has already been proved in Theorem 38. By the Hahn-
Banach Theorem, Ti1 = S{A) if and only if L(S{A)) = 0 implies L is identi-
identically zero, where L is a bounded linear functional. Let Ibea bounded linear
functional which is such that L{S{A)) = 0. Then there exists a martingale
N in BMO such that L = Ln- The local martingale N is locally bounded,
so by stopping we can assume it is bounded and that JVo = 0. (See the two
corollaries of Theorem 54.) Let us also assume it is not identically zero, and
let с be a bound for N. We can then define two new probability measures Q
and R by
2c ' 2c
ft Л О / Л \ Ч -М—v 1 S^L . 1 "I—fc » . 1 -M—v *
IS
not extremal in Л42(А), a contradiction. Therefore we must have that L is
identically zero and we have that Ti1 = S{A). As far as Ti2 is concerned, it
is a subspace of Ti1, hence Ti2 С <S(.4). But by construction of S(A), it is
contained in H2, and we have martingale representation. ?
5 Examples of Martingale Representation
In Sect. 3 we have already seen the most important example of martingale
representation, that of Brownian motion. In this section we give a method
to generate a family of examples which are local martingales with jumps,
and which have the martingale representation property. The limitations are
that the family of examples is one dimensional (so that we exclude vector-
valued local martingales such as n-dimensional Brownian motion), and that
the descriptions of the jumps are all of the same rather simple kind.
The idea is to construct a class of local martingales Ti such that for 1бН
we have both that Xq = 0 and the compensator of [X, X]t is At = t. If X has
the martingale representation property, then there must exist a predictable
process H such that
[X,X]t-t= / HsdXs. (*)
Jo
The above equation is called Emery's structure equation, and it is written
(in a formal sense) in differential notation as
In order to establish that solutions to Emery's structure equation actually
exists we write it in a form resembling a differential equation:
d[X,X]t = dt + <j>(Xt-)dXt {**)
Equation (**) is unusual and different from the stochastic differential equa-
equations considered later in Chap. V, since while the unknown is of course the

5 Examples of Martingale Representation 201
local martingale X (and part of the structure equation is to require that any
solution X be a local martingale), no a priori stochastic process is given in
the equation. That is, it is lacking the presence of a given stochastic driving
term such as, for example, a Brownian motion, a compensated Poisson pro-
process, or more generally a Levy process. Since therefore no probability space
is specified, the only reasonable interpretation of equation (**) is that of a
weak solution. That is, we want to show there exists a filtered probabil-
probability space (ft, J-, F, P) satisfying the usual hypotheses, and a local martingale
X, such that X verifies equation (**). It would also be nice to have weak
uniqueness which means that if X and Y are solutions of (**) for a given ф,
possibly defined on different filtered probability spaces, then X and Y have
the same distribution as processes. That means that for every A, a Borel
set on the function space of cadlag functions mapping R+ to R, we have
P{uj : t t-> Xt{ui) G A) = Q(uj : t i-> Xt(oj) e A), where P and Q are the
probability measures where X and У are respectively defined.
Inspired by knowledge of stochastic differential equations, it is natural to
conjecture that such weak solutions exist and are unique if the coefficient
ф is Lipschitz continuous.14 This is true for existence and was proven by
P. A. Meyer [179]; see alternatively [136]. Since the proof uses weak conver-
convergence techniques which are not within the scope of this book, we omit it.
Theorem 58 (Existence of Solutions of the Structure Equation). Let
ф : R —» К be Lipschitz continuous. Then Emery's structure equation
[X,X]t-t= [ ф{Ха.NХа (***)
Jo
has a weak solution with both (Xt)t>o and (/0 ф(Х3-)AХа)г>о local martin-
martingales.
The issue of uniqueness is intriguing. Emery has shown that one has
uniqueness when ф is linear, but uniqueness for others ф'з, including the Lip-
Lipschitz case, is open. The next theorem collects some elementary properties of
a solution X.
Theorem 59. Let X be a (weak) solution of (* * *). Then the following hold.
(i) E{X%} = E{[X,X]t} = t, and X is a square integrable martingale on
compact time sets.
(ii) All jumps of X are of the form AXt = </>(Xt_).
(Hi) X has continuous paths if and only if ф is identically 0, in which case X
is standard Brownian motion,
(iv) If a stopping time T is a jump time of X, then it is totally inaccessible.
Proof. We prove the statements in the order given. Since a solution X
and the integral term are both required to be local martingales, we know
Lipschitz continuity is defined and discussed in Chap. V.

202 IV General Stochastic Integration and Local Times
there exists a sequence (Tn)n>i of stopping times increasing to oo such that
/otAT" 4>{Xs-)dXs is in L\ Therefore E{[X,X}tATJ = E{tATn}, and apply-
applying the Monotone Convergence Theorem to each side of the equality in this
equation yields -E{pf, X]t} = t which further implies that X is a martingale
on [0,t] for each t < oo and that E{X?} = E{[X,X]t} = t.
For the second statement, recall that
s<t
and hence we have A[X,X]t = (AXtJ = ф{Хг-)АХи and dividing both
sides by AXt (when it is not zero) gives the result.
For the third statement, suppose ф is identically zero. Then by the second
statement X has no jumps and must be continuous. Since X is then a con-
continuous local martingale with [X, X\t = t, it is Brownian motion by Levy's
Theorem. For the converse, if we know that X is continuous, if it is non-trivial
it has paths of infinite variation since it is a local martingale. Thus so too does
the term /0 ф(Xs-)dXs. But notice that this term is the right side of equa-
equation (* * *) which is of finite variation, and we have a contradiction, so we
must have that ф is zero.
For the fourth statement, we implicitly stop X so that it is a uniformly
integrable martingale. Next let T be a jump time of X. Then T = Та Л Tg
where A is the accessible part of T and В is the totally inaccessible part of
T. Since Та can be covered by a countable sequence of predictable times with
disjoint graphs, we can assume Та is predictable without loss of generality.
Thus it suffices to show P(A) — 0. However since Та is predictable, and
X is a uniformly integrable martingale, we have E{AXta\Fta-} = 0. But
АХтА = ф(ХтА-) by part (ii) of this theorem, and ф(Хтл-) € У~тл- since
Та is predictable, which implies that the jump of X at Та is zero, which in
turn implies that P(A) =0. ?
Let us now consider a special class of structure equations where ф is
assumed to be an affine function. That is, we assume ф is of the form
ф(х) — a + fix. We analyze these special cases when a and /3 vary. Emery has
named the solutions corresponding to affine structure equations the Azema
martingales, since J. Azema's work on Markov processes and expansion of
filtrations led him to the amazing formula of "the" Azema martingale given
later in Sect. 7. Equation (**) now becomes
d{X,X}t = dt + (a + pXt^)dXu
and when /3 = 0, it reduces to
d[X, X\t = dt + otdXt,
and when in addition a = 0 we have seen that X is standard Brownian mo-
motion. Note that Levy's Theorem gives us weak uniqueness in this case, since

5 Examples of Martingale Representation 203
any solution with a = /3 = 0 must have the same distribution, namely that of
Wiener measure. We have much more, as we see in the next theorem. However
it has a long and difficult proof. Rather than present it, we refer the inter-
interested reader to the excellent treatment of M. Emery, in his original paper [69]
proving the result.
Theorem 60 (Emery's Uniqueness Theorem). Let X be a local martin-
martingale solution of the structure equation
d[X,X]t = dt + (a + pXt-)dXu Xo = x0.
Then X is unique in law. That is, any other solution Y must have the same
distribution as does X. Moreover X is a strong Markov process.
The uniqueness is especially significant in light of the next theorem. By
martingale representation we mean that every square integrable martin-
martingale can be represented as a stochastic integral with respect to one fundamen-
fundamental local martingale.
Theorem 61. Consider the equation
[X,X]t-t= I
Jo
on a filtered probability space (Q,J-,?,P) which satisfies the usual hypotheses.
Then X has martingale representation for its completed natural filtration if
and only if the law P is an extreme point of the convex set of all probabilities on
@, J-, F) for which X is a martingale and verifies the equation. Moreover if the
equation with fixed initial condition Xq has weak uniqueness of solutions, then
every solution X of the equation has martingale representation with respect
to the smallest filtration satisfying the usual hypotheses and to which X is
adapted.
Proof. By the Jacod-Yor Theorem (Theorem 57) we need to verify that P is
extremal in the set M2 of all probability measures such that X is a square
integrable martingale. It is clearly true if P is extremal. Suppose then that P
is not extremal, and let Q and R both be in Л42, such that P = AQ + A — X)R,
with 0 < Л < 1. Both Q and R are absolutely continuous with respect to P,
so under Q and R the terms [X, X] and JQ (fi(Xs-)dXs are the same a.s. (resp.
dQ and dR). Therefore X satisfies equation (®) for both Q and R as well as
P. Thus if P is not extremal in Л42 then it is also not extremal in the set of
probability measures such that X satisfies equation (®).
To prove the second statement, let О denote the canonical path space of
cadlag paths, with X being the projection process given by Xt(uj) = uj(t). We
have just seen that among the solutions of equation (<8>), the ones having mar-
martingale representation are those whose law constitutes an extremal probability
measure. But if weak uniqueness holds, the collection of all such probabilities
consists of only one, and thus extremality is trivial. ?

204 IV General Stochastic Integration and Local Times
We now examine several special cases as a and /3 vary. Let a = /3 = 0,
and we have seen that X is standard Brownian motion. Because of Theo-
Theorem 61 we conclude that (one dimensional) Brownian motion has martingale
representation, recovering a special case of Theorem 43.
Next suppose a — 1 and /3 = 0. In this case the equation of Theorem 60
with Xo = 0 becomes, in integral form,
Therefore Xt — [X,X]t — t and hence X is a finite variation martingale.
Moreover AXt = 1, so X only jumps up, with jumps always of size 1. Now let
N be a standard Poisson process with arrival intensity A = 1, and let Xt —
Nt~t, the compensated Poisson process. Then X satisfies the equation, and by
weak uniqueness all such X are compensated Poisson processes with A = 1.
We conclude that a compensated standard Poisson process has martingale
representation with respect to its natural (completed) filtration. For general
a (and not just a = 1), but still with C = 0, it is simple to check that
is the unique solution of the equation of Theorem 60 if N is a standard Poisson
process. Note that (as is well known) Xa converges (weakly) to Brownian
motion asa-tO.
We now consider the more interesting cases where /3^0. We repeat the
equation of Theorem 60 here (in integrated form) for ease of reference:
[X,X]t=xo + t +
Jo
Observe that X is a solution of the above equation if and only if X + j is a
solution of
with initial condition Xo = xo + § • Therefore without loss of generality we
can assume that a = 0 and we do this from now on.
We have two explicit examples for equation (®®). When /3 = -1, we take
Xt = -=Mt = V-sign (Bt)y/(t-gt),
where M is Azema's martingale15, В is standard Brownian motion, and
gt = sup{s < t : Bs = 0}. By Theorem 86 of Sect. 8 of this chapter, we know
that [X,-X]? = 0 and [X,-X]t = 2gt. Integration by parts gives /0 Xs-dXs —
(t - 9t) ~ 9t = t - [X,X]t, since [X,X]t = J23<t(AXaJ, because of course
[X, X]% = 0. This proves the assertion that for /3 = — 1, Xt = -4=Mt, where
15 Azema's martingale is treated in detail in Sect. 8 later in this chapter.

6 Stochastic Integration Depending on a Parameter 205
M is Azema's (original) martingale as presented in Sect. 8, and thus we have
martingale representation for Azema 's martingale.
Our second (and last) explicit example is for /3 = —2. In this case our
equation becomes
[X,X]t-t = -2 [ Xs-
Jo
and using integration by parts we obtain 2 /0 Xs-dXs = X"l — [X, X\t- Equat-
Equating terms gives Xf = t, and since E{Xt} = 0 for all t because xq = 0 we
deduce that
p(xt = уД) = p(xt = -Vt) = I
for all t > 0. The jumps of X occur from a change of sign, and they arrive
according to a Poisson process with intensity j^dt. Such a process can be seen
to be a martingale (as in [204]) by constructing a process X with a distribution
as above and with filtration F. Then for 0 < s < t,
where N is independent of the cr-algebra J-s and has a Poisson distribution
with parameter Л = \ ln(|). In this way it is easily seen to be a martingale.
We call it the parabolic martingale. Once again we are able to conclude:
we have martingale representation for the parabolic martingale.
6 Stochastic Integration Depending on a Parameter
The results of this section are of a technical nature, but they are needed for our
subsequent investigation of semimartingale local times. Nevertheless they have
intrinsic interest. For example, Theorems 64 and 65 are types of Fubini Theo-
Theorems for stochastic integration. A more comprehensive treatment of stochastic
integration depending on a parameter can be found in Strieker-Yor [219] and
Jacod [103]. Throughout this section (А, A) denotes a measurable space.
Theorem 62. Let Yn(a,t,tj) be a sequence of processes that are (i) A®
Б(К+) <8> T measurable, and (ii) for each fixed a the process Yn(a,t,uj) is
cadlag. Suppose Yn(a,t, •) converges in ucp for each a € A. Then there exists
an A® B(R+) <8>F measurable process Y = Y(a, t,ui) such that
(a) Y(a,t, ¦) — Ншп-юо Yn(a,t, ¦) with convergence in ucp;
(b) for each a e A, Y is a.s. cadlag.
Moreover there exists a subsequence nk{a) depending measurably on a such
that lim^a^oo Y^ = Yt uniformly in t on compacts, a.s.

206 IV General Stochastic Integration and Local Times
Proof. Let S^itj = supt<u |Уг(М, •) - Y^>(a,t,-)\. Since Yi is cadlag in t
the function (a,oj) >—> S^ j is A <8> J7 measurable. By hypothesis we have
Ит^-юо S">itj = 0 in probability. Let no(a) = 1, and define inductively
nk(a) = inf{m > max(&,rafc_i(a)) : sup P{S^ti>j > 2~k) < 2~~k}.
i,j>m
We then define
Zk(a,t,u)=Yn^(a,t,c).
Since each a t—> rifc(a) is measurable, so also is Zk. Define
t<u
then also (а,ш) н-> Т"^(о>) is jointly measurable, since Z* have cadlag paths
(in t). Moreover by our construction P(Tgkk+m > 2~k) < 2~k for any m > 1.
The Borel-Cantelli Lemma then implies that Ит^-юо Т*г^ = О almost surely,
which in turn implies that
lim Zl(a,t, •) exists a.s.,
t—>oo
with convergence uniform in t. Let Ла be the set where Z% converges uniformly
(note that Aa G Л <8> T and Р(Л°) = 1, each fixed a), and define
Then У is cadlag thanks to the uniform convergence, and it is jointly mea-
measurable. D
Theorem 63. Let X be a semimartingale with Xq =0 a.s. and let H(a, t, из) =
Ht{uj) be A®V measurable16 and bounded. Then there is a function Z(a,t,u>)
in A® B(R+) <S> T such that for each a & A, Z(a,t,uj) is a cadlag, adapted
version of the stochastic integral /0 HgdXs.
Proof. Let H = {H G bA<S>V such that the conclusion of the theorem holds}.
If К = K(t,uj) e hV and / = f(a) e ЪА, and if H(a,t,u) = f(a)K(t,w),
then
/ H(a,s,-)dXs= [ f(a)K(s,-)dX, = f(a) [ K(s,-)dXs,
o Jo Jo
and thus clearly H = fK is in Ц. Also note that 7i is trivially a vector space,
and that Я of the form H = fK generate ЪА <8> V.
Next let Hn ? Ti and suppose that Hn converges boundedly to a process
Я G ЪА <8> V. By Theorem 32 (for example) we have that Я" • X converges
uniformly in t in probability on compacts, for each a. Therefore H ? 7i, and
an application of the Monotone Class Theorem yields the result. ?
16 Recall that V denotes the predictable cr-algebra.

6 Stochastic Integration Depending on a Parameter 207
Corollary. Let X be a semimartingale (Xo = 0 a.s.), and let H(a,t,u>) =
Щ(и>) G A <g> V be such that for each a the process Ha G L(X). Then there
exists a function Z(a, t,uj) = Z? 6 Д<8>??(]ВЦ_) <g> .F such that for each a, Z?
is an a.s. cadlag version of /0 HfdXs.
Proof. By Theorem 32 the bounded processes Za<k = Hal^Ha.\<kyX converge
to HaX in мер, each a. But Za>k can be chosen cadlag and jointly measurable
by Theorem 63. The result now follows by Theorem 43. D
Theorem 64 (Pubini's Theorem). Let X be a semimartingale, Щ =
H(a,t,u>) be a bounded A®V measurable function, and let jjl be a finite mea-
measure on A. Let Z? = /0 HfdXs be A® B(R+) <8>T measurable such that for
each a, Za is a cadlag version of Ha ¦ X. Then Yt = fA Zf/j,(da) is a cadlag
version of H ¦ X, where Ht = JA H?n(da).
Proof. By pre-stopping we may assume without loss of generality that X G
H2, and because the result holds for the finite variation part of the canonical
decomposition of X by the ordinary Stieltjes Fubini Theorem, we may further
assume that X is a martingale with E{[X,X]^} < сю. Next suppose Щ
is of the form H(a,t,u>) = K(t,u>)f(a) where К G hV and / is bounded,
measurable. Then К G L(X) and / \f(a)\n(da) < сю. In this case we have
Zta = f(a)K ¦ X, and moreover
/ Z?n{da) = / f{a)K • Xn{da) = К ¦ X f f{a)n{da)
= (J' f(a)i*(da)K) ¦ X
= HX.
By linearity the same result holds for the vector space V generated by pro-
processes of the form K(t,u>)f(a) with К G ЪР and / bounded, measurable.
By the Monotone Class Theorem it now suffices to show that if Hn G V
and limn^oo#n = H, then the result holds for H. Let Z%t = Щ ¦ X, the
cadlag version. Then by Jensen's and the Cauchy-Schwarz inequalities,
[
a t
[ sup |
a t
= / ?{suP |Z» t - Z»|>(da)
Ja t
< 4 I E{{Zant00 - Z^fj^da) = 4 I E{[Zan - Z\ Zan - Z°]oo}Ai(do)
Ja Ja
by Doob's quadratic inequality for the martingales Z" and Za, and by Corol-
Corollary 3 of Theorem 27 of Chap. II. Continuing, the preceding equals
= 4 f E{ Г {HI, - H^2d{X,X]s}v(da),
Ja Jo

208 IV General Stochastic Integration and Local Times
and the above tends to 0 by three applications of the Dominated Convergence
Theorem.
We conclude from the preceding that
/ sup \Zl t - Z?\n(da) < oo a.s.
Ja t
and therefore JA \Z?\/j,(da) < oo for all t, a.s. Moreover
?{sup | / Zantlx{da) - f Z?n(da)\} < E{ f sup |Z»t - Z
t Ja Ja Ja t
which tends to 0. Therefore taking Hn^t = J H^tn(da) we have Hn ¦ Xt =
fA Z^t^{da) converges in ucp to f Z^/j,(da). Since Hn ¦ X converges to H ¦ X
by Theorem 32, we conclude H X = J Z?n(da). ?
The version of Fubini's Theorem given in Theorem 64 suffices for the
applications of it used in this book. Nevertheless it is interesting to determine
under what more general conditions a Fubini-type theorem holds.
Theorem 65 (Fubini's Theorem: Second Version). Let X be a semi-
martingale, let H^ = H(a,t,u>) be A®V measurable, let/j, be a finite positive
measure on A, and assume
Letting Z% = /0* H%dXs be A® B(R+) ® T measurable and Za cadlag for
each a, then Yt = fA Z^n(da) exists and is a cadlag version of H ¦ X, where
f
Proof. By pre-stopping we may assume without loss of generality that X G H2
and that \\Ha\\L2(dfl) is (H2,X) integrable. Let X = iV + Л be the canonical
decomposition of X. Then
E{ (H°Jv(da)d[N,N}s}+E{( \\H°\\L4dll)\dAs\J}
Jo Ja Jo
< oo.
Next observe that
лОО /*ОО
E{ \\H°\\L4dfl)\dAs\}>E{\\ / \Has
Jo Jo
/
o
= с j E{ Г \Hf\\dAs\Mda).
Ja Jo
Also

6 Stochastic Integration Depending on a Parameter 209
E{ Г j {Has?v{da)d[N,N]s} = f E{ Г{HasL[N,N}s}^da),
Jo Ja Ja Jo
and therefore ?{/0°° \H°\\dAs\} < oo and E{f™(H°Jd[N,N]s} < oo for
уц-almost all a G A. Whence Ha G L(X) for /j, almost all a G A.
Next define Hn = i71{|^|<n}, and the proof of Theorem 64 works as well
here. ?
The hypotheses of Theorem 65 are slightly unnatural, since they are not
invariant under the transformation
Я - -±-Ha pi -> фЫёа)
<p(a)
where <p is any positive function such that f ip(a)n(da) < сю. This can
be alleviated by replacing the assumption (JA(HaJyu(da)I/2 e L(X) with
(/ ^H(J) <"(^a)I</2 G L(X) for some positive <p e Ll{dn). One can also relax
the assumption on ц to be cr-finite rather than finite.
Example. The hypotheses of Theorem 65 are a bit strange, however they
are in some sense best possible. We give here an example of a parameterized
process (.Hra)a6A) a positive, finite measure jjl on A, and a semimartingale X
such that
(i) (a, t) —» Щ is A <g> V measurable,
(ii) Ha e L(X), each a ? A, and
(Hi) fA\H°\p{da) e ЦХ)
but such that if Z" = Jo H"dXs then JA Z^/j,(da) does not exist as a Lebesgue
integral. Thus a straightforward extension of the classical Fubini Theorem for
Lebesgue integration does not hold.
Indeed, let A = N = {1, 2,3,... } and let ц be any finite positive measure
on A such that ц({а}) > 0 for all a ? A. Let X be standard Brownian motion,
let to < t\ < ti < ¦ ¦ ¦ be an increasing sequence in [0,1], and define
Щ = -
a
Then
1 a
a=l
is in L2(dt), whence Ht = fA \H^\/j,(da) G L(X), and moreover if t > 1,
я • xt =
a=l
where the sum converges in I?. However if t > 1 then

210 IV General Stochastic Integration and Local Times
z? = Ha-xt = -MWrHta -ta_i
a
and
/
jA
= °° a-s-
a=l
because {ta—ta-i) 1^2(Xta -Xta_1) is an i.i.d. sequence and J2T=ia 1 = °°-
Note that this example can be modified to show that we also cannot replace
the assumption that
/
A
with the weaker assumption that {jA{Hf)pn{da))l/'p e L(X) for some p < 2.
7 Local Times
In Chap. II we established Ito's formula (Theorem 32 of Chap. II) which
showed that if / : R —> R is C2 and X is a semimartingale, then f(X) is again
a semimartingale. That is, semimartingales are preserved under C2 transfor-
transformations. This property extends slightly: semimartingales are preserved under
convex transformations, as Theorem 66 below shows. (Indeed, this is the best
one can do in general. If В = (Bt)t>o is standard Brownian motion ai}d
Yt = f(Bt) is a semimartingale, then / must be the difference of convex func-
functions. (See Cinlar-Jacod-Protter-Sharpe [34].) We establish a related result in
Theorem 71, later in this section.) Local times for semimartingales appear in
the extension of Ito's formula from C2 functions to convex functions (Theo-
(Theorem 70).
Theorem 66. Let f : R —» К be convex and let X be a semimartingale. Then
f(X) is a semimartingale and one has
f(Xt) - f(X0) = / f'(Xs_)dXs + At
Jo+
where f is the left derivative of f and A is an adapted, right continuous,
increasing pmcess. Moreover AAt = f(Xt) — f(Xt_) — f'(Xt-)AXt.
Proof. First suppose \X\ is bounded by n, and in H2, and that Xo = 0.
Let g be a positive C°° function with compact support in (—oo,0] such that
/!^5(s)^s = 1- Let fn{t) = n jTtx, f(t + s)g(ns)ds. Then /n is convex and
C2 and moreover f'n increases to /' as n tends to сю. By Ito's formula
U(Xt) - /„(Xo) = f fn{Xs-)dXs + Ant
Jo

7 Local Times 211
where
2 Jo
The convexity of / implies that An is an increasing process. Letting n tend
to oo, we obtain
-I'1'
Jo
f(xt) - f(x0) = / f'(xs_)dxs + A (*)
./o
where limn^oo Л" = At in L2, and where the convergence of the stochastic
integral terms is in H2 on [0, t].
We now compare the jumps on both sides of the equation (*). Since
f° f'(Xs-)dXs = 0 we have that Ao = 0. When t > 0, the jump of the
left side of (*) is f(Xt) — f{Xt-), while the jump of the right side equals
f'(Xt_)AXt + AAt. Therefore AAt = f(Xt) - f(Xt-) - f'{Xt-)AXt, and
the theorem is established for \X\ bounded by n and in H2.
Now let X be an arbitrary semimartingale with Xo — 0. By Theorem 13 we
know there exists a sequence of stopping times (Tn)n>i, increasing to сю a.s.
such that XT ~ e TL2 for each n. An examination of the proof of Theorem 13
shows that there is no loss of generality in further assuming that \XT ~\ < n,
also. Then let Yn = JO[0T™) and we have
f(Ytn) - f(Yon) = f f(y,n_)dY? + Ant,
Jo
which is equivalent to saying
f(Xt) - /(Xo) = Г f'{X,-)dX, + A?
Jo
on [0,Tn). One easily checks that (A"+1)T"- = (An)T"-, and we can define
A = An on [0,Tn),each n.
The above extends without difficulty to functions g : K2 —» R of the form
g(Xt,H) where H is an Tq measurable random variable and x ^ g(x,y) is
convex for every y. For general X we take Xt = Xt — Xo, and then f(Xt) =
f(Xt + Xo) = g(Xt, Xo), where g(x, y) = f(x + y). This completes the proof.
?
Notation. For x a real variable let x+, x~ be the functions x+ = max(:r,0)
and x~ = — min(:r,0). For х, у real variables, let x V у = max(x) y) and
x Л у = min(a;, у).
Corollary 1. Let X be a semimartingale. Then \X\, X+, X~ are all semi-
martingales.

212 IV General Stochastic Integration and Local Times
Proof. The functions f(x) — \x\, g(x) = x+, and h(x) = x~ are all convex, so
the result then follows by Theorem 66. ?
Corollary 2. Let X, Y be semimartingales. Then X V Y and X A Y are
semimartingales.
Proof. Since semimartingales form a vector space and xVy = ~(\x — y\+x + y)
and x Ay = T,(x + у — \x — y\), the result is an immediate consequence of
Corollary 1. ?
We can summarize the surprisingly broad stability properties of semi-
semimartingales.
Theorem 67. The space of semimartingales is a vector space, an algebra, a
lattice, and is stable under C2, and more generally under convex transforma-
transformations.
Proof. In Chap. II we saw that semimartingales form a vector space (The-
(Theorem 1), an algebra (Corollary 2 of Theorem 22: Integration by Parts), and
that they are stable under C2 transformations (Theorem 32: Ito's Formula).
That they form a lattice is by Corollary 2 above, and that they are stable
under convex transformations is Theorem 66. G
Definition. The sign function is defined to be
... /l, ifar>0,
sign(x) = i
1—1, if x < 0.
Note that our definition of sign is not symmetric. We further define
ho(x) = \x\ and ha(x) = \x — a\. (*)
Then sign(:r) is the left derivative of ho(x), and sign(:r—a) is the left derivative
of ha{x). Since ha(x) is convex by Theorem 66 we have for a semimartingale
X
ha(Xt) = \Xt -a\ = \X0-a\+ f sign(Xs_ - a)dXs + Aat, {**)
where A^ is the increasing process of Theorem 66. Using (*) and (**) as
defined above we can define the local time of an arbitrary semimartingale.
Definition. Let X be a semimartingale, and let ha and Aa be as defined in
(*) and (**) above. The local time at a of X, denoted Щ = La(X)t, is
defined to be the process given by
Lat =Aat- Y, iha(Xs) - K{XS_) - h'a(Xs_)AXs}.
0<s<t
Notice that by the corollary of Theorem 63 the integral fQ+ sign(Xs_ — a)dXs
in (**) has a version which is jointly measurable in (a,t,u>) and cadlag in

7 Local Times 213
t. Therefore so does (A")t>o, and finally so too does the local time L". We
always choose this jointly measurable, cadlag version of the local time, without
any special mention. We further observe that the jumps of the process Aa
defined in (**) are precisely ^23<t{ha(Xa) - ha(Xs_) — ha(Xs_)AXs} (by
Theorem 66), and therefore the local time (!<" )t>o is continuous in t. Indeed,
the local time La is the continuous part of the increasing process Aa.
The next theorem is quite simple yet crucial to proving the properties of
La that justify its name.
Theorem 68. Let X be a semimartingale and let La be its local time at a.
Then
i-t
(Xt -a)+ - (Xo -a)+ = / l{x,_>a}dXs+ 2^ l{xa->a}{Xs - a)~
Jo+ o<s<t
0<s<t
(Xt - a)' - (X
o -a)-=- l{Xs.<a}dXs + V
Jo+ o<s<t
Jo+ o<s<t
0<s<t
Proof. Applying Theorem 66 to the convex functions f(x) = (x — a)+ and
g(x) — (x — a)~ we get
f(Xt) = /(Xo) + Г f'(X,-)dX, + C+
Next let
g(Xt) = g(X0) + j g'{Xs.)dXs + C~
Jo+
D+ =C+- ^ {/№) " /№-) " f'(Xa-)AX.}t
0<s<t
0<s<t
and subtracting the formulas we get Ct+ - Ct~ — 0 and hence Df = Dt~. Also
Df + ДГ = Lf, so that Dt+ = Dt~ = |l/", and the proof is complete. ?
The next theorem, together with the "occupation time density" formula
(Corollary 1 of Theorem 70), are the traditional justifications for the termi-
terminology "local time."

214 IV General Stochastic Integration and Local Times
Theorem 69. Let X be a semimartingale, and let L% be its local time at the
level a, each a 6 №. For a.a. w, the measure in t, dL^(u), is carried by the set
{s:Xs_(to)=Xs{to) = a}.
Proof. Since L" has continuous paths, the measure dL^(u) is diffuse, and since
{s : Xs-(u>) = a} and {s : Xs-(w) = Xs(w) = a} differ by at most a countable
set, it will suffice to show that dL^(u>) is carried by the set {s : Xs^(u>) = a}.
Suppose 5, T are stopping times and that 0 < S < T such that [5, T) с
{(s,u) : A~s_(w) < a} = {A"_ < a}. Then X < a on [S,T) as well. Hence by
the first equation in Theorem 68 we have
(A -O)f -(Л -OM = / l{X,_>a}ttAs+ > l{Xs_>a}(^s -a)
/ Q '
^ i
S<s<T
However the left side of the above equation equals zero, and all terms on the
right side except possibly ^{Lj, — Las) also equal zero. Therefore ^{Lj, — Las) =
0, whence Щ,~Ь8.
Next for r€Q, the rationale, define the stopping times Sr(u>), r > 0, by
олл-/г' ifAV-M<a,
I сю, it Xr-(u>) > a.
Then define
Tr{to) = ini{t > Sr(uj) : Xt-(w) > a}.
Then [5,.,^) С {A"_ < a}, and moreover the interior of the set {X- <
a} equals (JreQ r>o(^'^)- As we have now seen, dLa does not charge the
interior of the set {X- < a}. This set is open on the left, hence it differs from
its interior by an at most countable set, and since dLa is diffuse it doesn't
charge countable sets. Thus dLa does not charge the set {A~_ < a} itself.
Analogously one can show dLa does not charge {A~_ > a}. Hence its
support is contained in the set {a~_ = a}, and we are done. ?
The next theorem gives a very satisfying generalization of Ito's formula
(Theorem 32 of Chap. II).
Theorem 70 (Meyer-Ito Formula). Let f be the difference of two convex
functions, let f be its left derivative, and let jjl be the signed measure (when
restricted to compacts) which is the second derivative of f in the generalized
function sense. Then the following equation holds:
f(Xt)-f(X0)= f f'(Xs_)dXs+
Jo+
0<s<t
1 Z
- /
г J-
(*)

7 Local Times 215
where X is a semimartingale and L" = Ц:{Х) is its local time at a.
Proof. Notice that equation (*) is trivially true if / is an affine function (i.e.,
f{x) — ax + b). Next let us first assume that ц is a signed measure with finite
total mass on a compact interval. Define a function g by
{x) = - J \x- y\n(dy).
It is well known that / and g differ by at most an affine function h(x) = ax+b.
Therefore without loss of generality we can assume that the function / in (*)
is of the form f(x) — \ j \x — y\/j,(dy). Then f'(x) = | J*sign(:r — y)n(dy) and
f"(x) — /j,(dx). Moreover if
J*= J2 \Xs-y\-\Xs.-y\-sign(Xs_~y)AXs,
0<s<t
then
0<s<t
Also, letting H\ = \Xt — y\ — \X0 - y\, one has
l- f Jytlx{dy) = J2 {/№) - f(Xs-) - f'(Xs_)AXs}. A)
0<s<t
. B)
Next consider Zt" = /0*+sign(Xs_ - y)dXs, and let Zy be the B(M) ®
T measurable version, which we know exists by Theorem 63. By
Pubini's Theorem (Theorem 64) we have
^ C)
Since Ц = Щ - J? - Z%, we have
\ I L^(da) = ~J HMda) -\J JtKda) - \ J Zatix{da) D)
and combining D) with A), B), and C), yields the formula (*).
It remains to reduce the general case to that of jjl supported on a compact
interval with finite total mass. By linearity it further suffices to treat the case
where / is convex. Define
[/(«) + f'(n)(x-n), ifx>n,
fn{x) = < f(x), if -n<x<n,
[f(-n)+f'(-n)(x + n), \ix<-n.
Then /n is also convex and its second (generalized) derivative jjin{dx) agrees
with jj, on \—n,ri) and is of finite total mass. Let У" = fn(X), and Tn =

216 IV General Stochastic Integration and Local Times
inf{* > 0 : \Xt\ > n}. Note that Yn = f{X) on [0,Tn), and that Щп = О for
all a, \a\ > n, by Theorem 69. Therefore / L?nn{da) = fLfn{da) for t < Tn,
and by the preceding (*) holds for Y = fn{X) = f(X) on [0,Tn). Therefore
(*) holds on [0,T"). Since the stopping times (Tn)n>i increase to сю a.s., we
have (*) holds on all of R+ x fi, and the theorem is proved. ?
The next formula gives an interpretation of semimartingale local time as
an occupation density relative to the random "clock" d[X,X}^.17
Corollary 1. Let X be a semimartingale with local time (La)aeK- Let g be
a bounded Borel measurable function. Then a.s.
/ Latg(a)da= / g(Xe-)d[X,X]ct.
J-oo Jo
Proof. Let / be convex and C2. Comparing (*) of Theorem 70 with Ito's
formula (Theorem 32 of Chap. II) shows that
/ LaJ"(a)da= / f»(Xa-)d[X,X]ct
J—oo JO
where jji{da) is of course f"(a)da. Since the above holds for any continuous
and positive function /", a monotone class argument shows that it must hold,
up to a P-null set, for any bounded, Borel measurable function g. D
We record here an important special case of Corollary 1.
Corollary 2. Let X be a semimartingale with local time (La)aeu- Then
[x,xyt= Г
J-c
Ltc
Corollary 3 (Meyer-Tanaka Formula). Let X be a semimartingale with
continuous paths. Then
ft
у у I I I • / у \ i у i 7" U
Jo+
Proof. This is merely Theorem 70 with f(x) = \x\, which implies n(da) =
2eo(da), point mass at 0. The formula also follows trivially from the definition
of L°. ?
If Xt ~ Bt is a standard Brownian motion, then Mt — fQ sign(Bs)dBs is a
continuous local martingale, and [M,M]t — /0 sign(BsJd[B,B]s = [B,B]t =
t. Therefore by Levy's Theorem (Theorem 39 of Chap. II) we know that Mt
17 Recall that [X, X]c denotes the path-by-path continuous part oft^ [X, X]t with
[X,X]c0 = 0.

7 Local Times 217
is another Brownian motion. We therefore have, where Bq = x for standard
Brownian motion,
\Bt\ = \BQ\+Pt+Lu (**)
where fit = /0 sign(Bs)dBs is a Brownian motion, and Lt is the local time of
В at zero. Formula (**) is known as Tanaka's Formula.
Observe that if f(x) = \x\, then /" = 25(x), where 5 is the "delta
function at 0," which of course is a generalized function, or "distribution."
Thus Corollaries 1 and 3 give the intuitive interpretation of local time as
Ц = /0* 5{Xs)d[X,X]s, and Ц = /0* S(XS - a)d[X,X]s, for continuous semi-
martingales.
Local times also allow us to give simple examples of functions / that are
only slightly more general than convex functions and are such that f(X) need
not be a semimartingale when X is one. Let f(x) = |ж|а, 0 < a < 1. Then
/ cannot be expressed as the difference of two convex functions because the
slope of / becomes vertical as x decreases to zero. The proof of this result is
simpler if we restrict a to 0 < a < 1/2.
Theorem 71. Let X be a continuous local martingale with Xq — 0, and let
0 < a < 1/2. Then Yt — \Xt\a is not a semimartingale unless X is identically
zero.
Proof. Let us suppose that Y = \X\a is a semimartingale. Let /3 = I/a > 2.
We then have
=Yf = C f YtldYs + ^^ f
Jo l Jo
since Y has continuous paths. Using Theorem 69 and the Meyer-Tanaka for-
formula (Corollary 3 of Theorem 70) we have, where Lt — L® is the local time
of X at 0,
Lt = l{xs=o}dLs
Jo
= - / l{xs=0}sign(Xs)dX5+ / l{Xs=0)d\X\s
Jo Jo
The integral Jo l{xs=o} sign(Xs)dXs is identically zero. For if T™ is a sequence
of stopping times increasing to сю a.s. such that XT" is a bounded martingale,
then
tAT" ЛЛТ"
/
/tAT ЛЛТ
l{Xs=0} sign(Xs)dXsJ} = E{ / l{x3=0}d[X, X]s}
Jo
L?ATnl{0}(a)da} = 0.

218 IV General Stochastic Integration and Local Times
Therefore Lt = L°t = /0* 1{Хв=0)й\Х\8. Since {Xs = 0} equals {Ys = 0}, this
becomes
= l{Y3=o}d\X\s
Jo
Vi{ys=0}(iF8 + t^ Г
A Jo
Jo
= C [t
Jo
= 0.
Using the Meyer-Tanaka formula again we conclude that
\Xt\ = f sign(Xs)dXs.
Jo
Since X is a continuous local martingale, so also is the stochastic integral on
the right side above (Theorem 30); it is also non-negative, and equal to zero
at 0. Such a local martingale must be identically zero. This completes the
proof. D
It is worth noting that if X is a bounded, continuous martingale with
Xo = 0, then Yt = \Xt\a, 0 < a < 1/2 is an example of an asymptotic
martingale, or AMART, which is not a semimartingale.18
We next wish to determine when there exists a version of the local time
L% which is jointly continuous in (a, t) *->¦ L" a.s., or jointly right continuous
in (a,t) и-> Lf. We begin with the classical result of Kolmogorov which gives
a sufficient condition for joint continuity. There are several versions of Kol-
mogorov's Lemma. We give here a quite general one because we will use it
often in Chap. V. In this section we use only its corollary, which can also be
proved directly in a fairly simple way.
Before the statement of the theorem we establish some notation. Let A
denote the dyadic rational points of the unit cube [0,1]™ in R™, and let Am
denote all x e A whose coordinates are of the form k2~m, 0 < к < 2т.
Theorem 72 (Kolmogorov's Lemma). Let (E,d) be a complete metric
space, and let Ux be an E-valued random variable for all x dyadic rationals
in R™. Suppose that for all x, y, we have d(Ux, Uy) is a random variable and
that there exist strictly positive constants e, C, /3 such that
Then for almost all и the function x н-> Ux can be extended uniquely to a
continuous function from M.n to E.
18 For a more elementary example of an asymptotic martingale that is not a semi-
semimartingale, see Gut [86, page 7].

7 Local Times 219
Proof. We prove the theorem for the unit cube [0,1]™ and leave the extension
to R™ to the reader. Two points x and у in Дт are neighbors if supj \хгуг\ —
2~m. We use Chebyshev's inequality on the inequality hypothesized to get
P{d(Ux,Uy) > 2~am} <
Let
Am = {u : 3 neighbors x,y e Am with d{Ux(u), Uy{u)) > 2~ат}.
Since each x € Дт has at most 3™ neighbors, and the cardinality of Дт is
2mn, we have
P{Am) < c2m{a?-0)
where the constant с = 3nC. Take a sufficiently small so that ae < /3. Then
Р(Лт) < c2~mS
where 5 = C—a.e > 0. The Borel-Cantelli Lemma then implies Р(Лт infinitely
often) = 0. That is, there exists an mo such that for m > mo and every pair
(u, v) of points of Дт that are neighbors,
d{Uu,Uv) <2-am.
We now use the preceding to show that x i—» Ux is uniformly continuous
on Д and hence extendable uniquely to a continuous function on [0,1]™. To
this end, let x,y ? A be such that \\x — y\\ < 2~k~1. We will show that
d(Ux, Uy) < c2~ak for a constant c, and this will complete the proof.
Without loss of generality assume к > mo- Then x = (x1,... ,xn) and
у = (у1,... ,yn) in Д with \\x — y\\ < 2~k~1 have dyadic expansions of the
form
xi = иг +^o}2-J'
where a*-, b^ are each 0 or 1, and u,v are points of A^ which are either equal
or neighbors.
Next set uo = и, щ = u0 + aic+i2~k~1, щ = щ + Ofe+22~fe'2, We also
make analogous definitions for vo,vi,vz, Then u,_i and щ are equal or
neighbors in Ak+i, each i, and analogously for «j_i and w,. Hence
j=k

220 IV General Stochastic Integration and Local Times
and moreover
d(Uu{u),Uv{u))<2-ak.
The result now follows by the triangle inequality. ?
Comment. If the complete metric space (E,d) in Theorem 72 is separable,
then the hypothesis that d(Ux, Uy) be measurable is satisfied. Often the metric
spaces chosen are one of R, Rrf, or the function space С with the sup norm
and these are separable.
A complete metric space that arises often in Chap. V is the space E = Vn
of cadlag functions mapping [0, сю) into R™, topologized by uniform conver-
convergence on compacts. While this is a complete metric space, it is not separable.
Indeed, a compatible metric is
oo
<*(/.») = ? ^AЛ sup \f(s)-g(s)\).
However if fa(t) = l[QiOO)(*), then d{fa,f0) = 1/2 for all a, /3 with 0 < a <
/3 < 1, and since there are uncountably many such a, /3, the space is not
separable. Fortunately, however, the condition that d{Ux, Uy) be measurable
is nevertheless satisfied in this case, due to the path regularity of the functions
in the function space Z>™. (Note that in many other contexts the space Vn is
endowed with the Skorohod topology, and with this topology T>n is a complete
metric space which is also separable; see for example Ethier-Kurtz [71] or
Jacod-Shiryaev [110].)
We state as a corollary the form of Kolmogorov's Lemma (also known as
Kolmogorov's continuity criterion) that we will use in our study of local times.
Corollary 1 (Kolmogorov's Continuity Criterion). Let {X^)t>o,aeun
be a parameterized family of stochastic processes such that 1i—> Xf is cadlag
a.s., each о ? R™. Suppose that
\\
s<t
for some e, /3 > 0, C{t) > 0. Then there exists a version X? of Xf which
is B(M.+) <g> B(M.n) <g> T measurable and which is cadlag in t and uniformly
continuous in о on compacts and is such that for all о € R™, t > 0,
X?=X<1 a.s.
(The null set {Xf ф X"} can be chosen independently of t.) In this sec-
section we will use the above corollary for parameterized processes Xa which
are continuous in t. In this case the process obtained from the corollary of
Kolmogorov's Lemma, Xa, will be jointly continuous in (a, t) almost surely.
In particular, Kolomogorov's Lemma can be used to prove that the paths
of standard Brownian motion are continuous.

7 Local Times 221
Corollary 2. Let В be standard Brownian motion. Then there is a version
of В with continuous paths, a.s.
Proof. Since Bt - Bs is Gaussian with mean zero and variance t — s, we know
that E{\Bt - Bs\4} < c(t - sJ. (One can give a cute proof of this moment
estimate using the scaling property of Brownian motion.) If we think of time
as the parameter and the process as being constant in time, we see that the
exponent 4 is strictly positive, and that the exponent on the right, 2, is strictly
bigger than the dimension, which is of course 1. Corollary 2 now follows from
Corollary 1. D
Hypothesis A. For the remainder of this section we let X denote a semi-
martingale with the restriction that 53o<s<t I^-Xal < °° a-s-> eacn * > 0-
Observe that if (?l,Jr,(J-'t)t>o,P) is a probability space where {J-t)t>o is
the completed minimal filtration of a Brownian motion В = (Bt)t>o, then
all semimariingales on this Brownian space verify Hypothesis A. Indeed, by
Corollary 1 of Theorem 43 all the local martingales are continuous. Thus if
X is a semimartingale, let X — M + A be a decomposition with M a local
martingale and A an FV process. Then the jump processes AX and AA are
equal, hence
0<s<t 0<s<t
since A is of finite variation on compacts.
Let X be a semimartingale satisfying Hypothesis A, and let
Jt =
0<s<t
that is, J is the process which is the sum of the jumps. Because of our hypoth-
hypothesis J is an FV process and thus also a semimartingale. Moreover Yt = Xt - Jt
is a continuous semimartingale with Yo = 0, and we let
Y = M + A
be its (unique) decomposition, where M is a continuous local martingale and
Л is a continuous FV process with Mo = Ao — 0. Such a process M is then
uniquely determined and we can write
M = XC,
the continuous local martingale part of X.
Notation. We assume given a semimartingale X satisfying Hypothesis A. If
Z is any other semimartingale we write
= Г
Jo+
»->o}
dZs (a?

222 IV General Stochastic Integration and Local Times
This notation should not be confused with that of Kolmogorov's Lemma (The-
(Theorem 72). It is always assumed that we take the В(Ж)®В(М.+)®3Г measurable,
cadlag version of Za (cf., Theorem 63).
Before we can prove our primary regularity property (Theorem 75), we
need two preliminary results. The first is a very special case of a family of
martingale inequalities known as the Burkholder-Davis-Gundy inequalities.
Theorem 73 (Burkholder's Inequality). Let X be a continuous local mar-
martingale with Xo =0, 2 < p < oo, and T a finite stopping time. Then
where Cp = {gP(Hfcii)}P/2, „ah I + A = 1.
Proof. By stopping, it suffices to consider the case where X and [X, X] are
bounded. By Ito's formula we have
P{P 1} f
\XT\P=p[ signfJQI^r1^ + P{P~ 1} f \Xs\r-4[X,X)s.
Jo * Jo
By Doob's inequalities (Theorem 20 of Chap. I) we have (with - + | — 1)
< qpE{\XTf}
ll Г \Xsr2d[x,x)s}
Jo
with the last inequality by Holder's inequality. Since Е{(Хт)р}х~* < оо, we
divide both sides by it to obtain
and raising both sides to the power p/2 gives the result. ?
Actually much more is true. Indeed for any local martingale (continuous
or not) it is known that there exist constants cp, Cp such that for a finite
stopping time T
E{(XWYlp < cpE{[X,X}pT/2}^p < CpE{{X*TYY'i>
for 1 < p < oo. See Sect. 3 of Chap. VII of Dellacherie-Meyer [46] for these
and related results. When the local martingales are continuous some results
even hold for 0 < p < 1 (see, e.g., Barlow-Jacka-Yor [9, Table 4.1 page 162]).

7 Local Times 223
Theorem 74. Let X be a semimartingale satisfying Hypothesis A. There ex-
exists a version of(Xc)f such that (a,t,u>) i—> (Xc)"(w) is B(M.)®V measurable,
and everywhere jointly continuous in (a, t).
Proof. Without loss of generality we may assume X — Xo € H2. If it is not,
we can stop X — Xo at Tn—. The continuous local martingale part of
is then just (XC)T". Suppose -oo < о < b < oo, and let
at{a,b) = E{{ f l{b>Xa_>a}d[X,X)csJ}.
Jo
By Corollary 1 of Theorem 70 we have
at{a,b) -
Г
J a
. .2 ,. 1
b —
2 f 1
ЧЬ-с
by the Cauchy-Schwarz inequality. The above implies
By the definition,
bt
and therefore
Е{{ЦJ} <_
<
at(a, b) < (b- a)
<A?<\Xt-XQ\-
Z2E{\Xt-X0\2}l
сЩХ-Х0\\Ъ+4:
2 sup -B{(L");
ue(a,b)
- / sign(Xs_ ¦
-2E{([ sign(J
Jo+
Г ud J\
a Ja *
pb
I Ja *
- lt)dXs
te_ - «)dXsJ}
¦ (Л — Л0)\\пз
X0\\^2 <oo,
and the bound is independent of u. Therefore
at{a,b)<{b-aJr,
for a constant Г < oo, and independent of t. Next using Burkholder's inequal-
inequality (Theorem 73) we have
/•OO
E{sup\(Xc)bs - (Xc)as\4} < C4E{( / l{b>Xs__>a}d[X,X}csJ}
s Jo
< Ci sup at(a, b)
t
< С4Г{Ъ - aJ.
The result now follows by applying Kolmogorov's Lemma (the corollary of
Theorem 72). D

224 IV General Stochastic Integration and Local Times
We can now establish our primary result.
Theorem 75. Let X be a semimartingale satisfying Hypothesis A. Then there
exists a B(M.)®V measurable version of(a,t,u>) >—* L^(u>) which is everywhere
jointly right continuous in a and continuous in t. Moreover a.s. the limits
Ц~ — limf,_,aif,<aLj exist.
Proof. Since X satisfies Hypothesis A, the process Jt = J2o<s<t ^^* is an
FV semimartingale, and Y = X — J is a continuous semimartingale. We let
Y = M + A be the (unique) decomposition of Y, with Mo = Ao = 0.
Then X - M + A + J. Further define
0<s<t 0<s<t
Observe that \S?\ < ?0<s<t \AX*\ < oo. By Theorem 68 we have
(Xt - o)+ - (Xo - a)+ = {M)at + (i)? + (J)at + S? + ±Ц.
Consider
We have
and
/'
Jo+
a<b
0+
l{x._>b}dAe
l{x._>b)dAe
where the convergence is uniform in t on [0, r], and r > 0. We have analogous
results for (J)f and also for Sf, because it is dominated by J2o<s<t l^^sl <
oo. Since we already know that (M)" is continuous, the proof is complete. ?
The next three corollaries are simple consequences of Theorem 75. Re-
Recall that for a semimartingale X satisfying Hypothesis A, we let Jt =
J2o<s<t ^Xs, Y = X - J, and Y = M + A be the (unique) decomposition of
Y, with Ao = 0.
Corollary 1. Let X be a semimartingale satisfying Hypothesis A. Let X =
M + A + J be a decomposition with M and A continuous and J the jump
process of X, and let (L")t>0 be its local time at the level a. Then
at-Lat~ =2 f l{Xs_
Jo
= 2 f l{Xs=
Jo
dAs.

7 Local Times 225
An example of a semimartingale satisfying Hypothesis A but having a
discontinuous local time is JQ = \Bt\ where В is standard Brownian motion
with Bo = 0. Here La{X) = La(B) + L~a(B) for о > 0, L°(X) = L°(B), and
La(X) — 0 for о < 0. J. Walsh [227] has given a more interesting (but more
complicated) example.
Corollary 2. Let X be a semimartingale satisfying Hypothesis A. Let A be
as in Corollary 1. The local time (La) is continuous in t and is continuous at
о = oo if and only if
/ l{Xs=ao}\dAs\ = 0.
Jo
Observe that if X = B, a Brownian motion (or any continuous local mar-
martingale) , then A = 0 and the local time of X can be taken everywhere jointly
continuous.
Corollary 3. Let X be a semimartingale satisfying Hypothesis A. Then for
every (a, t) we have
and
Lt =
lim- / l{a<xs<a+s}d[X,X}cs, a.s.
^° ? Jo
I ft
lim)-/ 1{a-s<xs<a}d[X,X}cs, a.s.
?^° e Jo
The lack of symmetry in the above formula stems from the definition of
local time, where we defined sign(a:) in an asymmetric way:
x > 0,
x < 0.
A symmetrized result follows trivially, namely
T a j_ то,—
^Уо 4\xs~a\<e}d[X,X\l, a.s.
Corollary 3 is intuitively very appealing, and justifies thinking of local time
as an occupation time density. Also if X = B, a Brownian motion, then
Corollary 3 becomes the classical result
as, a.s.
Local times have interesting properties when viewed as processes with
"time" fixed, and the space variable "a" as the parameter. The Ray-Knight
Theorem giving the distribution of (Ьу~а)о<а<ъ the Brownian local time
sampled at the hitting time of 1, as the square of a two dimensional Bessel
diffusion, is one example.

226 IV General Stochastic Integration and Local Times
The Meyer-Ito formula (Theorem 70) can be extended in a different di-
direction, which gives rise to the Bouleau-Yor formula, allowing non-convex
functions of semimartingales. The key idea is that the function a i—» Ьц in-
induces a measure on R if L is the local time of a semimartingale X satisfying
Hypothesis A and U is a positive random variable.
Theorem 76. Let X be a semimartingale satisfying Hypothesis A, U a pos-
positive random variable, La the local times of X. Then the operation
n
J
where f(x) = J2 fi^(a,i,ai+1](x)> can be extended uniquely to a vector measure
on В(Ж) with values in L°.
Proof. Recall that L° denotes finite-valued random variables. By Theorem 68
we know that
1 _ - /"*
-Lat = (Xt -a) -(X0-a) + l{xs_<
1 Jo+
0<s<t 0<s<t
We write Sf for the last two sums on the right side of the equation above.
Note first that (Хц — a)~ — (Xo — a)~ is Lipschitz continuous in о and
hence absolutely continuous in a. Therefore it induces a measure in a. Also
the function а м 5^ is cadlag and of finite variation in a, and moreover
f \daSu\ < 2j2o<s<u\^Xs\- Finally consider the stochastic integral. Let
1 ~Xu)-Jo
= i
и
0+
и
Therefore by the Dominated Convergence Theorem for stochastic integrals
(Theorem 32) we have that Yli fi(Xy'+1 — X^) extends to a measure with
values in L° and moreover / f{a)daXfj = J0+f(Xs^)dXs, for / bounded,
Borel measurable on R. Thus daLfj can be defined as the sum of three L°-
valued measures on (R, Z?(R)), and the theorem is proved. D
In the proof of Theorem 76 we proved the following result, which is im-
important enough to state as a corollary.

8 Azema's Martingale 227
Corollary. Let X be a semimartingale satisfying Hypothesis A, U a positive
random variable, and X" = Jo l{x3-<a}dXs. Then daXy can be defined as
an L°-valued measure, and for / e Ы?(Й) we have
/ f{a)daXb = I f{Xs_)dXs.
Ju Jo+
Combining Theorem 76 and its corollary yields the Bouleau-Yor formula.
We leave its proof to the reader.
Theorem 77 (Bouleau-Yor Formula). Let X be a semimartingale satisfy-
satisfying Hypothesis A, U a positive random variable, f a bounded, Borel function,
and F(x) = /oz f(u)du. Then
F(Xu) - F(X0) = / f(Xs_)dXs - \ ff(a)daLau
+ Y, {F(Xs)-F(Xs_)-f(Xs_)AXs}.
0<s<U
Combining Theorem 77 with the Meyer-Ito formula (Theorem 70) we ob-
obtain the following relationship.
Corollary. Let A" be a semimartingale satisfying Hypothesis A, and let T be
a finite-valued random variable. Let La be the local times of X and let / be
a C1 function. Then
/ f'{a)LaTda =- f{a)daLaT.
J—oo J—oo
Remark. The Meyer-Ito formula (Theorem 70) and the Bouleau-Yor formula
can be extended in interesting ways. This was first done in [77], which inspired
subsequent work by N. Eisenbaum [62] and others (e.g., [6]). See [80] for a
compilation of selected results in these areas and also new ones.
8 Azema's Martingale
In order to prove the regularity properties of local times in Sect. 7, we needed
Hypothesis A: a semimartingale X satisfies Hypothesis A if X is such that
^0<s<t|AA"s| < oo a.s., each t > 0. This hypothesis was needed to prove
Theorems 74 through 77 and their corollaries. We present here a counterex-
counterexample, known as Azema's martingale, that shows that these results do not
hold for general semimartingales. In particular, Theorems 81 and 82 show
that Azema's martingale has a local time that is zero at every level a except
a = 0, and therefore there is no regular version in the space variable.
Let (п, T,F, (Bt)t>o,P) be a standard Brownian motion, Bo = 0, with
the filtration F = {Tt)t>o completed (and hence right continuous). We define

228 IV General Stochastic Integration and Local Times
/l, ifa:>0,
sign(x) = <
v ; [-1, ifx<0.
Set g% = cr{sign(Bs); s < t}, and let G = (&)t>o denote the filtration (?to)t>o
completed. (It is then right continuous as a consequence of the strong Markov
property of Brownian motion.)
Let Mt = E{Bt\Qt},t > 0. Then for s<t,
E{Mt\gs} = E{Bt\gs} = E{E{Bt\rs}\gs} = E{B8\gs} = ms,
and MisaG martingale. We always take the cadlag version of M.
Definition. The process Mt — E{Bt\Qt} is called Azema's martingale.
Fundamentally related to Azema's martingale is the process which gives
the last exit from zero before time t. We define
gt = sup{s < t : Bs - 0}.
To study the process g we need the reflection principle for Brownian motion,
which we proved in Chap. I (Theorem 33 and its corollary) and which we
recall here for convenience.
Theorem 78 (Reflection Principle for Brownian Motion). Let В be
standard Brownian motion, Bo = 0, and с > 0. Then
P(supBs > c) = 2P(Bt > c).
s<t
An elementary conditioning argument provides a useful corollary.
Corollary. Let В be standard Brownian motion, Bo = 0, and 0 < s < t.
Then
P( sup Bu > 0, Bs < 0) = 2P(Bt >0,Bs< 0).
s<u<t
Theorem 79 is known as Levy's arcsine law for the last exit from zero.
Theorem 79. The process gt is G adapted, and P(gt < s) = ^arcsiny^,
0 < s <t.
Proof. For s<twe have almost surely
{<?*<*} = ( П {s«>o» LJ( П {в»<°})
s<u<t s<u<t
from which it follows that gt is Qt adapted.
To calculate the distribution of gt we observe for s < t (using the symmetry
ofB),

8 Azema's Martingale 229
P(gt >s) = 2P{gt >s,Bs< 0)
= 2P( sup Bu >0,Bs< 0)
s<«<t
= 4P(Bt>0,Bs<0),
by the corollary of Theorem 78. Since (Bs,Bt) are jointly Gaussian, there
exist X, Y that are independent N@,1) random variables with
Bs = y/sX and Bt = л/sX + ^t-sY.
Then {Bs < 0} = {X < 0}, and {Bt > 0} = {Y > --j?Lx}. To calculate
P({Y > -^Lx,X < 0}), use polar coordinates:
P(Bt > 0, Bs < 0) = —
7Г —arcsin
1 ,7Г /7
= —( arcsinA/-)
1 1 /I
= arcsin \ -.
4 2тг V *
Therefore 4P(Bt > 0,Bs < 0) = 1 - ^ arcsin yf and P(gt < s) =
- arcsin л/Т. D
Theorem 80. Azema's martingale M is given by Mt = sign(??t)y^J\/t — gt,
t>0.
Proof. By definition,
Mt = Е{Б*|а*} = E{sign(Bt)\Bt\\Gt} = si
However ?{|Bt||(?t} = JB{|Bt||9t; sign(Bs), s < *}. But the process sign(Bs) is
independent of \Bt\ for s < gt, and sign(??s) = sign(Bt) f°r 9t < s <t. Hence
sign(Bs) is constant after gt. It follows that
E{\Bt\\Gt} = E{\Bt\\gt}.
Therefore if we can show 2?{|.Bt||9t = s} = \f\^/t — s, the proof will be
complete.
Given 0 < s < t we define
T = t Л inf{u > s : Bu = 0}.
Note that T is a bounded stopping time. Then

230 IV General Stochastic Integration and Local Times
E{\Bt\;gt<s} = 2E{Bt;gt<s and Bs > 0}
= 2E{Bt;T = t and Bs > 0}
= 2E{BT;Bs>0}
= 2E{Bs;Bs>0},
since BT = 0 on {T ф t}, and the last step uses that В is a martingale, and
s < T. The last term above equals E{\BS\} = \j\\ft- Therefore
?P(9t < s)
since j^P{gt < s) — —,1 , by Theorem 79. D
Let T{® = a{Ms; s < t), the minimal filtration of Azema's martingale M,
and let H = CHt)t>o be the completed filtration.
Corollary. The two filtrations H and G are equal.
Proof. By definition Mt G Qt, each t > 0. Hence Tit С ^t. However, by
Theorem 80, sign(Mt) = sign(Bt), so that Qt С Tit- ?
Next we turn our attention to the local times of Azema's martingale. For
a semimartingale (Xt)t>0, let (L^(X))t>0 = La{X) denote its local time at
the level a.
Theorem 81. The local time at zero of Azema's martingale is not identically
zero.
Proof. By the Meyer-Ito formula (Theorem 70) for f(x) = \x\ we have
\Mt\ = f sign(Ms_)dMs +L°t(M) + Y, {\Ms\ - \Ms-\- sign(Ms_)AMs}.
J° 0<s<t
Set Yt = \Mt\. Since sign(Ms_)AMs = AYS, we have that \Mt\ - Ц(М)
is a martingale. If L°(M) were the zero process, then Y = \M\ would be a
non-negative martingale with Yo = 0, hence identically zero. Since Y is not
identically zero, the theorem is proved. D
Theorem 82. The local times at all levels except zero of Azema's martingale
are 0. That is, Ц(М) = 0, t > 0, if а ф 0.

8 Azema's Martingale 231
Proof. By Theorem 69 we know that the measure dL^(u>) on K+ is carried by
the set {s : Ms_(o>) = М3(ш) = a}, a.s., for each a. However if а ф 0, this set
is countable. Since s н-> L" is a.s. continuous, it cannot charge a countable set.
Therefore dLas is the zero measure for а ф 0. Since Lg = 0, we have L" = 0
for а ф 0. D
Theorems 81 and 82 together show that Azema's martingale provides a
counterexample to a general version of Theorem 75, for example. Therefore a
hypothesis such as Hypothesis A is necessary, and also we see that Azema's
martingale does not satisfy Hypothesis A. Since all semimartingales for the
Brownian motion nitration F do satisfy Hypothesis A, as noted in Sect. 7, we
conclude the following.
Theorem 83. Azema's martingale M is not a semimartingale for the Brown-
Brownian filtration F.
While the local time at zero, L°(M), is non-trivial by Theorem 81, more
information is available. It is actually the same as the Brownian local time,
as Theorem 85 below shows. First we need a preliminary result.
Theorem 84. The local time L°(B) of Brownian motion is G adapted.
Theorem 84 will be proved if one can express L°(B) as the a.s. limit of
measurable functionals of the zero set of Brownian motion. Such a result is
classical: see, e.g., Kingman [126, page 730].
Theorem 85. The local times at zero of Brownian motion and Azema's mar-
martingale are the same. That is, L°(B) = L°(M).
Proof. As we saw in the proof of Theorem 81, \Mt\ - L®(M) is a martingale.
Since the process L°(M) is non-decreasing, continuous and adapted, it is
natural. If A is another natural process such that \M\ - A is a martingale,
then L°(M) = Aby (for example) Theorem 30 of Chap. III. Thus, it suffices
to show that \M\ — L°(B) is a martingale for the filtration G.
By Tanaka's formula, \Bt\ - L^(B) = iVt, where iV is an F martingale.
Then
E{\Bt\ - L°t(B)\Gt} = E{Nt\gt}
and by Theorem 84
E{\Bt\\gt} - L°t(B) is a martingale.
Since
E{\Bt\\gt} = E{sign(Bt)Bt\gt} = sign(Bt)E{Bt\gt}
= sign(Bt)Mt
= \щ,
we have \Mt\ — L^(B) is a martingale, and the theorem is proved. D

232 IV General Stochastic Integration and Local Times
Corollary 1. The process \M\ - L°(B) is a G martingale.
Proof. This corollary is proved at the end of the proof of Theorem 85. D
Corollary 2. The process l{B(>o}\/fV^ ~ 9t ~ \L®{B) is a G martingale.
Proof. Since 1{ms^>o}M~ — 1{ms_<o}^s+ = 0> it follows from Theorem 68
that Mt+ — ^L®(M) is a martingale. However Mt+ = 1{.в4>о}\/|лА — 9t by
Theorem 80, and L?(M) = L?(B) by Theorem 85, and the theorem follows.
D
The next theorem shows that Azema's martingale is quadratic pure jump.
Theorem 86. For Azema's martingale M, [M,M]C = 0, and [M,M]t — \gt-
Proof. By Corollary 2 of Theorem 70,
/•OO
[M,M]ct= / Latda.
J — OO
By Theorem 82, [M,M]C = 0. We conclude that [M,M}t = *?0<s<t(AMsJ,
and it follows from Theorem 80 that [M, M}t = |gt. ~ П
Theorem 86 allows us to compute the compensation of [M, M]; that is, the
unique predictable, increasing process A such that [M, M] — A is a martingale.
This process A is denoted (M,M) in the literature, and it was defined in
Sect. 4 of Chap. III.
Theorem 87. Let M be Azema's martingale. Then [M,M]t — jt is a mar-
martingale. That is, {M,M)t = ~t.
Proof. Recall that Mt2 - [M, M]t is a martingale (cf., Theorem 27 of Chap. II).
By Theorems 80 and 86,
Since the above is a martingale, it remains only to observe that \gt - \t is a
martingale. D
Note that if Xt = sign(Bt)\/2V* ~ fft> then X is a quadratic pure jump
martingale with {X, X)t = t. Recall that if N is a Poisson process with
intensity one, then Nt - t is a martingale. Since [N, N]t = Nt, we have
[iV, N]t — t is a martingale; thus {N, N)t = t. Another example of a martin-
martingale Y with (У, Y)t = t is Brownian motion. Since [B,B]t = t is continuous,

9 Sigma Martingales 233
9 Sigma Martingales
We saw with Emery's example (which precedes Theorem 34) a stochastic
integral with respect to an Ti2 martingale that was not even a local martingale.
Yet "morally" we feel that it should be one. It is therefore interesting to
consider the space of stochastic integrals with respect to local martingales.
We know these are of course semimartingales, but we want to characterize
the class of processes that arise in a more specific yet reasonable way.
By analogy, recall that in measure theory a er-finite measure is a measure /j,
on a space E, B) such that there exists a sequence of measurable sets УЦ е В
such that (Ji Ai = S and ц(Лг) < oo, each i. Such a measure can be thought
of as a special case of a countable sum of probability measures. It is essentially
a similar phenomenon that brings us out of the class of local martingales.
Definition. An Kd-valued semimartingale X is called a sigma martingale if
there exists an Rd-valued martingale M and a predictable K+-valued process
H e L(M) such that X = H-M.
In the next theorem, H will always denote a strictly positive predictable
process.
Theorem 88. Let X be a semimartingale. The following are equivalent:
(i) X is a sigma martingale;
(ii) X = H ¦ M where M is a local martingale;
(iii) X = H • M where M is a martingale;
(iv) X = H ¦ M where M is a martingale in H1 ¦
Proof. It is clear that (iv) implies (iii), that (iii) implies (ii), and that (iii)
obviously implies (i), so we need to prove only that (i) implies (iv). Without
loss assume that Mo = 0. Since A" is a sigma martingale it has a representation
of the form X = H ¦ M where M is a martingale. We know that we can
localize M in Ti1 by Theorem 51, so let Tn be a sequence of stopping times
tending to oo a.s. such that MT" e H1 for each n. Set To = 0 and let Nn =
1(т„_1,т„] ¦ MT". Let an be a strictly positive sequence of real numbers such
that iZn^W^Wn1 < °°. Define N = ^ZnanNn and one can check that N
is an Jil martingale. Set J = 1{#=о} + Н^2пО!Й11(т„_1,т„]- Then it is simple
to check that X = J ¦ N and that J is strictly positive. D
Corollary 1. A local sigma martingale is a sigma martingale.
Proof. Let A" be a local sigma martingale, so that there exists a sequence of
stopping times tending to oo a.s. such that XTn is a sigma martingale for
each n. Since XT" is a sigma martingale, there exists a martingale Mn in
H1 such that XTn = Hn ¦ Mn, for each n, where Hn is positive predictable.
Choose фп positive, predictable so that \\фпНп ¦ MT"||Wi < 2"n. Set Го = О
and ф° = ф^1щ. Then ф = ф° + ^2п>1Фп1(тп~1,т„] ls strictly positive and
predictable. Moreover ф ¦ X is an Ti1 martingale, whence X = 4 • ф • X, and
the corollary is established. ?

234 IV General Stochastic Integration and Local Times
Corollary 2. A local martingale is a sigma martingale.
Proof. This is simply a consequence of the fact that a local martingale is
locally a martingale, and trivially a martingale is a sigma martingale. D
The next theorem gives a satisfying stability result, showing that sigma
martingales are stable with respect to stochastic integration.
Theorem 89. If X is a local martingale and Я e L(X), then the stochastic
integral H ¦ X is a sigma martingale. Moreover if X is a sigma martingale
(and a fortiori a semimartingale) and H e L{X), then H ¦ X is a sigma
martingale.
Proof. Clearly it suffices to prove the second statement, since a local mar-
martingale is already a sigma martingale. But the second statement is simple.
Since A" is a sigma martingale we know an equivalent condition is that there
exists a strictly positive predictable process ф and a local martingale M such
that X = ф ¦ M. Since H is predictable, the processes Я1 = Я1{#>0} + 1
and H2 = — Я1{я<0} + 1 are both strictly positive and predictable. Moreover
H = Я1 - H2. The processes Я1 and H2 are both in L{X) because H is. By
associativity and linearity of the stochastic integral,
H ¦ X = Н{ф ¦ M) = (Нф) ¦ M = Н1ф ¦ M - Н2ф ¦ M,
and since Нгф are both strictly positive for i = 1, 2 we have by Theorem 88
that Нгф ¦ M is a sigma martingale for each « = 1,2. Sigma martingales form
a vector space, and we are done. ?
We also would like to have sufficient conditions for a sigma martingale to
be a local martingale.
Theorem 90. A sigma martingale which is also a special semimartingale is
a local martingale.
Proof. Any sigma martingale is a semimartingale by definition; here we also
assume it is special. Thus it has a canonical decomposition X = M + A where
M is a local martingale and A is a process of finite variation on compacts
which is also predictable. We want to show A = 0. We assume Ao = 0.
Choose H predictable, H > 0 everywhere, such that H ¦ X is a martingale,
which we can easily do using Theorem 88. Then (Я Л 1) ¦ X = (^-) • (Я • X)
is a local martingale, and hence without loss of generality we can assume H
is bounded. Then Я • A is a finite variation predictable process, and we have
H ¦ A = H ¦ X — H ¦ M which is a local martingale. Thus we conclude that
H ¦ A = 0. If we replace H with JH where J is predictable, \J\ < 1, and is
such that /0 JsdAs = JQ \dAs\, then we conclude that /0 Hs\dAs\ — 0, and
since H > 0 everywhere, we conclude that A = 0 establishing the result. D
We now have that any criterion that ensures that the semimartingale X
is special, will also imply that the sigma martingale A" is a local martingale.

Bibliographic Notes 235
Corollary 1. If a sigma martingale X has continuous paths, then it is a local
martingale.
Corollary 2. If A" is a sigma martingale and if either Xf = sups<t \XS\ or
Yt = sups<t \AXS\ is locally integrable, then X is a local martingale.
Note that as a corollary to Corollary 2 we have the following.
Corollary 3. If A" is a sigma martingale with bounded jumps, then A" is a
local martingale.
Bibliographic Notes
The extension by continuity of stochastic integration from processes in L to
predictable processes is presented here for essentially the first time. How-
However the procedure was indicated earlier in Protter [201, 202]. Other ap-
approaches which are closely related are given by Jacod [103] and Chou-Meyer-
Stricker [31] (see an exposition in Dellacherie-Meyer [46, page 381]).
The "H" in the space Ti2 of semimartingales comes from Hardy spaces,
and this bears explaining. When the semimartingale X G H2 is actually a
martingale, the space of Hp martingales has a theory analogous to that of
Hardy spaces. While this is not the case for semimartingales, the Hp norms
for semimartingales are, in a certain sense, a natural generalization of the W
norms for martingales, and so the name has been preserved. The 7ip norm
was first introduced by Emery [64], though it was implicit in Protter [197].
Most of the treatment of stochastic integration in Sect. 2 of this chapter is
new, though essentially all of the results have been proved elsewhere with dif-
different methods. The author benefited greatly throughout by discussions with
S. Janson. Theorem 1 (that H2 is a Banach space) is due to Emery [66]. The
equivalence of the two pseudonorms for Ti2 (the corollary of Theorem 24)
is originally due to Yor [241], while the fact that U2{Q) С Н2(Р) if $ is
bounded is originally due to Lenglart [144]. Theorem 25 is originally due to
Jacod [103, page 228]. The concept of a predictable cr-algebra and its impor-
importance to stochastic integration is due to Meyer [166]. The local behavior of
the stochastic integral was first investigated by McShane for his integral [154],
and then independently by Meyer [171], who established Theorem 26 and its
corollary for the semimartingale integral. The first Dominated Convergence
Theorem for the semimartingale integral appearing in print seems to be in
Jacod [103].
The theory of martingale representation dates back to Ito's work on mul-
multiple stochastic integrals [96], and the theory for M2 presented here is largely
due to Kunita-Watanabe [134]. A more powerful theory (for M1) is presented
in Dellacherie-Meyer [46]. Absolutely continuous spaces are defined here for
the first time. The author benefited from discussions with Yan Zeng on this
subject. That the dual of Ti} is BMO for martingales is due to the combined

236 Exercises for Chapter IV
efforts of many researchers, culminating in Meyer's paper [170]; we follow the
treatment in [87]. The Jacod-Yor Theorem is from [111], and Sect. 9 largely
follows Emery [69].
The theory of stochastic integration depending on a parameter is due to
the fundamental work of Strieker-Yor [219], who used a key idea of Doleans-
Dade [48]. The Fubini Theorems for stochastic integration have their origins
in the book of Doob [55] and Kallianpur-Striebel [120] for the ltd integral,
Kailath-Segall-Zakai [119] for martingale integrals, and Jacod [103] for semi-
martingales. The counterexample to a general Fubini Theorem presented here
is due to Janson.
The theory of semimartingale local time is of course abstracted from Brow-
nian local time, which is due to Levy [149]. It was related to stochastic inte-
integration by Tanaka (see McKean [153]) for Brownian motion and the ltd inte-
integral. The theory presented here for semimartingale local time is due largely to
Meyer [171]. See also Millar [181]. The measure theory needed for the rigorous
development of semimartingale local time was developed by Strieker-Yor [219],
and the Meyer-Ito formula (Theorem 70) was formally presented in Yor [242],
as well as in Jacod [103]. Theorem 71 is due to Yor [244], and has been ex-
extended by Ouknine [188]. A more general version (but more complicated) is
in Qinlar-Jacod-Protter-Sharpe [34]. The proof of Kolmogorov's Lemma given
here is from Meyer [177]. The results proved under Hypothesis A are all due
to Yor [243] except the Bouleau-Yor formula [22]. See Bouleau [21] for more
on this formula.
Azema's martingale is of course due to Azema [4], though our presenta-
presentation of it is new and it is due largely to Janson. For many more interesting
results concerning Azema's martingale, see Azema-Yor [5], Emery [69], and
Meyer [179].
Sigma martingales date back to the work of Emery [67], and to Chou [29].
The importance of sigma martingales was clarified through the work in math-
mathematical finance of Delbaen and Schachermayer [38]. See [110] for a more
comprehensive treatment.
Exercises for Chapter IV
Exercise 1. Let A" be a semimartingale. Show that X is special if and only
if the increasing process Ct — [X, X}t is locally integrable; that is, there
exists a sequence of stopping times Tn increasing to infinity a.s. such that
E{[X,Х]т„} < oo for each time Tn.
* Exercise 2. Let A" be a special semimartingale with canonical decomposi-
decomposition Xt = Xo + Mt + At. Show that the following two inequalities hold:
E{[A,AU} < ЕЦХ^и) and

Exercises for Chapter IV 237
Exercise 3. Let A2, T, P) be a complete probability space. Let Xn be a
sequence of random variables such that P(|Xn| < сю) = 1, all n. Show that
there exists another probability Q which is equivalent to P in the sense that
the null sets are the same for both measures, with j^ bounded, and every
random variable Xn is in Ll(dQ).
Exercise 4. Let X be a semimartingale. Show that there is a probability
measure Q, equivalent to P, such that under Q, X has a decomposition X =
M + A, where M is a martingale with EQ([M,M]t) < oo for each t, 0 <
t < oo, and A is a predictable process of finite variation on compacts with
Eq(J0 |cL4,j|) < oo for each t, 0 < t < oo. (Hint: Use Exercises 1, 2, and 3.)
*Exercise 5. Let Xn be a sequence of semimartingales. Show that there
exists one probability measure Q, equivalent to P, such that under Q, each
Xn has a decomposition Xn = Mn + An, where Mn is a martingale with
Ео([Мп, Mn]t) < oo for each t, 0 < t < oo and each n, and An is a predictable
process of finite variation on compacts with Eq(J0 \dA™\) < oo for each t,
0 < t < oo and each n.
Exercise 6. Let S denote the space of all square integrable martingales with
continuous paths a.s., on a given filtered complete probability space satisfying
the usual conditions. Show that S is a stable subspace. (This exercise was
referred to in Definition 3.)
Exercise 7. Let Z be a Levy process with ?^{|Zt|} < oo and E{Zt} = 0 for
all t > 0. Show that Z is a martingale, and that Zc is either 0 or Brownian
motion, where Zc is defined in Exercise 6. Conclude that any martingale Levy
process Z that has no Brownian component is purely discontinuous.
Exercise 8. Let N be a standard Poisson process with arrival intensity A.
Show that the martingale Nt — Xt is purely discontinuous. (Note that this gives
an example of a purely discontinuous martingale whose sample paths are not
purely discontinuous in the sense that they do not change only by jumps.)
Exercise 9. Let M be a square integrable martingale. Show that [M, M}% =
[Mc,Mc]t = [Mc,M]t, all t > 0, almost surely.
Exercise 10 (example of orthogonal projection). Let T be a stopping
time. Show that the collection S of all square integrable martingales stopped
at T is a stable subspace. Also show that if M is an arbitrary square integrable
martingale, then its orthogonal decomposition with respect to this subspace
is M = MT + (M — MT). Give a description of the orthocomplement of S.
Exercise 11 (example of orthogonal projection). Let T be a totally
inaccessible stopping time and let M be a square integrable martingale. Let
Ut = ДМт1{4>т} and let L be the martingale of U minus its compensator.
We denote L by ПМ. Show that ПМ and M - ПМ are orthogonal. Let S
denote the space of all square integrable martingales which are continuous at
the instant T. Show that S is a stable subspace, and that ПМ represents the
orthogonal projection of M onto the orthocomplement of S.

238 Exercises for Chapter IV
* Exercise 12. Let T be an arbitrary stopping time, T > 0, and let X be
a random variable in L1. Show that M = Xl{t>x} is a uniformly integrable
martingale if and only if E{X\J~t-} = 0.
*Exercise 13. Let T be a stopping time, T > 0, and let M be a square
integrable martingale. Show that the decomposition
Mt = [(MT - E{MT\TT-})l{t>T}] + [Mt - (MT - Е{Мт\Рт-}I{г>т})]
is an orthogonal decomposition. Show that one of the stable subspaces in
question is the space of all square integrable martingales M such that Mt is
Tt- measurable. (Hint: Use Exercise 12.)
*Exercise 14. Let Z be a Levy process on a complete probability space
(Sl,T, P) and let F be its minimal filtration completed (and satisfying the
usual hypotheses). Show that (Sl,J-, F, P) is an absolutely continuous space.
Exercise 15. Consider the structure equation d[X,X]t = dt + /3Xt-dXt.
Show that this equation has a scaling property: if X is a martingale solu-
solution, then so too is jX\2t for every А ф 0. Note that by the uniqueness in
distribution of the solution, the processes (Xt)t>o and (j-XA2t)t>o have the
same distribution.
* Exercise 16. Let X be a martingale solution of the structure equation of
Exercise 15. Show that Xе = 0, where Xе is defined in Exercise 6.
Exercise 17. Let M be a local martingale in 7ip for some p > 1. Show
that M is a uniformly integrable martingale. Show further that if p = 1 then
E{MQO] < c||M||fti where the constant с is universal. That is, с does not
depend on M.
Exercise 18. Suppose that M is a bounded martingale. Show that
This can improved to replace \/5 with 2.
Exercise 19. Show that there are martingales with finite BMO norm which
are not bounded. (Hint: Let M be a bounded martingale and Я be a pre-
predictable process such that |#| < 1, and let N = H ¦ M. Then it is easy to see
that N is in BMO, and to construct such an N that is not bounded.)
Exercise 20. Prove this version of the Fundamental Theorem of Local Mar-
Martingales. Let M be a uniformly integrable martingale. Show that there exist N
and U such that N is in BMO, U is of integrable variation, and M = U + V.
If M is a local martingale this holds locally.
Exercise 21. Show that an arbitrary local martingale is a semimartingale by
using the Burkholder-Davis-Gundy inequalities instead of the Fundamental
Theorem of Local Martingales. (See [61].)

Exercises for Chapter IV 239
Exercise 22. Let M be a local martingale. If T is a predictable stopping time,
then Ijq^^s) € ЪР so that Mj~ = Jo l[olr)(s)dMs is a local martingale. Give
an example of a local martingale M and a stopping time T such that MT~ is
not a local martingale.
Exercise 23. Let X be a continuous semimartingale. Show that
[X, X]. (Hint: Use local times.)
* Exercise 24. Let В be standard Brownian motion and let a > 0. Show that
/0 \Bt\~adt < oo a.s. if and only if a < 1. (Hint: Use local times.)
Exercise 25. Let X be a semimartingale. Show that L%(X) = ±L°(X+)
where X+ — max(X, 0) and L\(X) denotes the local time of X at time t and
level 0.
Exercise 26. Let A be adapted, continuous, and of finite variation on com-
compacts, and let В be standard Brownian motion. Let A denote Lebesgue mea-
measure on the line. Show that A(s : Bs = As) = 0 a.s.
Exercise 27. Let X be a continuous semimartingale. Show that L0. € Tt ,
each t > 0, where T\ denotes the smallest right continuous completed fil-
filtration generated by the process
Exercise 28. Let X be a semimartingale. Show that for each fixed (u>, t) the
section x i—> L*(X) has compact support.
Exercise 29 (Skorohod's Lemma). This result will be useful for the next
two exercises. Let x be a real valued continuous function on [0, сю) such that
x@) > 0. Show there exists a pair of functions (y, a) on [0, сю) such that
x — у + а, у is positive, and a is increasing, continuous, a@) = 0, and the
measure das induced by a is carried by the set {s : y(s) = 0}. Show further
that the function a is given by a(t) = sups<t(—x(s) V 0).
Exercise 30. Let M be a continuous local martingale with Mo = 0. Let
\Mt\ = Nt + L%(M). Show that N is a continuous local martingale with
No = 0, and that L°t(M) = sups<t(-Ns).
*Exercise 31. Let X be a continuous semimartingale, and suppose Y satisfies
the equation dYt = -s\gn(Yt)dXt. Show that \Yt\ = Xt* - Xt, where X* =
supu<tXu. (Hint: Use Tanaka's formula and Exercise 29.)
Exercise 32. Suppose that X and Y are continuous semimartingales such
that LPt(Y -X) = 0. Show that
L°t(X V Y) = f l{YtmdL°s(X) + f l{Xa<0]dL°s(Y).
Jo Jo
Exercise 33. Let X and Y be continuous semimartingales. Show that
L°t(X AY)
= f l{Ys>0]dL°s(X) + f l{Xs>0}dL°s(Y) + f l{Xs=Ys=o}dLos(Y+-X+).
Jo Jo Jo

240 Exercises for Chapter IV
Exercise 34 (Ouknine's Formula). Let X and Y be continuous semi-
martingales. Show that
L°t{X V Y) + L°t(X Л Y) = L°t(X) + L°t(Y).
* Exercise 35 (Emery-Perkins Theorem). Let В be standard Brownian
motion and set Lt = L^(B). Let X = В + cL, с € R fixed. Assume that
there exists a set D С М+ х п such that (i) D is predictable for Tx; (ii)
P(w : (t,w) e D) = 1 each t > 0; and (iii) P(w : (Tt(w),w) € D) = 0 each
t > 0, where Tt = inf(s > 0 : Ls > t).
(a) Show that Bt - J* lD(s)dXs. Also show that Bt 6fx, each t > 0, and
conclude that JFB = Tx = JTs+ci, all cEM.
(b) Let
„ИтХ
™~* fc=i
Use the fact that s i—> (BsATt,LsATt) and s н-> (—BTt_s,LiTt — Ьтг„3) have
the same joint distribution to show that D satisfies (i), (ii), and (iii).
* Exercise 36. With the notation of Exercise 35, let Y = \B\+cL. Show that
TY С JTlBl,for all с
*Exercise 37. With the notation of Exercise 36, Pitman's Theorem states
that У is a Bessel process of order 3, which in turn implies that for TY, with
c=l,
f
J0
for an ^"y-Brownian motion /3. Use this to show that TY ф T^B\ for с = 1.
(Note: One can show that TY = T^B\ for all с ф 1.)
* Exercise 38 (Barlow's Theorem). Let X solve the equation
Xt = Xo + [ u(Xs)dBs + [ b(Xs)ds
J J
[ [
o Jo
where В is a standard Brownian motion. The above equation is said to have
a unique weak solution if whenever X and У are two solutions, and if
fix is defined by цх{Л) = Р{ш '¦ t i—> Xt(u>) € A}, then fix — Hy- The
above equation is said to have a unique strong solution if whenever both
X and Y are two solutions on the probability space on which the Brownian
motion В is denned, then P{u : f н> 1((ш) = f и У4(ш)} = 1. Assume that
the equation has a unique weak solution with the property that if X, Y are
any two solutions defined on the same probability space, then for all t > 0,
L®(X — Y) — 0. Show that the equation has a unique strong solution.

Exercises for Chapter IV 241
Exercise 39. Consider the situation of Exercise 38, and suppose that a and
b are bounded Borel. Further, suppose that (c(x) — u(y)J < p(\x — y\) for all
x, y, where p : [О, сю) —+ [0, сю) is increasing with J* -4^-du = +oo for every
e > 0. Show that if X, Y are two solutions, then L%(X - Y) = 0.
*Exercise 40. Let Б be a standard Brownian motion and consider the equa-
equation
Xt = / sign(Xs)dBs.
Jo
Show that the equation has a unique weak solution, but that if X is a solution
then so too is —X. (See Exercise 38 for a definition of weak and strong solu-
solutions.) Hence, the equation does not have a unique strong solution. Finally,
show that if X is a solution, then X is not adapted to the nitration TB. (Hint:
Show first that Bt = JQ l^Xa^oyd\Xs\.)
Exercise 41. Let X be a sigma martingale and suppose X is bounded below.
Show that X is in fact a local martingale.
*Exercise 42. Show that any Levy process which is a sigma martingale is
actually a martingale (and not just a local martingale).
Exercise 43. Let (Un)n>i be i.i.d. random variables with P(U\ = 1) =
P(UX = -1) = 1/2, and let X = Е„>1 2""t/«l{t>9n} where (qn)n>i is an
enumeration of the rationale in @,1). Let Ht = X)n>i ^^-{t>gn} ап<^ show
that X ? H2 and H € L(X), but also that X is of finite variation and
Y = H ¦ X has infinite variation. Thus the space of finite variation processes
is not closed under stochastic integration!
* Exercise 44. Let Xn be a sequence of semimartingales on a filtered complete
probability space (fi, T, F, P) satisfying the usual hypotheses. Show there ex-
exists one probability Q, which is equivalent to P, with a bounded density such
that under Q each Xn is a semimartingale in H2.
Exercise 45. Let Б be a standard Brownian motion and let Lx be its local
time at the level x. Let
Л = {u> : dLxs (u>) is singular as a measure with respect to ds]
where of course ds denotes Lebesgue measure on R+. Show that Р(Л) = 1.
Conclude that almost surely the paths s i—> Lx are not absolutely continuous.
(Hint: Use Theorem 69.)
Exercise 46. Let В be a standard Brownian motion and let N be a Poisson
process with parameter A = 1, with В and N independent as processes. Let
L be the local time process for В at level 0, and let X be the vector process
Xt — (Bt, NLt). Let F be the minimal filtration of X, completed in the usual
way.
(a) Show that X is a strong Markov process.

242 Exercises for Chapter IV
(b) Let T be the first jump time of X. Show that the compensator of the
process Ct = l{t>T} is the process At = LtAT-
Note that this gives an example of a compensator of the first jump time of a
strong Markov process which has paths that are almost surely not absolutely
continuous.
**Exercise 47. Let Z be a Levy process. Show that if Z is a sigma martingale,
then Z is a martingale. (Note: This exercise is closely related to Exercise 29
of Chap. I.)

V
Stochastic Differential Equations
1 Introduction
A diffusion can be thought of as a strong Markov process (in Rn) with contin-
continuous paths. Before the development of Ito's theory of stochastic integration
for Brownian motion, the primary method of studying diffusions was to study
their transition semigroups. This was equivalent to studying the infinitesimal
generators of their semigroups, which are partial differential operators. Thus
Feller's investigations of diffusions (for example) were actually investigations
of partial differential equations, inspired by diffusions.
The primary tool to study diffusions was Kolmogorov's differential equa-
equations and Feller's extensions of them. Such approaches did not permit an
analysis of the paths of diffusions and their properties. Inspired by Levy's in-
investigations of sample paths, Ito studied diffusions that could be represented
as solutions of stochastic differential equations1 of the form
Xt = Xo + f o{Xs)dBs + f b(Xs)ds
Jo Jo
(*)
where Б is a Brownian motion in R™, a is an n x n matrix, and b is an n-vector
of appropriately smooth functions to ensure the existence and uniqueness of
solutions. This gives immediate intuitive meaning. If (Ft)t>o is the underlying
filtration for the Brownian motion B, then for small e > 0
-XI- eb\Xt)){Xl+? - X{ - eV(Xt))\Ft} = (aa'f(Xt)s + o(e),
where a' denotes the transpose of the matrix a.
Ito's differential "dB" was found to have other interpretations as well. In
particular, "dB" can be thought of as "white noise" in statistical communi-
communication theory. Thus if ?t is white noise at time t, Bt = /0 ?sds, an equation
1 More properly (but less often) called "stochastic integral equations."

244 V Stochastic Differential Equations
which can be given a rigorous meaning using the theory of generalized func-
functions (cf., e.g., Arnold [2]). Here the Markov nature of the solutions is not as
important, and coefficients that are functionals of the paths of the solutions
can be considered.
Finally it is now possible to consider semimartingale driving terms (or
"semimartingale noise"), and to study stochastic differential equations in full
generality. Since "dB" and adf are semimartingale differentials, they are
always included in our results as special cases. While our treatment is very
general, it is not always the most general available. We have at times preferred
to keep proofs simple and non-technical rather than to achieve maximum
generality.
The study of stochastic differential equations (SDEs) driven by general
semimartingales (rather than just by dB, dt, dN, and combinations thereof,
where N is a Poisson process) allows one to see which properties of the so-
solutions are due to certain special properties of Brownian motion, and which
are true in general. For example, in Sect. 6 we see that the Markov nature of
the solutions is due to the independence of the increments of the differentials.
In Sects. 8 and 10 we see precisely how the homeomorphic and diffeomorphic
nature of the flow of the solution is a consequence of path continuity. In Sect. 5
we study Fisk-Stratonovich equations which reveal that the "correction" term
is due to the continuous part of the quadratic variation of the differentials. In
Sect. 11 we illustrate when standard moment estimates on solutions of SDEs
driven by Brownian motion and dt can be extended to solutions of SDEs
driven by Levy processes.
2 The HJ3 Norms for Semimartingales
We defined an 7i2 norm for semimartingales in Chap. IV as follows. If X is a
special semimartingale with Xq = 0 and canonical decomposition X = N + A,
then
\ /
Jo
/0
We now use an equivalent norm. To avoid confusion, we write H_ instead
of TL2. Moreover we will define H_p, 1 < p < oo. We begin, however, with a
different norm on the space E> (i.e., the space of adapted cadlag processes).
For a process H € D we define
H* = sup \Ht
t
Occasionally if H is in L (adapted and caglad) we write ||i?||sp as well, where
the meaning is clear.

2 The Яр Norms for Semimartingales 245
If A is a semimartingale with paths of finite variation, a natural definition
of a norm would be \\A\\P = || Jo°° |cL4s|||?P, where |cL4s(a;)| denotes the total
variation measure on R+ induced by s i—> As(ui). Since semimartingales do
not in general have such nice paths, however, such a norm is not appropriate.
Throughout this chapter, we will let Z denote a semimartingale with Zq = 0,
a.s. Let Z be an arbitrary semimartingale (with Zq = 0). By the Bichteler-
Dellacherie Theorem (Theorem 43 of Chap. Ill) we know there exists at least
one decomposition Z = N + A, with N a local martingale and A an adapted,
cadlag process, with paths of finite variation (also Щ = Aq = 0 a.s.). For
1 < p < oo we set
/>OO
jp(N,A) = \\[N,N}1Ji2+ / |dA,|||bP.
Definition. Let Z be a semimartingale. For 1 < p < oo define
where the infimum is taken over all possible decompositions Z — N + A where
N is a local martingale, A ? D with paths of finite variation on compacts,
and Aq — No = 0.
The corollary of Theorem 1 below shows that this norm generalizes the
IP norm for local martingales, which has given rise to a martingale theory
analogous to the theory of Hardy spaces in complex analysis. We do not pursue
this topic (cf., e.g., Dellacherie-Meyer [46]).
Theorem 1. Let Z be a semimartingale (Zo = 0). Then \\[Z, Z}H2\\LP <
и?, A <P< oo).
Proof. Let Z = M + A, Mo = Ac, = 0, be a decomposition of Z. Then
s
OO
/•OO
<{M,M]l?2+ \dA.\,
Jo
where the equality above holds because A is a quadratic pure jump semi-
1 /9
martingale. Taking Lp norms yields \\[Z,Z]^ \\lp < jp{M, A) and the result
follows. ?
Corollary. If Z is a local martingale (Zo = 0), then \\Z\\h_p = \\{Z, Z]U2\\lp-

246 V Stochastic Differential Equations
Proof. Since Z is a local martingale, we have that Z = Z+0 is a decomposition
of Z. Therefore
By Theorem 1 we have \\[Z, Z]H2\\lp < \\Z\\hp, hence we have equality. D
Theorem 2 is analogous to Theorem 5 of Chap. IV. For most of the proofs
which follow we need only the case p = 2. Since this case does not need
Burkholder's inequalities, we distinguish it from the other cases in the proof.
Theorem 2. For 1 < p < oo there exists a constant cp such that for any
semimartingale Z, Zq = 0,
Proof. A semimartingale Z is in ГО, so ||Z||s" makes sense. Let Z — M + A
be a decomposition with Mo = Aq — 0. Then
(-O
/
JO
using (o + 6)p < 2p-1(ap + ЬР).
In the case p = 2 we have by Doob's maximal quadratic inequality that
) = 4E{[M,MU}.
For general p, 1 < p < сю, we need Burkholder's inequalities, which state
for a universal constant cp which depends only on p and not on the local
martingale M. For continuous local martingales and p > 2 we proved this
using Ito's formula in Chap. IV (Theorem 73). For general local martingales
and for all finite p > 1 see, for example, Dellacherie-Meyer [46, page 287].
Continuing, letting the constant cp vary from line to line we have
/•OO
/ \dAs\f}
Jo
лОО
+ ( \dAa\y}
Jo
<cp{jp(M,A)f,
and taking p-th roots yields the result. D
Corollary. On the space of semimartingales, the H_p norm is stronger than
the 5P norm, 1 < p < oo.
Theorem 3 (Emery's Inequality). Let Z be a semimartingale, H € L,
l + l = I A < p < oo, l<q <oo). Then

)\\Lr
Г
2 The M_p Norms for Semimartingales 247
Proof. Let H ¦ Z denote (Jo HsdZs)t>Q. Recall that we always assume Z§ = 0
a.s., and let Z = M + A be a decomposition of Z with Mo = Ao = 0 a.s. Then
H ¦ M + H ¦ A is a decomposition of H ¦ Z. Hence
\\н-г\\ш<зЛн-м,н-А).
Next recall that [Я • M, H ¦ M] = / H2sd[M, M]s, by Theorem 29 of Chap. II.
Therefore
/¦OO ЛОО
jr(H-M,H-A) = \\( H2d[M,M}sI'2 + \Hs\\dAs\\\Lr
Jo Jo
<\\H*OO([M,M}1JJ+ Г \dA
Jo
< \\H*OO\\LP\\(\M,M}^2+ Г \dAs\)\\L4
Jo
= \\H\\gpjq(M,A),
where the last inequality above follows from Holder's inequality. The foregoing
implies that
\\H-Z\\&r<\\H\\gPjg(M,A)
for any such decomposition Z = M + A. Taking infimums over all such de-
decompositions yields the result. ?
For a process XeD and a stopping time T, recall that
X = -Х"Д[о,г) + -Xrl[r,oo)>
XT~ = -Х"Д[0)г) +-Х"г-1[г,оо)-
A property holding locally was defined in Chap. I, and a property holding
prelocally was defined in Chap. IV. Recall that a property тг is said to hold
locally for a process X if Хтп\{хп>0) has property тг for each n, where Tn
is a sequence of stopping times tending to oo a.s. If the process X is zero at
zero (i.e., Xo = 0 a.s.) then the property тг is said to hold prelocally if XT"~
has property тг for each n.
Definition. A process X is locally in 5P (resp. H_p) if there exist stopping
times (T")n>i increasing to oo a.s. such that XTnl{Tn>oy is in ^p (resp. Др)
for each n, 1 < p < oo. If Xo = 0 then X is said to be prelocally in S_p (resp.
?LP) if XT"- is in if (resp. Kp) for each n.
While there are many semimartingales which are not locally in H?, all
semimartingales are prelocally in H_p. The proof of Theorem 4 below closely
parallels the proof of Theorem 13 of Chap. IV.
Theorem 4. Let Z be a semimartingale (Zo = 0). Then Z is prelocally in
Hp,l<p<oo.

248 V Stochastic Differential Equations
Proof.. By the Fundamental Theorem of Local Martingales (Theorem 25 of
Chap. Ill) and the Bichteler-Dellacherie Theorem (Theorem 43 of Chap. Ill)
we know that for given e > 0, Z has a decomposition Z = M+A, Mo = Ao = 0
a.s., such that the jumps of the local martingale M are bounded by e. Define
inductively
Tk : [M, M)l/2 + / |<Me| > * + 1}.
./o
The sequence (Tk)k>x are stopping times increasing to oo a.s. Moreover
- AMTkl[TkiOo)) = N + C
is a decomposition of ZTk~. Also, since [М,М]тк = [М,М]тк- + (АМткJ,
we conclude
joo(N,C) = ||[7V,7V]V2+ Г \dC.\U
Jo
= \\([М,М}Тк- + (AMrJ2I/2 + / * \dC.\\\L*
Jo
(fc + e)||L~ <oo.
Therefore ZTk~ € K°° and hence it is in 2?P as well, 1 < p < oo. ?
Definition. Let Z be a semimartingale in H_°° and let a > 0. A finite sequence
of stopping times 0 = To < Tx < ¦ ¦ ¦ < Tk is said to a-slice Z if Z = ZTk~
and || (Z — ZTi)Ti+1~\\H°° <a, 0<i<k — I. If such a sequence of stopping
times exists, we say Z is a-sliceable, and we write Z ? S(a).
Theorem 5. Let Z be a semimartingale with Z$ = 0 a.s.
(i) For a > 0, if Z € S(a) then for every stopping time T, ZT € S(a) and
ZT~ e SBa).
(ii) For every a > 0, there exists an arbitrarily large stopping time T such
thatZT~ ?S(a).
Proof. Since ZT~ = MT + (AT- -AMTl[Tt0o)), and since ||ZT||g=o < ||Z||^~
always, one concludes ||ZT~||#°° < 2||Z||#°°, so that (i) follows.
Next consider (ii). If semimartingales Z and Y are a-sliceable, let T?
and T? be two sequences of stopping times respectively a-slicing Z and Y.
By reordering the points T? and Tj and using (i), we easily conclude that
Z + Y is 8a-sliceable. Next let Z = M + A, Mo = Ao = 0 a.s., with the
local martingale M having jumps bounded by the constant /3 = a/24. By the
preceding observation it suffices to consider M and A separately.

3 Existence and Uniqueness of Solutions 249
For A, let To = 0, Tk+1 = inf{* > Tk : /^ \dAs\ > a/8 or /„* |<Me| > *}.
Then ATk~ € <S(a/8) for each k, and the stopping times (Tk) increase to oo
a.s.
For M, let Ro = 0, Rk+l = inf{t > Rk : [M,M]t - [M,M}Rk >
01 or [M,M]t > k}. Then MRk~ e R°°, each k, and moreover
(M - MR-)R>+*- = MR^ - MR* - (AMRk+1l{Rk+1>Rk})llRk+uOo).
Hence
1 - [M,M]HJ1/2 +
2 + [M,M}Rk+l_ - [M,M]HJ1/2 + \AMRk+1\\\b
Thus for each k, MRk~ € <S(A + \/2)/3), and since C = a/24, the result
follows. ?
3 Existence and Uniqueness of Solutions
In presenting theorems on the existence and uniqueness of solutions of stochas-
stochastic differential equations, there are many choices to be made. First, we do not
present the most general conditions known to be allowed; in exchange we are
able to give simpler proofs. Moreover the conditions we do give are extremely
general and are adequate for the vast majority of applications. For more gen-
general results the interested reader can consult Jacod [103, page 451]. Second,
we consider only Lipschitz-type hypotheses and thus obtain strong solutions.
There is a vast literature on weak solutions (cf., e.g., Stroock-Varadhan [220]).
However, weak solutions are more natural (and simpler) when the differen-
differentials are the Wiener process and Lebesgue measure, rather than general semi-
martingales.
A happy consequence of our approach to stochastic differential equations
is that it is just as easy to prove theorems for coefficients that depend not
only on the state Xt of the solution at time t (the traditional framework), but
on the past history of the process X before t as well.
We begin by stating a theorem whose main virtue is its simplicity. It is a
trivial corollary of Theorem 7 which follows it. Recall that a process H is in
L if it has caglad paths and is adapted.
Theorem 6. Let Z be a semimartingale with Zo = 0 and let / : R+ x п x R —>
R be such that
(i) for fixed x, (t,ui) н-> f(t,w,x) is in L; and

250 V Stochastic Differential Equations
(ii) for each (t,oj), \f(t,w,x)—f(t,oj,y)\ < К(ш)\х—у\ for some finite random
variable K.
Let Xq be finite and J-o measurable. Then the equation
Xt=X0+ [ f{s,-,Xs_)dZs
Jo
admits a solution. The solution is unique and it is a semimartingale.
Of course one could state such a theorem for a finite number of differentials
dZ?, 1 < j< d, and for a finite system of equations.
In the theory of (non-random) ordinary differential equations, coefficients
are typically Lipschitz continuous, which ensures the existence and the unique-
uniqueness of a solution. In stochastic differential equations we are led to consider
more general coefficients that arise, for example, in control theory. There are
enough different definitions to cause some confusion, so we present all the def-
definitions here in ascending order of generality. Note that we add, for technical
reasons, the non-customary condition (ii) below to the definition of Lipschitz
which follows.
Definition. A function / : R+ x R™ —> R is Lipschitz if there exists a (finite)
constant к such that
(i) \f(t,x)-f(t,y)\ < k\x-y\, each teM.+ , and
(ii) t i-> f(t,x) is right continuous with left limits, each x € I".
/ is said to be autonomous if f(t,x) = f(x), all t > 0.
Definition. A function / : R+ x п x R™ —> R is random Lipschitz if /
satisfies conditions (i) and (ii) of Theorem 6.
Let D™ denote the space of processes X = (X1,..., Xn) where each PeD
A <i <n).
Definition. An operator F from D™ into D1 = D is said to be process
Lipschitz if for any X, Y in D™ the following two conditions are satisfied:
(i) for any stopping time T, XT" = Yr~ implies F(X)T~ = F(Y)T-, and
(ii) there exists an adapted process К ? L such that
\\F(X)t-F(Y)t\\<Kt\\Xt-Yt\\.
Definition. An operator F mapping Dn to D1 = D is functional Lipschitz
if for any X, Y in D™ the following two conditions are satisfied:
(i) for any stopping time T, XT" = YT~ implies F(X)T~ = F(Y)T~, and
(ii) there exists an increasing (finite) process К = (Kt)t>o such that \F(K)t -
F(Y)t\ < Kt\\X - Y||* a.s., each t > 0.

3 Existence and Uniqueness of Solutions 251
Note that if g(t,x) is a Lipschitz function, then f(t,x) = g{t—,x) is ran-
random Lipschitz. A Lipschitz, or a random Lipschitz, function induces a process
Lipschitz operator, and if an operator is process Lipschitz, then it is also
functional Lipschitz.
An autonomous function with a bounded derivative is Lipschitz by the
Mean Value Theorem. If a function / has a continuous but not bounded
derivative, / will be locally Lipschitz; such functions are defined and con-
considered in Sect. 7 of this chapter.
Let A = (j4t)t>o be continuous and adapted. Then a linear coeffi-
coefficient such as f(t,oj,x) = At(ui)x is an example of a process Lipschitz co-
coefficient. A functional Lipschitz operator F will typically be of the form
F(X) = f(t,w;Xs, s < t), where / is defined on [0,t] xdx D[0,t] for each
t > 0; here -D[0,?] denotes the space of cadlag functions defined on [0,t]. An-
Another example is a generalization of the coefficients introduced by Ito and
Nisio [101], namely
F(X)t= f g(u,u>,Xu)iJ,(u!,du)
Jo
for a random signed measure /i and a bounded Lipschitz function g with
constant C(w). In this case, the Lipschitz process for F is given by Kt(ui) =
C(a>)||/x(u;)t||, where ||/x(a»)t|| denotes the total mass of the measure fi{w,du)
on [0,t].
Lemmas 1 and 2 which follow are used to prove Theorem 7. We state and
prove them in the one dimensional case, their generalizations to n dimensions
being simple.
Lemma 1. Let 1 < p < oo, let J € 5P, let F be functional Lipschitz with
F@) = 0, and suppose supt |/ft(w)| < к a.s. Let Z be a semimartingale in
Я°° such that \\Z\\h°° < ^Tk • Then the equation
Xt = Jt+ f F(X)s_dZs
Jo
has a solution in S_p. It is unique, and moreover
Proof. Define A : gp -> gp by Л(ХL = Jt + /0* F(X)sdZs. Then by Theo-
Theorems 2 and 3 the operator is 1/2 Lipschitz, and the fixed point theorem gives
existence and uniqueness. Indeed
\\X\\gP <\\J\\gP + \\ I F(X)s^dZs\\§r

252 V Stochastic Differential Equations
Since \\F(X)\\gP = \\F(X) - F@)\\&, we have ||X||r < ||J||r + ?
which yields the estimate. ?
Lemma 2. Let 1 < p < oo, let J € 5P, let F be functional Lipschitz with
F@) = 0, and suppose supt |^t(w)| < к < oo a.s. Let Z be a semimartingale
such that Z € ^г^Га-)- Then the equation
Xt = Jt
/'
./о
has a solution in 5P. It is unique, and moreover ||X||gp < C(k, Z)\\ J||gp, where
C(k, Z) is a constant depending only on к and Z.
Proof. Let z = \\Z\\h-» and j = \\J\\sp. Let 0 = T$,Ti,... ,Te be the slicing
times for Z, and consider the equations, indexed by i = 0,1,2,...,
X = JTi~ + JF(X)s_dZj--: г = 0,1,2,.... (t)
Equation (г) has the trivial solution X = 0 since J°~ = Z°~ = 0 for all t,
and its S_p norm is 0. Assume that equation (г) has a unique solution Хг,
and let Xх = ||X*||sp. Stopping next at T, instead of T, —, let Y% denote the
unique solution of Y% = JTi + JF(Yl)s^.dZji, and set y% = ||F*||sp. Since
Yl = Хг + {AJTi + Р(Хг)т,-Агт,}1[т.<оо), we conclude that
Гг || ^ \\vi\\ i oil Til i IIZTYVMII II ^ll
IISf -S ЦЛ ||gp + Z|| J ||<Jp + \\r (Л. )\\Sp\\Zi \\Hf°
< xi + 2j + kx'z
= x*(l + kz) + 2j;
hence
We set for U e D, DtU = (U - ит')т^-. Since each solution X of
equation (г + 1) satisfies XTi = Y% on [0,Ti+1), we can change the unknown
by U = X - (Yi)T'+1-, to get the equations U = DiJ+J F(Yi + U)s-dDiZs.
However since F{Yi + 0) need not be 0 we define Gt(-) = F(Yi + •) - F^),
and thus the above equation can be equivalently expressed as
U = {DiJ+ f FfX^a-dDiZ,) + I Gi{U)s_dDiZs.
We can now apply Lemma 1 to this equation to find that it has a unique
solution in S_p, and its norm ul is majorized by
«' < 2\\DiJ + f
)

3 Existence and Uniqueness of Solutions 253
We conclude equation (г + 1) has a unique solution in SJ' with norm хг+1
dominated by (using (*))
xi+l <иг + уг < Aj + 2y* < 8j + 2A + kz)x\
Next we iterate from г = 0 to ? — 1 to conclude that
Finally, since Z = ZTt ~, we have seen that the equation X = J+f F(X)s-dZs
has a unique solution in S_p, and moreover X = Xе + J — JTi~. Therefore
\\X\\& < xe + 2j, and hence C(k, Z) < 2 + ЧЩ^1}- ?
Theorem 7. Given a vector of semimartingales Z = (Z1,... ,Zd), Zo = 0
processes J' eD, 1 < г < n, and operators Fj which are functional Lipschitz
A < i < n, 1 < j < d), the system of equations
A < г < n) has a solution in D™, and it is unique. Moreover if (Jl)i<n is a
vector of semimartingales, then so is (Xl)i<n.
Proof The proof for systems is the same as the proof for one equation pro-
provided we take F to be matrix-valued and X, J and Z to be vector-valued.
Hence we give here the proof for n = d — 1. Thus we will consider the equation
f
Jo
F(X)s.dZs.
Assume that maxjj supt Kl'J(ui) < к < оо a.s. Also, by considering the equa-
equation
Xt = (Jt+ f F@)s_dZs) + I G(X)s_dZs,
Jo Jo
where G(X) = F(X) - F@), it suffices to consider the case where F@) = 0.
We also need Lemmas 1 and 2 only for p = 2. In this case C2 = y/8.
Let T be an arbitrarily large stopping time such that JT~ e ^2 and such
that ZT~ € S( .]gk )¦ Then by Lemma 2 there exists a unique solution in 52
of
X(T)
= J?~ + f
Jo
By the uniqueness in ^2 one has, for R>T, that X(R)T~ = X(T)T-, and
therefore we can define a process lonllx [0,oo) by X = X(T) on [0,T).
Thus we have existence.

254 V Stochastic Differential Equations
Suppose next Y is another solution. Let S be arbitrarily large such that
(X — Y)s~ is bounded, and let R — min(Sr, T), which can also be taken
arbitrarily large. Then XR~ and YR~ are both solutions of
= JH~+
Jo
and since ZR~ e S( \- ), we know that XR~ = YR~ by the uniqueness
established in Lemma 2. Thus X = Y, and we have uniqueness.
We have assumed that maxij supt К}'3 (ш) < к < oo a.s. By proving
existence and uniqueness on [O,toL for to fixed, we can reduce the Lips-
chitz processes K\'3 to the random constants Kl'^(oj), which we replace with
K(w) = maxjj¦ KI'oj(lo). Thus without loss of generality we can assume we
have a Lipschitz constant K(oj) < oo a.s. Then we can choose a constant с
such that P[K < c) > 0. Let П„ = {К < с + n}, each n = 1,2, 3,.... Define
a new probability Pn by Pn(A) = P(A П П„)/Р(П„), and note that Pn <C P.
Moreover for n > m we have Pm <C Pn- From now on assume n > m in the
rest of this proof. Therefore we know that all P semimartingales and all Pn
semimartingales are Pm semimartingales, and that on Qm a stochastic inte-
integral calculated under Pm agrees with the the same one calculated under Pn,
by Theorem 14 of Chap. II. Let Yn be the unique solution with respect to Pn
which we know exists by the foregoing. We conclude Yn = Ym on flm, a.s.
(dPm). Define
Yt =
n=l
and we have Y = Yn a.s. (dPn) on Qn, and hence also a.s. (dP) on Qn, each
n. Since fl = U^L1(f2n \ iin-i) a-s. (dP), we have that on Qn:
Yt = Jt+ f F(Yn)s_dZs
Jo
= Jt+ f F{Y)s^dZs
Jo
a.s. (dP), for each n. This completes the proof. ?
Theorem 7 can be generalized by weakening the Lipschitz assumption on
the coefficients. If the coefficients are Lipschitz on compact sets, for example,
in general one has unique solutions existing only up to a stopping time T;
at this time one has limsupt^T |Xt| = oo. Such times are called explosion
times, and they can be finite or infinite. Coefficients that are Lipschitz on
compact sets are called locally Lipschitz. Simple cases are treated in Sect. 7
of this chapter (cf., Theorems 38, 39, and 40), where they arise naturally in
the study of flows.

3 Existence and Uniqueness of Solutions 255
We end this section with the remark that we have already met a funda-
fundamental stochastic differential equation in Chap. II, that of the stochastic
exponential equation
I
Jo
Z0=0.
There we obtained a formula for its solution (thus a fortiori establishing the
existence of a solution), namely
0<s<t
The uniqueness of this solution is a consequence of Theorem 7, or of Theo-
Theorem 6.
A traditional way to show the existence and uniqueness of solutions of
ordinary differential equations is the Picard iteration method. One might well
wonder if Picard-type iterations converge in the case of stochastic differential
equations. As it turns out, the following theorem is quite useful.
Theorem 8. Let the hypotheses of Theorem 7 be satisfied, and in addition
let (X°y = Hl be processes in D A < i < n). Define inductively
m+l\i _
{X,
and let X be the solution of
xi = Ji + ? / FWs-dzi (i < i < n).
3 = 1 J°
Then Xm converges to X in ucp.
Proof We give the proof for d = n = 1. It is easy to see that if итт^оо Хт =
X prelocally in S_ , then Итт^оо Xm = X in ucp; in any event this is proved
in Theorem 12 in Sect. 4. Thus we will show that limm^oo Xm = X prelocally
in ;S2. We first assume supt Kt < a < oo a.s. Without loss of generality we
can assume Z e S(a), with a - -57^, and that J € §^. Let 0 — To < T\ <
• • • < Tfc be the stopping times that a-slice Z. Then (Xm)Tl~ and XTl~ are
in S2 by Lemma 1. Then
by Theorems 2 and 3. Therefore

256 V Stochastic Differential Equations
so that lim^oo \\(Xm+1 - X)Tl-\\s2 = 0. We next analyze the jump at Tx.
Since
we have
MX™*1 -xf% <
where z = ||?||я~ < oo. Therefore
\\(Xm+1 - X)T- ||i2 < _J_
Next suppose we know that
where 7 = IKX1 - Х)Т"~\\з2. Then
f
JTt
JTt
where ^+1 = (Z - Z7''O'^1-. Therefore by iterating on ?, 0 < I < k,
\\(Xm+1 -Xf^'Wg, < \\(Xm+l -X)T'\\s2 + aV8a\\{Xm -Xf'+^
2Ш-1
Note that the above expression tends to 0 as m tends to oo. Therefore Xm
tends to X prelocally in S2 by a (finite) induction, and hence limm^oo Xm —
X in ucp.
It remains to remove the assumption that supt Кt < a < oo a.s. Fix a
t < oo; we will show ucp on [0,t]. Since t is arbitrary, this will imply the
result. As we did at the end of the proof of Theorem 7, let с > 0 be such that
P{Kt < c) > 0. Define Пп = {w : Kt(w) < c + n}, and Pn(A) = Р(Л\Пп).
Then Pn <C P, and under Pn, limm^ooXm = X in ucp on [0, t]. For e > 0,
choose N such that n > N implies P(&n) < ?- Then
- X)l >5)< Pn((Xm - X)*t >5) + e,
hence Итт^оо P((Xm — XI > S) < e. Since e > 0 was arbitrary, the limit
must be zero. ?

4 Stability of Stochastic Differential Equations 257
4 Stability of Stochastic Differential Equations
Since one is never exactly sure of the accuracy of a proposed model, it is
important to know how robust the model is. That is, if one perturbs the model
a bit, how large are the resulting changes? Stochastic differential equations
are stable with respect to perturbations of the coefficients, or of the initial
conditions. Perturbations of the differentials, however, are a more delicate
matter. One must perturb the differentials in the right way to have stability.
Not surprisingly, an H? perturbation is the right kind of perturbation. In this
section we will be concerned with equations of the form
f
n+ / Fn(Xn)s-dZ™ (*n)
Jo
w
JO
' F(X)s^dZs, (*)
/o
where Jn, J are in D, Zn, Z are semimartingales, and Fn, F are functional
Lipschitz, with Lipschitz processes Kn, K, respectively. We will assume that
the Lipschitz processes Kn, К are each uniformly bounded by the same con-
constant, and that the semimartingale differentials Zn, Z are always zero at 0
(that is, Zq = 0 a.s., n > 1, and Zq = 0 a.s.).
For simplicity we state and prove the theorems in this section for one
equation (rather than for finite systems), with one semimartingale driving
term (rather than a finite number), and for p = 2. The generalizations are
obvious, and the proofs are exactly the same except for notation. We say a
functional Lipschitz operator F is bounded if for all H G D, there exists a
non-random constant с < oo such that F{H)* < c.
Theorem 9. Let J, Jn e D; Z, Zn be semimartingales; and F, Fn be func-
functional Lipschitz with constants K, Kn, respectively. Assume that
(i) J, Jn are in g2 (resp. Щ2) and limn^oo Jn = J in ?2 (resp. ??*);
(ii) Fn are all bounded by the same constant c, and limn^oo Fn(X) = F(X)
in S2, where X is the solution of (*); and
(Hi) max.(supn Kn, К) < о < oo a.s. (a not random), Z e ^{^js~)i (^™)«>i
are in H2, and limn^oo Zn = Z in H_2.2
Then Нтп-юо Xn = X in S2 (resp. in H2), where Xn is the solution of (*n)
and X is the solution of (*).
Proof. We use the notation H ¦ Zt to denote JQ HsdZs, and H ¦ Z to denote
the process (H • Zt)t>o- We begin by supposing that J, (Jn)n>i are in 52 and
J™ converges to J in 52. Then
2 S(a) is defined in Sect. 2 on page 248.

258 V Stochastic Differential Equations
X - Xn = J - Jn + (F(X) - Fn(X))_ ¦ Z
+ (Fn(X) - P(I"))_ • Z + Fn(Xn)_ ¦ (Z - Zn).
Let Yn = (F(X) - Fn(.X"))_ • Z + Fn(Xn) ¦ (Z - Zn). Then
X -Xn = J-Jn + Yn + {Fn(X) - Fn(I"))_ • Z. (**)
For U G D define Gn by
Gn(U) = Fn(X) - Fn{X - U).
Then Gn{U) is functional Lipschitz with constant a and Gn@) = 0. Take
U = X — Xn, and (**) becomes
U = (J - Jn + Yn) + Gn(U)- ¦ Z.
By Lemma 2 preceding Theorem 7 we have
II* - Xn\\ga < C(a, Z)\\J -Jn + Yn\\g2 < C(a, Z){\\J - Jn\\g* + \\Yn\\g2}.
Since C(a, Z) is independent of n and Игг^-^ \\J - J™||52 = 0 by hypothesis,
it suffices to show limn^oo ||УИ||52 = О. But
\\Yn\\g, < \\{F{X) - Fn(X))_ ¦ Z\\& + \\Fn(Xn)_ ¦ (Z - Zn)\\g,
by Theorem 2 and Emery's inequality (Theorem 3). Since Ц^Ця00 < oo by
hypothesis and since
Xm^ \\F{X) - F"(X)||r = Jiit^ \\Z - Zn\\m = 0,
again by hypothesis, we are done.
Note that if we knew J™, J G Я2 and that Jn converged to J in Я2, then
\\X - Xn\\^ < || J - JI^ + ||yn||^2 + C\\X - Xn\\g, |
We have seen already that limn-^oo \\X — Xn\\g2 = 0, hence it suffices to show
limn-xx, ||^п||н;2 = 0. Proceeding as in (* * *) we obtain the result. D
Comment. Condition (ii) in Theorem 9 seems very strong; it is the per-
perturbation of the semimartingale differentials that make it necessary. In-
Indeed, the hypothesis cannot be relaxed in general, as the following example
shows. We take О = [0,1], P to be Lebesgue measure on [0,1], and {Ft)t>o
equal to T, the Lebesgue sets on [0,1]. Let ф{?) = min(?, 1), t > 0. Let
fn(u>) > 0 and set Ztn(u>) = <t>(t)fn(u>), u> G [0,1], and Zt{J) = 4>(t)f(u).

4 Stability of Stochastic Differential Equations 259
Let Fn(X) = F{X) = X, and finally let J? = Jt = 1, all t > 0. Thus the
equations (*n) and (*) become respectively
/
/o
Xt = l+ [ Xs_dZs,
J
= l+ [
Jo
which are elementary continuous exponential equations and have solutions
We can choose /„ such that limn^oo E{f%} = 0 but lim^oo E{f%} ф 0 for
p > 2. Then the Z™ converge to 0 in H2 but Xй does not converge to X = 1
(since / = 0) in ?p, for any p > l.Tndeed, limn^oo E{f%} Ф 0 for p > 2
implies linin-^oo ?{e*^" } ф 1 for any ? > 0.
The next result does not require that the coefficients be bounded, because
there is only one, fixed, semimartingale differential. Theorem 10, 11, and 13
all have H_2 as well as S2 versions as in Theorem 9, but we state and prove
only the 52 versions.
Theorem 10. Let J, Jn G D; Z be a semimartingale; F, Fn be functional
Lipschitz with constants K, Kn, respectively; and let Xn, X be the unique
solutions of equations (*n) and (*), respectively. Assume that
(i) Jn, J are in 52 and Нп^-^ Jn = J in S2;
(ii) lim Fn(X) = F(X) in S2, where X is the solution of (*); and
n—>oo —
(Hi) max(supn Kn,K) < a < oo a.s. for a non-random constant a, and Z €
Then limn^oo Xn = X in Q2 where Xn is the solution of (*n) and X is
the solution of (*).
Proof Let Xn and X be the solutions of equations (*n) and (*), respectively.
Then
X - Xn = J - Jn + (F(X) - Fn{X))_ ¦ Z + (Fn(X) - Fn(Xn))_ ¦ Z.
We let Yn = (F(X) - Fn(X))_ ¦ Z, and we define a new functional Lipschitz
operator Gn by
Gn{U) = Fn{X) - Fn(X - U).
Then Gn@) = 0. If we set U = X - Xn, we obtain the equation
U = J-Jn + Yn + Gn{U)- ¦ Z.

260 V Stochastic Differential Equations
Since Z e K°°, by Emery's inequality (Theorem 3) we have Yn -> 0 in Я2,
and hence also in 52 (Theorem 2). In particular ||yn||s2 < oo, and therefore
by Lemma 2 in Sect. 3 we have
\\U\\g. <C{a,Z)\\J - Jn + Yn\\g.,
where C(o, Z) is independent of n, and where the right side tends to zero as
n —> oo. Since U = X — Xn, we are done. G
We now wish to localize the results of Theorems 9 and 10 so that they hold
for general semimartingales and exogenous processes Jn, J. We first need
a definition, which is consistent with our previous definitions of properties
holding locally and prelocally (defined in Chap. IV, Sect. 2).
Definition. Processes Mn are said to converge locally (resp. prelocally)
in ?p (resp. Rp) to M if Mn, M are in ?p (resp. Др) and if there exists a
sequence of stopping times Tk increasing to oo a.s. such that limn^^, \\(Mn —
M)Tkl{Tk>o}h» = 0 (resp. lim^oo \\(Mn - M)Tk~\\gJ, = 0) for each к > 1
(resp. if replaced by Rp).
Theorem 11. Let J, Jn ? D; Z be a semimartingale (Zo — 0); and F, Fn
be functional Lipschitz with Lipschitz processes K, Kn, respectively. Let Xn,
X be solutions respectively of
[ Fn(Xn)s_dZs, (*n)
f F{X)s
Jo
(i) Jn converge to J prelocally in S2;
Xt = Jt+ f F{X)s_dZs.
Assume that
(ii) Fn(X) converges to F(X) prelocally in S2 where X is the solution of(*);
and
(Hi) max(supn Kn, К) < о < oo a.s. (a not random).
Then linin^oo Xn = X prelocally in 52 where Xn is the solution of (*n) and
X is the solution of (*).
Proof. By stopping at T- for an arbitrarily large stopping time T we can
assume without loss of generality that Z G ?(^7g~) by Theorem 5, and that
J™ converges to J in S2 and F(Xn) converges to F(X) in 52, by hypothesis.
Next we need only to apply Theorem 10. ?
We can recast Theorem 11 in terms of convergence in ucp (uniform conver-
convergence on compacts, in probability), which we introduced in Sect. 4 of Chap. II
in order to develop the stochastic integral.

4 Stability of Stochastic Differential Equations 261
Corollary. Let Jn, J G D; Z be a semimartingale (Zo = 0); and F, Fn be
functional Lipschitz with Lipschitz processes K, Kn, respectively. Let X, Xn
be as in Theorem 11. Assume that
(i) J™ converges to J in ucp,
(ii) Fn(X) converges to F(X) in ucp, and
(iii) max(supn Kn,K) < a < oo a.s. (o not random).
Then lim Xn = X in ucp.
n—»oo
Proof. Recall that convergence in ucp is metrizable; let d denote a distance
compatible with it. If Xn does not converge to 0 in ucp, we can find a subse-
subsequence n' such that infn/ d(Xn , 0) > 0. Therefore no sub-subsequence (Xn")
can converge to 0 in ucp, and hence Xn cannot converge to 0 prelocally in
S2 as well. Therefore to establish the result we need to show only that for
any subsequence n', there exists a further subsequence n" such that Xn' con-
converges prelocally to 0 in §_ . This is the content of Theorem 12 which follows,
so the proof is complete. G
Theorem 12. Let Hn, Я G D. For Hn to converge to H in ucp it is necessary
and sufficient that there exist a subsequence n' such that linv^oo Hn' — H,
prelocally in S2.
Proof. We first show the necessity. Without loss of generality, we assume that
H = 0. We construct by iteration a decreasing sequence of subsets (Nfc) of
N= {1,2,3,...}, such that
lim sup |Я™| = О a.s.
"^°°0<S<fc
By Cantor's diagonalization procedure we can find an infinite subset N' of N
such that
Iw^ sup |Я™| =0 a.s.,
each integer к > 0. By replacing N with N' we can assume without loss of
generality that Я™ tends to 0 uniformly on compacts, almost surely. We next
define
Tn=inf{t>0:|H?|>l},
Sn = inf Tm.
m>n
Then Tn and Sn are stopping times and the Sn increase to oo a.s. Indeed,
for each к there exists N(w) such that for n > N(u>), sup0<g<fc |Я"(и>)| < 1.
Hence Tn{w) > к and S^^(u;) > k. Next, define
— (/3 ) .

262 V Stochastic Differential Equations
Then
{Lm)* < sup |tf™
0<s<n
which tends to 0 a.s. Moreover, when ra > n,
(Lmy< sup |ЯП< sup |ЯП<1.
0<s<Sn 0<s<Tm
Hence by the Dominated Convergence Theorem we have that (Lm)* tends to
0 in L2, which implies the result.
For the sufficiency, suppose that Hn does not converge to H in ucp. Then
there exist t0 > 0, 8 > 0, e > 0, and a subsequence n' such that P{(Hn —
Н)ь> > 5} > ? for each "'• Then II (Я"' - я)?011ь2 > Sy/ё for all n'. Let Tk
tend to oo a.s. such that (Hn' - H)T ~ tends to 0 in S2, each k. Then there
exists К > 0 such that P(Tk < t0) < Щ- for all k> K. Hence
hm ||(Я"-Я)?0|ия< lim
We conclude 5-^/i < ^^, a contradiction. Whence Hn converges to H in
ucp. ?
Recall that we have stated and proven our theorems for the simplest case
of one equation (rather than finite systems) and one semimartingale driving
term (rather than a finite number). The extensions to systems and several
driving terms is simple and essentially only an exercise in notation. We leave
this to the reader.
An interesting consequence of the preceding results is that prelocal ^p
and prelocal H_p convergence are not topological in the usual sense. If they
were, then one would have that a sequence converged to zero if and only if
every subsequence had a sub-subsequence that converged to zero. To see that
this is not the case for S2 for instance, consider the example given in the
comment following Theorem 9. In this situation, the solutions Xn converge to
X in ucp. By Theorem 12, this implies the existence of a subsequence n' such
that ИгПп'-юоХ™' = X, prelocally in 52. However we saw in the comment
that Xn does not converge to X in S2. It is still a priori possible that Xn
converges to X prelocally in ^2, however. In the framework of the example a
stopping time is simply a non-negative random variable. Thus our counter-
counterexample is complete with the following real analysis result (see Protter [199,
page 344] for a proof). There exist non-negative functions fn on [0,1] such
that НгПп-^оо Jo /„(xJrfx = 0 and limsupn_>oo/yl(/n(x))p(ix = +oo for all
p > 2 and all Lebesgue sets Л with strictly positive Lebesgue measure. In
conclusion, this counterexample gives a sequence of semimartingales Xn such
that every subsequence has a sub-subsequence converging prelocally in S_p,
but the sequence itself does not converge prelocally in 5P, A < p < oo).

4 Stability of Stochastic Differential Equations 263
Finally, we observe that such non-topological convergence is not as un-
unusual as one might think at first. Indeed, let Xn be random variables which
converge to zero in probability but not a.s. Then every subsequence has a sub-
subsequence which converges to zero a.s., and thus almost sure convergence
is also not topological in the usual sense.
As an example of Theorem 10, let us consider the equations
X? = J?+ [ Fn(Xn)sdWs + f Gn{Xn)sds,
Jo Jo
Xt = Jt+ I F{X)sdWs + f G{X)sds
Jo Jo
where W is a standard Wiener process. If (Jn — J)j converges to 0 in L2, each
t > 0, and if Fn, Gn, F, G are all functional Lipschitz with constant К < oo
and are such that {Fn{X) - F{X))*t and {Gn{X) - G{X))t converge to 0 in
L2, each t > 0, then (Xn - X)% converges to 0 in L2 as well, each t > 0. Note
that we require only that Fn{X) and Gn(X) converge respectively to F(X)
and G(X) for the one X that is the solution, and not for all processes in D.
One can weaken the hypothesis of Theorem 9 and still let the differentials
vary, provided the coefficients stay bounded, as the next theorem shows.
Theorem 13. Let Jn, J e D; Z, Zn be semimartingales (Zg = Zo = 0 a.s.);
andF, Fn be functional Lipschitz with Lipschitz processes К, Кп, respectively.
Let Xn, X be solutions of (*n) and (*), respectively. Assume that
(i) Jn converges to J prelocally in S2;
(ii) Fn(X) converges to F(X) prelocally in S_2, and the coefficients Fn, F
are all bounded by с < oo;
(Hi) Zn converges to Z prelocally in H2; and
(iv) max(supn Kn,К) < о < oo a.s. (a not random).
Then lim Xn = X prelocally in S2.
n-»oo =
Proof. By stopping at Г— for an arbitrarily large stopping time T we can
assume without loss that Z G ^{wj5~) by Theorem 5, and that J™ converges
to J in ;g2, Fn(X) converges in g2 to F(X), and Zn converges to Z in Я2,
all by hypothesis. We then invoke Theorem 9, and the proof is complete. D
The assumptions of prelocal convergence are a bit awkward. This type of
convergence, however, leads to a topology on the space of semimartingales
which is the natural topology for convergence of semimartingale differentials,
just as ucp is the natural topology for processes related to stochastic integra-
integration. This is exhibited in Theorem 15.
Before defining a topology on the space of semimartingales, let us recall
that we can define a "distance" on D by setting, for Y, Z еШ

264 V Stochastic Differential Equations
r(Y) = ? 2""Я{1 Л sup
and d(Y, Z) — r{Y — Z). This distance is compatible with uniform convergence
on compacts in probability, and it was previously defined in Sect. 4 of Chap. II.
Using stochastic integration we can define, for a semimartingale X,
f(X) = sup r(tf • X)
where the supremum is taken over all predictable processes bounded by one.
The semimartingale topology is then defined by the distance d(X, Y) =
f(X - Y).
The semimartingale topology can be shown to make the space of semi-
martingales a topological vector space which is complete. Furthermore, the fol-
following theorem relates the semimartingale topology to convergence in HJ". For
its proof and a general treatment of the semimartingle topology, see Emery [66]
or Protter [201].
Theorem 14. Let 1 < p < oo, let Xn be a sequence of semimartingales, and
let X be a semimartingale.
(i) If Xn converges to X in the semimartingale topology, then there exists a
subsequence which converges prelocally in Hp'.
(ii) If Xn converges to X prelocally in H_p, then it converges to X in the
semimartingale topology.
In Chap. IV we established the equivalence of the norms ||^||н2 an(i
suP|h|<i 11-^" -^Ils2 ш the corollary to Theorem 24 in Sect. 2 of that chapter.
Given this result, Theorem 14 can be seen as a uniform version of Theorem 12.
We are now able once again to recast a result in terms of ucp convergence.
Theorem 13 has the following corollary.
Corollary. Let J™, JeB; Zn, Z be semimartingales (Z? = Zo = 0); and
jP™, F be functional Lipschitz with Lipschitz processes K, Kn, respectively.
Let Xn, X be solutions of (*n) and (*), respectively. Assume that
(i) J™ converges to J in ucp,
(ii) Fn(X) converges to F(X) in ucp where X is the solution of (*), and
moreover all the coefficients Fn are bounded by a random с < oo,
(iii) Zn converges to Z in the semimartingale topology, and
(iv) max(supn Kn, K) < a < oo a.s.
Then lim Xn = X in ucp.
П—ЮО
Proof. Since Zn converges to Z in the semimartingale topology, by Theo-
Theorem 14 there exists a subsequence n' such that Zn converges to Z prelocally
in Я2. Then by passing to further subsequences if necessary, by Theorem 12
we may assume without loss that J™ converges to J and Fn(X) converges

4 Stability of Stochastic Differential Equations 265
to F(X) both prelocally in S2, where X is the solution of (*). Therefore, by
Theorem 13, Xn converges to X prelocally in S2 for this subsequence. We
have shown that for the sequence (Xn) there is always a subsequence that
converges prelocally in S2. We conclude by Theorem 12 that Xn converges to
X in ucp. П
The next theorem extends Theorem 9 and the preceding corollary by re-
relaxing the hypotheses on convergence and especially the hypothesis that all
the coefficients be bounded.
Theorem 15. Let Jn, J G D; Zn, Z be semimartingales (Z? = Zo = 0); and
Fn, F be functional Lipschitz with Lipschitz process K, the same for all n.
Let Xn, X be solutions respectively of
[ F"(Xn)s_dZs", (*n)
o
Xt = Jt+ [ F(X)s.dZs. (*)
Jo
Assume that
(i) Jn converges to J in ucp;
(ii) Fn{X) converges to F(X) in ucp, where X is the solution of (*); and
(Hi) Zn converges to Z in the semimartingale topology.
Then Xn converges to X in ucp.
Proof. First we assume that supt Kt{uj) < a < oo. We remove this hypothesis
at the end of the proof. By Theorem 12, it suffices to show that there exists
a subsequence n' such that Xn converges to X prelocally in S2. Then by
Theorem 12 we can assume with loss of generality, by passing to a subsequence
if necessary, that Jn converges to J and Fn(X) converges to F(X) both
prelocally in 52. Moreover by Theorem 14 we can assume without loss, again
by passing to a subsequence if necessary, that Zn converges to Z prelocally
in H2, and that Z G <S( .Jg )¦ Thus all the hypotheses of Theorem 13 are
satisfied except one. We do not assume that the coefficients Fn are bounded.
However by pre-stopping we can assume without loss that |F(X)| is uniformly
bounded by a constant с < oo.
Let us introduce truncation operators Tx defined (for x > 0) by
TX(Y) = min(x,sup{-x,Y)).
Then Tx is functional Lipschitz with Lipschitz constant 1, for each x > 0.
Consider the equations
Ytn =
/ (Ta+c+1Fn)(Yn)s-dZ^.
Jo

266 V Stochastic Differential Equations
Then, by Theorem 13, Yn converges to X prelocally in S2. By passing to yet
another subsequence, if necessary, we may assume that Fn(X) tends to F(X)
and Yn tends to X uniformly on compacts almost surely. Next we define
Sk = inf{t > 0 : 3m > jfc : \Ytm - Xt\ + \Fm(X)t - F(X)t\ > 1}.
The stopping times Sk increase a.s. to oo. By stopping at Sk —, we have for
n> к that (Yn - X)* and {Fn{X) - F{X))* are a.s. bounded by 1. (Note
that stopping at Sk- changes Z to being in S{-^-jw~) instead of S{ j4g-) by
Theorem 5.) Observe that
\Fn{Yn)\ < \Fn(Yn) - Fn{X)\ + \Fn(X) - F(X)\ + \F(X)\
< a{Yn - X)* + (Fn{X) - F{X))* + F(X)*
< a + 1 + c,
whence (Ta+c+1Fn)(Yn) = Fn(Yn). We conclude that, for an arbitrarily
large stopping time R, with Jn and Zn stopped at R, Yn is a solution of
Ytn =
s:
which is equation (*n). By the uniqueness of solutions we deduce Yn = Xn
on [0, R). Since Yn converges to X prelocally in 52, we thus conclude Xn
converges to X prelocally in S2.
It remains only to remove the hypothesis that supt Kt{uS) < о < oo. Since
we are dealing with local convergence, it suffices to consider supg<t Ks < Kt,
for a fixed t. Since Kt < oo a.s., let a > 0 be such that P(Kt < a) > 0, and
define
^m = {w : Kt{uj) < a + m}.
Then Om increase to О a.s. and as in the proof of Theorem 7 we define Pm
by Pm(A) = Р{А\Пт), and define J^ = ^|nm, the trace of Tt on Пт. Then
Pm <c P, so that by Lemma 2 preceding Theorem 25 in Chap. IV, if Zn
converges to Z prelocally in Я2(Р), then Zn converges to Z prelocally in
H_ (Pm) as well. Therefore by the first part of this proof, Xn converges to X
in ucp under Pm, each m > 1.
Choose e > 0 and m so large that -P(O?J < e. Then
lim P((Xn - X)*t >5)< lim Pm((Xn - X)* > S) + Р(Пст)
< lim Pm({Xn - X)\ > 6) + г
and since e > 0 was arbitrary, we conclude that Xn converges to X in ucp on
[0, t]. Finally since t was arbitrary, we conclude Xn converges to X in ucp. ?

4 Stability of Stochastic Differential Equations 267
Another important topic is how to approximate solutions by difference
solutions. Our preceding convergence results yield two consequences (Theo-
(Theorem 16 and its corollary).
The next lemma is a type of Dominated Convergence Theorem for stochas-
stochastic integrals and it is used in the proof of Theorem 16.
Lemma (Dominated Convergence Theorem). Let p,q,r be given such
that - + - = ?, where 1 < r < oo. Let Z be a semimartingale in H4, and let
Hn e§* such^hat \Hn\ < Y € §?, all n > 1. Suppose lim^^ Щ_{ш) = О,
all (t,u>). Then
lim
Proof. Since Z e H_q, there exists a decomposition of Z, Z = N + A, such
/
Jo
/ |e||U <oo.
Jo
Let Cn be the random variable given by
cn = (Г°(явп_)МадвI/2 + Г \н?_\\ма
Jo Jo
The hypothesis that \Hn\ <Y implies
/ \dAa\) a.s.
/ \
Jo
/
o
However
/»OO /»OO
/ \dAa\)hr < \\Y*\\lA\IN,n№ + / |dAe|||L,
Jo Jo
,A) <oo.
Thus Cn is dominated by a random variable in Lr and hence by the Dominated
Convergence Theorem Cn tends to 0 in Lr. D
We let <7ra denote a sequence of random partitions tending to the identity.3
Recall that for a process Y and a random partition а = {0 = Го < Т\ < ¦ ¦ ¦ <
Tfen}, we define
к
Note that HY is adapted, cadlag (i.e, Y € Щ, then (Y°)s>o is left continuous
with right limits (and adapted, of course). It is convenient to have a version
of Ya e D, occasionally, so we define
3 Random partitions tending to the identity are defined in Chap. II, preceding
Theorem 21.

268 V Stochastic Differential Equations
Theorem 16. Let J € 5 , let F be process Lipschitz with Lipschitz process
К < a < oo a.s. and F@) € S2. Let Z be a semimartingale in S( j- ), and
let X(a) be the solution of
rt
Xt = Jt+ F(X°+)°dZs, (*CT)
Jo
for a random partition a. If an is sequence of random partitions tending to
the identity, then X(an) tends to X in S2, where X is the solution of (*) of
Theorem 15.
Proof. For the random partition an (n fixed), define an operator Gn on D by
Gn{H) = F{Ha"+)a"+.
Note that Gn(H) € © for each Я e D and that СП(Я)_ =
Then Gn is functional Lipschitz with constant К and sends S2 into itself, as
the reader can easily verify. Since F@) € S2, so also are Gn@) e S2, n > 1,
and an argument analogous to the proof of Theorem 10 (though a bit simpler)
shows that it suffices to show that /0 Gn(X)s_dZs converges to /0 F(X)s_dZs
in S2, where X is the solution of
Xt = Jt+ I F(X)s-dZs.
I F(X)s-t
Jo
Towards this end, fix (t,oj) with t > 0, and choose e > 0. Then there exists
5 > 0 such that \Хи(ш) - Xt-(w)\ < e for all и € [t - 25, t). If mesh((r) < 6,
then also \X?+(w) - Xu(co)\ < 2e for all и e [t - 5, t). This then implies that
\F(Xa+)(oj) - F(X)(oj)\ < 2ae. Therefore (F{Xa"+) - F(X))°"(uj) tends to
0 as mesh(crn) tends to 0. Since, on @,oo),
lim
mesh(CTn)—
we conclude that
lim
mesh(an)—>0
where convergence is pointwise in (t, u>). Thus
lim G»(X)-=F(X)-.
meshfcr,,)—>0
However we also know that
< F@)' + aX*
which is in S_ and is independent of an. Therefore using the preceding lemma,
the Dominated Convergence Theorem for stochastic integrals, we obtain the
convergence in S2 of /0 Gn{X)s-dZs to /0 F(X)s-dZs, and the proof is com-
complete. D

4 Stability of Stochastic Differential Equations 269
Remark. In Theorem 16 and its corollary which follows, we have assumed
that F is process Lipschitz, and not functional Lipschitz. Indeed, Theorem 16
is not true in general for functional Lipschitz coefficients. Let Jt = l{t>i}>
Zt - t Л 2, and F(Y) = Yil{t>i}. Then X, the solution of (*) of Theorem 15
is given by Xt — (t Л 2)l{t>ij, but if a is any random partition such that
Tfc ф 1 a.s., then (X(a)a+)~t = 0 for t < 1, and therefore F{X(a)a+) = 0,
and X{o~)t = Jt = l{t>i}. (Here X(a) denotes the solution to equation (*a)
of Theorem 16.)
Corollary. Let J e D; F be process Lipschitz; Z be a semimartingale; and
let an be a sequence of random partitions tending to the identity. Then
lim X(an) = X in ucp
n—>oo
where X(an) is the solution of (*a) and X is the solution of (*), as in Theo-
Theorem 16.
Proof. First assume К < a < oo, a.s. Fix t > 0 and e > 0. By Theorem 5 we
can find a stopping time T such that ZT~ € S( L ), and P(T < ?) < e. Thus
without loss of generality we can assume that Z € S( ^ ). By letting 5fc =
inf{? > 0 : | Jt\ > k}, we have that Sk is a stopping time, and lim/^oo Sk = oo
a.s. By now stopping at Sk— we have that J is bounded, hence also in S_2,
and Z e S(tjs-)- ^n analogous argument gives us that F@) can be assumed
bounded (and hence in ^2) as well; hence Z e ^(тАт-). We now can apply
Theorem 16 to obtain the result.
To remove the assumption that К < a < oo a.s., we need only apply
an argument like the one used at the end of the proofs of Theorems 7, 8
and 15. D
Theorem 16 and its corollary give us a way to approximate the solution of
a general stochastic differential equation with finite differences. Indeed, let X
be the solution of
Xt = Jt+ f F(X)s-dZ
J
f F(X)s
Jo
where Z is a semimartingale and F is process Lipschitz. For each random
partition an = {0 = Tff < T™ < ¦ ¦ • < T? }, we see that the random variables
Х(ап)т? verify the relations (writing a for an, X for X(an), Tk for Tg)
Xt0 = Jo,
ХТк+1 = ХТк + JTk+1 - JTk + F(X°+)Tk(ZTk+1 - ZTk).
Then the solution of the finite difference equation above converges to the
solution of (*), under the appropriate hypotheses.
As an example we give a means to approximate the stochastic exponential.

270 V Stochastic Differential Equations
Theorem 17. Let Z be a semimartingale and let X = ?{Z), the stochastic
exponential of Z. That is, X is the solution of
Xs-dZs.
Let an be a sequence of random partitions tending to the identity. Let
Then lim Xn — X in ucp.
n—>oo
Proof. Let Yn be the solution of
f
Jo
equation (*a) of Theorem 16. By the corollary of Theorem 16 we know that
Yn converges to X — ?{Z) in ucp. Thus it suffices to show Yn = Xn.
Let an = {0 = Ton < If < ¦ ¦ • < TfcnJ. On (Tf ,T?+1] we have
Ytn = Y?r
Inducting on i down to 0 we have
for T? < t < T?+v Since ZTi*+i - ZT? = 0 for all j > i when T? <t<T?+1,
we have that Yn = Xn, and the theorem is proved. D
5 Fisk-Stratonovich Integrals and Differential Equations
In this section we extend the notion of the Fisk-Stratonovich integral given
in Chap. II, Sect. 7, and we develop a theory of stochastic differential equa-
equations with Fisk-Stratonovich differentials. We begin with some results on the
quadratic variation of stochastic processes.
Definition. Let H, J be adapted, cadlag processes. The quadratic covari-
covariation process of H, J denoted [H, J] = ([H, J]t)t>o, if it exists, is defined
to be the adapted, cadlag process of finite variation on compacts, such that
for any sequence an of random partitions tending to the identity,

5 Fisk-Stratonovich Integrals and Differential Equations 271
lim Sa (tf,J) = lim tfoJo + Y(HT^ - HT")(JT"^ - JT")
n—*oo n—>oo ^—'
г
= [tf,J]
with convergence in ucp, where on is the sequence 0 = Tff < T™ < • • • < Tg .
A process H in ID is said to have finite quadratic variation if [H, H]t exists
and is finite a.s., each t > 0.
If H, J, and H+J in ID have finite quadratic variation, then the polarization
identity holds:
[tf, J] = ^([tf + J, tf + J] - [tf,tf] - [J, J]).
For X a semimartingale, in Chap. II we defined the quadratic variation of X
using the stochastic integral. However Theorem 22 of Chap. II shows every
semimartingale is of finite quadratic variation and that the two definitions are
consistent.
Notation. For H of finite quadratic variation we let [H, H]c denote the con-
continuous part of the (non-decreasing) paths of [H,H]. Thus,
[tf,tf]t = [tf,tf]tc+ ? A[H,H]S,
0<s<t
where Д[#, H}t = [Я, H]t - [H, H]t-, the jump at t.
The next definition extends the definition of the Fisk-Stratonovich integral
given in Chap. II, Sect. 7.
Definition. Let H ? D, X be a semimartingale, and assume [H, X] exists.
The Fisk-Stratonovich integral of H with respect to X, denoted /0 #s_ о
dXs, is defined to be
f tfs_ о dXs = f Hs-dXs + \ [Я, X\%.
Jo Jo z
To consider properly general Fisk-Stratonovich differential equations, we
need a generalization of Ito's formulas (Theorems 32 and 33 of Chap. II).
Since Ito's formula is proved in detail there, we only sketch the proof of this
generalization.
Theorem 18 (Generalized Ito's Formula). Let X = (X1,.. .,Xn) be an
n-tuple of semimartingales, and let f : WL+ x Q x W1 —> WL be such that
(i) there exists an adapted FV process A and a function g such that
t,u>,x) = / g(s,w,
Jo
f(t,u>,x) = / g(s,oj,x)dAs,
Jo
(s,oj) \—> g(s,cj,x) is an adapted, jointly measurable process for each x,
and Jo supxe^ \g(s,oj,x)\\dAs\ < oo a.s. for compact sets K.

272 V Stochastic Differential Equations
(ii) the function g of (i) is C2 in x uniformly in s on compacts. That is,
sup{\g(s,oj,y) -y2
<rt(W)||x-y||)||x-y||2
a.s., where rt : f!xl| —> R+ is an increasing function with\imuiort(u) =
0 a.s., provided к ranges through a compact set (rt depends on the compact
set chosen).
(in) the partial derivatives fXi, fxiXj, 1 < i, j < n all exist and are continu-
continuous, and moreover
fXt(t,w,x)= / gXi(s,w,x)dAs,
Jo
fXiXj(t,ui,x)= / gXiXj(s,u,x)dAs.
Jo
Then
/t n -t
g(s,ui,Xs)dAs + y2 .
t=i Jo
1 n rt
0<s<t
n
- g(s,u>,Xs)AAs - ^J/Xi(s-,w,Xs_)AXJ}.
Proof. We sketch the proof for n = 1. We have, letting 0 = to < t\ < ¦ ¦ ¦ <
tm — t be a partition of [0,t], and assuming temporarily \X\ < к for all s < t,
к a constant,
f(t,w,Xt)-f(p,w,X0)
m — \ m—1
fe=0 fe=0
m~1 rtk+i m-1
-1 ptk+1 m-1
= 52 9(u,uj,Xtk+1)dAu+J2f(tk,uj,Xtk+1)~f(tk,oj,Xtk) (*)
fe=0 Jtk k=0
Consider first the term Y^k=o /*к+1 Э^т^тXtk+1)dAu. The integrand is not
adapted, however one can interpret this integral as a path-by-path Stieltjes

к к
5 Fisk-Stratonovich Integrals and Differential Equations 273
integral since A is an FV process. Expanding the integrand for fixed (u,oj)
by the Mean Value Theorem yields
g(u,w,Xtk+1) = д(и,ш,Хи) + gx(u,u>,Xu)(Xtk+1 - Xu)
where Xu is in between Xu and Xtk+1. Therefore
g(u,oj,Xtk+1)dAu
/•tk+l ftk + 1
= Y1 I 9(u,oj,Xu)dAu + V / gx(u,u,Xu)(Xtk+1 - Xu)dAu,
к Jtk к Jtk
and since A is of finite variation and X is right continuous, the second sum
tends to zero as the mesh of the partitions tends to zero. Therefore letting тгп
denote a sequence of partitions of [0, t] with limn_,oo meshGrn) = 0,
JLim^ Y^ / 9(u,oj,Xtk+1)dAu= / g(u,u),Xu)dAu(uj) a.s.
Next consider the second term on the right side of (*), namely
m-l
V ^ xf-t. v \ ?/4. v \
/ ^ J{tk,uj,Ji.tk+x) — j[tk,uJ,Ji.tk).
fe=0
Here we proceed analogously to the proof of Theorem 32 of Chap. II. Given
e > 0, t > 0, let A(e, t) be a set of jumps of X that has a.s. a finite number
of times s, and let В = B(e, t) be such that ^g?B(AXsJ < e2, where A U В
exhaust the jumps of X on @,t]. Then
m-l
2_^/(?fc,w,^tk+i) - f(tk,w,xtk)
fe=0
) - f(tk,w,Xtk) + 2_;f(tk,w,Xtk+1) - f(tk,w,Xtk)
k,A k,B
where J2k,A denotes J2tke*n l{An(tk,tk+1]^}- Thus
- J[tk,w,Xtk) = у f(s-,u,Xs) - /(s-,w,As_).
k,A
By Taylor's formula, and letting AkX denote Xtk+1 — Xtk,

274 V Stochastic Differential Equations
, w,Xtk+1)~ f(tk, u>, Xtk) (**)
k,B
к,А
k,B
By Theorems 21 and 30 of Chap. II, the first sums on the right above con-
converge in ucp to /0 fx(s-,u>,Xs-)dXs and \ /0 fxx(s-,u>,Xs^)d[X,X]s, re-
respectively. The third sum converges a.s. to
seA
By condition (ii) on the function g, we have
limsup Yl R(tk,u,Xth,Xtk+1)<rt{w,e+)[X,X]t
n tke-wn,B
where
rt(u),e+) = limsup rt(u>,d).
Sle
Next we let e tend to 0; then rt(cj,e+)\X,X]t tends to 0 a.s., and finally
combining the two series indexed by A we see that
{f(s-,oj,Xs) - f(s-,oj,Xs_) - fx(s-,uj,Xs_)AXs
s?A(e,t)
tends to the series
0<s<t
-\fxx{s-,u,Xs_){AXsJ}
which is easily seen to be an absolutely convergent series (cf., the proof of The-
Theorem 32 of Chap. II). Incorporating - J2o<s<t \fxx{s-,u),Xs-){AXsJ into
\ /o fxx{s-,u,Xs-.)d{X,X]s yields the term \ f* fxx(s-,w,Xa-)d[X,X]ca.
We further note that f(s,u>,Xs) - f(s—,w,Xs) = g(s,u>,Xs)AAs, and
since J3o<s<t5(s'w'^s)Ayls is absolutely convergent, the theorem is proved.
The assumption that \XS\ < к for s < t is removed as it was in Theorem 32
of Chap. II. D

5 Fisk-Stratonovich Integrals and Differential Equations 275
Note that a consequence of Theorem 18 is that for / satisfying the hy-
hypotheses, we have f(t, -,Xt) is a semimartingale when X = (X1,... ,Xn) is
an n-tuple of semimartingales. Also, an important special case is when the
process As = s; in this case the hypotheses partly reduce to assuming / is
absolutely continuous in t.
The next theorem allows an improvement of the Fisk-Stratonovich change
of variables formula given in Chap. II (Theorem 34).
Theorem 19. Let X = (X1,... ,Xd) be a vector of semimartingales and let
f : R+ x п x Rd -> R be such that
(i) there exists an adapted FV process A and a function g such that
f(t,u,x) = / g(s,u>,x)dAs
Jo
where (s,oj) >—> <?(s,w,x) is an adapted and jointly measurable process for
each x.
(ii) for each compact set K, JQ supxeK |#(s,w,x)||cL4s| < oo a.s.
(Hi) fxi exists and is continuous in x and fXi(t,u,x) = JQ gXi(s,co,x)dAs.
Let
Yt = f(t,oj,X\...,Xd).
Then Y is an adapted process of finite quadratic variation, and moreover
г j 0<s<t
Proof. First assume that for fixed (t, w) the function / and all its first partials
are bounded functions of x. Then by optional stopping at times T— we can
assume without loss of generality that / and all its first partials (in x) are
bounded functions. Let fu be a sequence of functions on R+ x п х Rd which are
C2 on M.d such that fk and its first partials converge uniformly on M.d respec-
respectively to / and its corresponding first partials. Moreover we can take all the
fk bounded and Lipschitz continuous with Lipschitz constant c, independent
of к. For simplicity take d = 1. Let an be a sequence of random partitions
tending to the identity. We write, for a process Z,
where
an = {0 = Ton < T\n < • • • < 2?}.
Since fk and their first partials converge to / and its first partials uni-
uniformly, we have fk - fe is Lipschitz with constant eke, which tends to 0 as к, ?
tend to oo. Therefore, letting r denote an for a given n and writing fk(X) for
fk{t,u,Xt), we have

276 V Stochastic Differential Equations
\ST(fk(X)) - ST(fe(X))\
< sT((fk - h){X)) + 2{sT(fk(X))sT((fk - h){X))Y'2
2ceM)ST{X)
by the Lipschitz properties. Since fk{X) and fe(X) are semimartingales, we
know (by Theorem 22 of Chap. II) that San(fk(X)) and San(fi(X)) converge
in ucp respectively to [fk(X),fk(X)} and [fe(X),fe(X)]. By restricting our
attention to an interval (s,t), we then have from (*) that
\{[fk(X)Jk(X)]t - lfk(X),fk(X)]s} - {\fe(X),fe(X)]t - [ft{X)Jt(X)]a}\
2ceke){[X,X}t-[X,X}s}.
Since this is true for all 0 < s < t < oo, we deduce that
f \d[fk(X)Jk(X)}s - d[fe(X),fe(X)]a\ < {e2u + 2ceu)[X,X\t.
Jo
Therefore d[fk(X),fk(X)] is a Cauchy sequence of random measures, con-
converging in total variation norm on @,oo), a.s. In addition, by Ito's formula
(Theorem 18) and Theorem 29 of Chap. II we have that
from Theorem 23 of Chap. II we also know that
Therefore
[fk(X),fk(X)}t= [^±(з-,о;,
°X 0<s<t
and since we have a.s. convergence in total variation norm, we can pass to the
limit to obtain
Vt= [()[];
^ o<s<t
Our identification of the limit removes the dependence on the representation
of Y by /, because
t - Vt\ < \SaM(X))t - SaMk(X))t\ + \Vt - [fk(X),fk(X)]
+ \saMk(X))t-[fk(X),fk(X)]t\,
and by taking к large enough the first two terms on the right side can be taken
small in probability independently of n; one then takes n large enough in the

5 Fisk-Stratonovich Integrals and Differential Equations 277
third term to make it small in probability, and we have limn_,oo Scrn (f(X))t =
Vt, in probability. Therefore Vt = [У,^]< and this completes the proof for /
and its first partials bounded functions of x, and d = 1. The proof for general
d is exactly analogous. For general / satisfying the hypotheses, let gm : Md —>
Md be C°° such that gm(x) — x if |x| < m, and gm has compact support.
Defining Ytm = f(t,uj,gm(Xt)), let T"» = inf{t > 0 : Yt{tj) ф У/7»)-
Then Tm increase to oo a.s. Also, since the quadratic variation is a path
property, [Ym,Ym]t(uj) = [Y,Y]t(w), for all t < Tm(w). Therefore Y is of
finite quadratic variation, and since by the preceding (for d = 1)
= f* ^-(s-,LJ,gm(Xs-))g>m(Xs-)d[X,X}cs+ ? Af(s,w,gm(Xs)J
J° 0<s<t
which, on [0,Tm), is equal to
= f^{s-,u>,Xs-)d[X,X]cs+ ? Af(s,XsJ;
J° 0<s<t
whence the result. D
Note that the subspace of processes Y that have a representation Y =
f(Xl,... ,Xd), where X = (X1,... ,Xd) is a finite-dimensional semimartin-
gale (d is not fixed) and / € C1, is a vector subspace of the space of processes
of finite quadratic variation. As an immediate consequence of Theorem 19 we
have the following extension of Theorem 34 of Chap. II.
Theorem 20. Let X be a semimartingale and let f be C2. Then
f(Xt)-f(X0)= f f'(Xt.)odXa+ J2 {f(Xs)-f(Xs-)-f'(Xs_)AXs}.
J0+ n^.^^
0<s<t
Proof. Note that /' is C1, so that f'(X) is in the domain of definition of the
F-S integral by Theorem 19. Also by definition we have
f f(
/
By Theorem 19 and polarization we know that
[f{X),Xft= f f\
Jo
and thus the result follows by Ito's formula (Theorem 32, Chap. II). D
Theorem 20 has an obvious generalization to the multi-dimensional case.
We omit the proof which is an analogous consequence of Theorem 33 of
Chap. II.

278 V Stochastic Differential Equations
Theorem 21. Let X = (X1,... ,Xn) be an n-tuple of semimartingales, and
let f : Mn —> R have second order continuous partial derivatives. Then /(X)
is a semimartingale and the following formula holds:
/(Xt)
As a corollary of Theorem 21, we have the Stratonovich integration by
parts formula.
Corollary 1 (Stratonovich Integration by Parts Theorem). Let X and
Y be semimartingales. Then
XtYt-X0Y0= f Xs^odYs+ f Ys_odXs+ V AXSAYS.
Jo+ Jo+ o<s<t
Proof. Let f(x, y) = xy and apply Theorem 21. D
Corollary 2. Let X and Y be semimartingales, with at least one of X or Y
continuous. Then
XtYt - X0Y0 = f *5_ odYs + f Ys_odXs.
Jo+ Jo+
Recall that since Xo- = 0 by convention for a cadlag process X, we did
not really need to write Jo, ; the formula also holds for Jo.
For stochastic differential equations with Fisk-Stratonovich differentials we
are limited as to the coefficients we can consider, because the integrands must
be of finite quadratic variation. Nevertheless we can still obtain reasonably
general results. We first describe the coefficients.
Definition. A function / : M+ x Cl x Rd —» M is said to be Fisk-Stratonovich
acceptable if
(i) there exists an adapted FV process A and a function g such that
f(t,w,x) = / g(s,uj,x)dAs
Jo
where (s,w) i—» g(s, lj,x) is an adapted, jointly measurable process for
each x;
(ii) for each compact set K, Jo supxeA: |<?(s,w,x)||cL4.s| < oo a.s.;
(iii) fXi is C1 and
fXl(t,w,x)= / gXi(s,w,x)dAs;
Jo

5 Fisk-Stratonovich Integrals and Differential Equations 279
(iv) for each fixed (t, ш), the functions x i-> f(t, ш, х) and хи (J? •/)(*, u>, x)
are all Lipschitz continuous with Lipschitz constant K(w), К < oo a.s.
A < i < d).A
We will often write "F-S acceptable" in place of "Fisk-Stratonovich ac-
acceptable."
Theorem 22. Given a vector of semimartingales Z = (Z1,... ,Zk), semi-
martingales Jl A < г < d), and F-S acceptable functions /j A < i < d, 1 <
j < k), then the system of equations
x; = Jt + Y] f) (s-, w, xs_) о ^ (*)
has a unique semimartingale solution. Moreover the solution X of (*) is also
the (unique) solution of
T~> Jo
s
E [\-l§;-f™)(s-,u>,Ks-)dlZ\Ziys (**)
j,n=l ro=l ^°
Proof. We note that equation (**) has a unique solution as a trivial conse-
consequence of Theorem 7. Since X is a d-dimensional semimartingale, we know
that fj(s—, ¦, Xs_) is in the domain of definition of the F-S integral by The-
Theorem 19. Further, as a consequence of Theorem 19, we have that5
d /-t
/j(.,u,,X.),Z']tc= E f ^i{s^,
and the equivalence of (*) and (**) follows. Therefore the existence of a unique
semimartingale solution of (**) is equivalent to the existence of a unique
semimartingale solution of (*), and we are done. Note that if Jm is of finite
variation, the terms involving [Jm, Zi]c disappear. ?
4 By (-gjj -f)(t,w,x.) we mean the usual product of functions J^-(t,o;,x) -f(t,w,x.).
5 Since we are taking the continuous parts of the quadratic variations, we need not
write f(s— ,ш,Ха-), etc.

280 V Stochastic Differential Equations
In Chap. II we studied the stochastic exponential of a semimartingale. The
F-S integral allows us to give a version that has a more natural appearance.
Theorem 23. Let Z be a semimartingale, Zo = 0. The unique solution of
the equation
Xt = Xo + / Xs- о dZs
Jo
is given by
Xt = Xoexp{Zt} Ц A + AZs)e~AZ>,
0<s<t
and it is called the Fisk-Stratonovich exponential, Xq?fs{Z).
Proof. By Theorem 22 the equation above is equivalent to
Xt=X0+ f Xs^dZs + \ f Xs_d{Z, Z\%
Jo z Jo
xs_d(zs + hz,z\cs).
Jo z
Therefore
[Z,Zf)
= ?{Z)?{\[Z,Z\C)
where the second equality is by Theorem 38 of Chap. II. Using the formula for
the stochastic exponential established in Theorem 37 of Chap. II, the result
follows. ?
The most interesting case is when the semimartingale is continuous.
Corollary 1. Let Z be a continuous semimartingale, Z$ = 0. Then the unique
solution, ?-p.$(Z), of the exponential equation
Xt = 1 + / X, о dZ,
/
Jo
is given by ?F.s(Z)t = exp{Zt}.
Corollary 2. Let В be a Brownian motion with Bq = 0. Then the unique
solution of
rt
Xs о dД,
/o
is given by Xt = Xo exp{i?t}.
f
Jo

5 Fisk-Stratonovich Integrals and Differential Equations 281
The simplicity gained by using the F-S integral can be surprisingly helpful.
As an example let Z = (Z1,..., Zn) be an n-dimensional continuous semi-
martingale and let x = (x1,..., xn)' be a column vector in Rn. (Here' denotes
transpose.) Let / be the identity n x n matrix and define
(The matrix a(x) represents projection onto the hyperplane normal to x.)
Note that x'a(x) = 0. We want to study the system of F-S differential equa-
equations
Xt = x0 + / a(Xs)odZs, \<i<n. (* * *)
Jo
where X.= (X1,...,Xn)' and Z = (Z\...,Zn)'.
Theorem 24. The solution X of equation (* * *) above exists, is unique, and
it always stays on the sphere of center 0 and radius ||xo||.
Proof. Since a is singular at the origin, we need to modify it slightly. Let g(x)
be a C°° function equal to a(x) outside of a ball centered at the origin, No,
such that x0 ф No- Let Y be the solution of
Yt=x0+ / g(Ys)odZs.
Then Y exists and is unique by Theorem 22. Let T = inf{? > 0 : Yt G No}-
Since Yq = Xo and Y is continuous, P(T > 0) = 1. However for s < T,
g(Ys) = a(Ys). Therefore it suffices to show that the function Ы ||Yt|| is
constant, since ||Yo|| = ||xo||. This would imply that Y always stays on the
sphere of center 0 and radius ||xo||, and hence P(T — oo) = 1 and g(Y) =
a(Y) always. To this end, let /(x) = Х)"=1(ж4J, where x = (ж1,... ,xn)'.
Then it suffices to show that df(Yt) =0,(t> 0). Note that / is C2. We have
that for t < T,
df(Yt) = 2Y'todY = 2Y; • a(Yt) о dZt = 0.
Therefore ||У^|| = ||жо|| for t < T and thus by continuity T = oo a.s. ?
Corollary. Let В = (В1,... ,Bn)' be n-dimensional Brownian motion, let
a(x) — I — f^p, and let X be the solution of
Xt =xo+ / a(Xs)odB3.
Jo
Then X is a Brownian motion on the sphere of radius ||xo||.

282 V Stochastic Differential Equations
Proof. In Theorem 36 of Sect. 6 we show that the solution X is a diffusion. By
Theorem 24 we know that X always stays on the sphere of center 0 and radius
||xo||. It thus remains to show only that X is a rotation invariant diffusion,
since this characterizes Brownian motion on a sphere. Let U be an orthogonal
matrix. Then UB is again a Brownian motion, and thus it suffices to show
d(UX) = a(UX)od(UB).
The above equation shows that UX is statistically the same diffusion as is X,
and hence X is rotation invariant. The coefficient a satisfies
a{Ux) = Ua(x)U',
and therefore
d(UX) = UodX = Ua(X) odB = Ua(X)U'Uо<ffi
= a{UX)od{UB),
and we are done. ?
The F-S integral can also be used to derive explicit formulas for solutions
of stochastic differential equations in terms of solutions of (non-random) or-
ordinary differential equations. As an example, consider the equation
Xt=x0+ [ f(Xs) о dZs + [ g(Xs)ds, (*4)
Jo Jo
where Z is a continuous semimartingale, Zo = 0. (Note that Jo g(Xs) о ds =
Jo g(Xs)ds, so we have not included Ito's circle in this term.) Assume that /
is C2 and that /, g, and //' are all Lipschitz continuous. Let и — u(x, z) be
the unique solution of
du
-K-(x,z) = f(u(x,z))
u(x,0) — x.
Then J^ = f'(u(x,z))%±, and §?(ж,0) = 1, from which we conclude that
д-(ж, z) = exp{ / f'(u(x, v))dv}.
Ox Jo
Let Y = {Yt)t>o be the solution of
ft fZ.
Yt = x0+ exp{- / f(u(Ys, v))dv}g(u(Ys, Zs))ds
Jo Jo
g(u(x Zs))
which we assume exists. For example if ^ц ' — is Lipschitz, this would
suffice.

5 Fisk-Stratonovich Integrals and Differential Equations 283
Theorem 25. With the notation and hypotheses given above, the solution X
of (*4) is given by
Xt=u(Yt,Zt).
Proof. Using the F-S calculus we have
u(Yt,Zt) = u(xo,O) + f ^{Ys,Zs)odYs + f ^-{Ys,Zs)odZs
ft fZ, rt
= xo+ exp{/ f'(u(Ys,v))dv}odYs+ f{u{Ys,Zs))odZs.
Jo Jo Jo
Since
we deduce
dY» rZ
= exp{-/ f'(u(Ys,v))dv}g(u(Ys,Zs)),
as Jo
u(YuZt) = x0+ [ g(u(Ys,Zs))ds+ [ f(u(Ys,Zs)) odZs.
Jo Jo
By the uniqueness of the solution, we conclude that Xt = u(Yt, Zt). D
We consider the special case of a simple Stratonovich equation driven by
a (one dimensional) Brownian motion B. As a corollary of Theorem 25 we
obtain that the simple Stratonovich equation
Xt=x0+ [ f(Xs)odBs
Jo
has a solution Xt — h~l{Bt + /i(Xo)), where
-^r-rds + С
The corresponding Ito stochastic differential equation is
dXt = l-f{Xt)f{Xt)dt + f(Xt)dBt.
By explicitly solving the analogous ordinary differential equation (without
using any probability theory) and composing it, we can obtain examples of
stochastic differential equations with explicit solutions. This can be useful
when testing simulations and numerical solution procedures. We give a few
examples.
Example. The equation
dXt = -\a2Xtdt + aJl -
has solution Xt = sin(aBt + arcsin(X0)).

284 V Stochastic Differential Equations
Example. The following equation has only locally Lipschitz coefficients and
thus can have explosions: for m ф 1,
\
dXt = -a2mX;m~xdt + aXmdBt
has solution Xt = (X^~m - a(m -
Example. Our last example can also have explosions:
dXt = Xt(l + X*)dt + A + X*)dBt
has the solution Xt — tan(? + Bt + arctan(Xo)).
The Fisk-Stratonovich integrals also have an interpretation as limits of
sums, as Theorems 26 through 29 illustrate. These theorems are then useful
in turn for approximating solutions of stochastic differential equations.
Theorem 26. Let H be cadlag, adapted, and let X be a semimartingale.
Assume [H,X] exists. Let an = {0 = To™ < Tf < • ¦ • < T?n} be a sequence
of random partitions tending to the identity. If H and X have no jumps in
common (i.e., Ylo<s<t AHSAXS = 0, all t > 0), then
equals the F-S integral /0 i?5_ о dXs in ucp.
Proof. It follows easily from the definition of [H, X] at the beginning of this
section that A[H,X]t = AHtAXt and lim^oo Е(ят^+1 - HTn){XT?+* -
XT?) = [H,X] — HqXo- Thus if H and X have no jumps in common we
conclude [H,X] = [H,X]C +H0X0. Observing that
the result follows from Theorems 21 and 22 of Chap. II. ?
Corollary 1. If either H or X in Theorem 21 is continuous, then
lim Y\{HTn+HTn ){XT^ -XT?) = [ #5_ odXs,
where Ho- = 0, in ucp.

5 Fisk-Stratonovich Integrals and Differential Equations 285
Corollary 2. Let X and Y be continuous semimartingales, and let an =
{0 = Ttf < T™ < • ¦ • < Tg } be a sequence of random partitions tending to
the identity. Then
,- odX
s,
with convergence in ucp.
Proof. By Theorem 22 of Chap. II, any semimartingale Y has finite quadratic
variation. Thus Corollary 2 is a special case of Theorem 26 (and of Corol-
Corollary 1). ?
Theorem 27. Let H be cadlag, adapted, of finite quadratic variation, and
suppose J2o<s<t |Ajffs| < oo a.s., each t > 0. Let X be a semimartingale,
and let an = ¦pTlo<i<fcn be a sequence of random partitions tending to the
identity. Assume [H, X] exists. Then
f
In
и
lim {T \(HT~ +#t- - T AHS)(XT^ -XT")} = f tfs_ odXs,
г р+1
with convergence in ucp.
Proof. First observe that Ht = Ht — Ylo<s<t AHS defines a continuous process
of finite quadratic variation and that [H, X] = [H, X]c + H0X0. Next we note
that
\{HTr + HTr+i - Y ^ 52
Therefore by Theorems 21 and 22 of Chap. II, or directly from the definition
as in the proof of Theorem 26, we have
= lim > Hxn(X г+1 — X г )+ lim -
n—>oo -^—J г n—>oo Z
i
o+ z
= / Я5_ оdX5 + I[H, X}c = / Я5_ о dXs, where Яо_ = Яо,
Jo * Jo
and the result follows. D

286 V Stochastic Differential Equations
For the general case we have a more complicated result. Let H be a cadlag,
adapted process of finite quadratic variation. For each e > 0 we define
0<s<t
Then JE is also a cadlag, adapted process, and it has paths of finite variation
on compacts.
Theorem 28. Let H be cadlag, adapted, of finite quadratic variation, and let
X be a semimartingale. Assume [H, X] exists. Let an = {TJ1} be a sequence
of random partitions tending to the identity. Let JE be as defined above. Then
Jo
with convergence in ucp.
Proof. Let HE = H — JE. Then the jumps of HE are bounded by s. Moreover,
Therefore the first limit at t > 0 is
f
Hs-dXs+\[H,X]ct+l- f
2 2t? Jo 2o<s<t
and letting e tend to 0 gives the result. That is, since
\AHESAXS\ < [Н*,НЕI/2[Х,Х}\/2 < со,
and since [HE,H% = Eo<s<t(AHsJ, we have lim^o Eo<5<t |АЯ|ДХв| -
0. П
The next result is useful, for example, for approximating integrals of the
form fQ f(Bs) °dBs, where В is a Brownian motion. It can be generalized to
Y a continuous, adapted process of finite quadratic variation, but we do not
do so here.
Theorem 29. Let X be a semimartingale and Y a continuous semimartin-
semimartingale, and let f beC1. Let ц be a probability measure on [0,1] let a = J Xfi(dX),
and let an = {T™}i>o be a sequence of random partitions tending to the iden-
identity. Then

5 Fisk-Stratonovich Integrals and Differential Equations 287
lim V / f(YTr.
« f f'(Y3)d[Y,X}3,
Jo+
0+ J0+
with convergence in ucp. In particular if a = 1/2 then the limit is the F-S
integral /0+ f(Y3) о dX3.
Proof. We begin by observing that
~\,-XT")
Г
jo
V
f
The first sum on the right side of the above equation tends to /0+ f(Ys-)dX3
in ucp. Using the Fundamental Theorem of Calculus, the second sum on the
right above equals
J2 f MdA) / ds\f'{YTr + Xs(YTr+1 -Y^))(YTn -Г
i Jo Jo
which in turn equals
\(YTr+1 - YTn)(XT^ - XT")|,
i
where
Fan(aj) < 8ир{|/'(Утг + Xs(YTr+i - уг„)) - f(YTr)\}.
Since /' and Y are continuous, FOn (w) tends to 0 on compact time intervals.
Also, on [0,t],
-XT")\ < [F,F]t1/2[X,X]t1/2,
and the result follows by Theorem 30 of Chap. II. G
Corollary. Let X be a semimartingale, К be a continuous semimartingale,
and /gC1. Let an = {1Т}о<1<кп be a sequence of random partitions tending
to the identity. Then

288 V Stochastic Differential Equations
with convergence in ucp.
Proof. Let n(d\) = ?{1/2} (dX), point mass at 1/2. Then a = /0 Xfi(dX) = 1/2,
and we need only apply Theorem 29. D
For Brownian motion, the Fisk-Stratonovich integral is sometimes defined
as a limit of the form
l\mTf(BiJ±h±1)(Bti+1 - Bti) = j f(Bs)odBs;
2 Jo
that is, the sampling times are averaged. Such an approximation does not hold
in general even for continuous semimartingales (see Yor [238, page 524]), but
it does hold with a supplementary hypothesis on the quadratic covariation, as
Theorem 30 reveals.
Theorem 30. Let X be a semimartingale and Y be a continuous semimartin-
gale. Let ц be a probability measure on [0,1] and let a = J X/j,(dX). Further
suppose that \X, Y]t = Jo Jsds; that is, the paths of [X, Y] are absolutely con-
continuous. Let an = {i™} be a sequence of non-random partitions tending to the
identity. Let f be C1. Then
/
f(Ys)dXs+a f f'(Ys)d[Y,X}s,
o+ Jo+
with convergence in ucp. In particular if a = 1/2 then the limit is the F-S
integral /0+ f(Ys) о dXs.
Proof. We begin with a real analysis result. We let t± denote i, + A(*i+i - ti),
where the ti are understood to be in an. Suppose a is continuous on [0, t].
Then
> / a(s)ds — > a(ti)X(ti+i — ti) < > /
iJu V i Ju
which tends to 0. Therefore
lim У^ / a(s)ds = X a(s)ds. (*)
»-»Y^ Jo
Moreover since continuous functions are dense in L1([0,t},ds), the limiting
result (*) holds for all a in L1([0,t},ds).

5 Fisk-Stratonovich Integrals and Differential Equations 289
Next suppose Я is a continuous, adapted process, and set
H 'n =
Then taking limits in ucp we have
limУ^Ни{[Х,У]г" - [X,Y}U} = lim
71 n
o
and using integration by parts, this equals
lim{#A'n ¦ (XY) - (HX'nX-) ¦ Y - (Hx'nY) ¦ X}.
71
By Theorem 21 of Chap. II, this equals
However since [X, Y]t = /0 Jsds, by the result (*) we conclude
u{[X,Y]ti - [X,Y]*<} (**)
= A / Hsd[X,Y]s.
Jo
We now turn our attention to the statement of the theorem. Using the
Mean Value Theorem we have
» - X*') (* * *)
U)+Yl f\(d\)f'(Yti)(Yt} -Yti
i Jo
The first sum tends in ucp to / f(Ys)dXs by Theorem 21 of Chap. II. The
second sum on the right side of (* * *) can be written as

290 V Stochastic Differential Equations
The first sum above converges to
/ /x(dA)A / f'(Y3)d[Y,X]s
Jo Jo
by (**)) and the second sum can be written as
ту\.п v"
Л • Л,
where
Then limn KX'n ¦ X converges to 0 locally in ucp by the Dominated Conver-
Convergence Theorem (Theorem 32 of Chap. IV).
Finally consider the third sum on the right side of (***). Let
Fn(oj) = sup sup \f(Yti + a(Ytx - Yu)) - f(Yu)\.
ti€crn s€[0,l]
Then
r-l
-X
= 0
since lim Fn = 0 a.s. and the summations stay bounded in probability. This
completes the proof. Since Y is continuous, [У, X] = [Y, X]c, whence if а = 1/2
we obtain the Fisk-Stratonovich integral. D
Corollary. Let X be a semimartingale, Y a continuous semimartingale, and
/ be C1. Let [X, Y] be absolutely continuous and let on = {?"} be a sequence
of non-random partitions tending to the identity. Then
lim "?f(yh±h±i)(XU+1-XU)= I f(Ys)°dXs
with convergence in ucp.

6 The Markov Nature of Solutions 291
Proof. Let /j.(dX) = ?{1/2}(dX), point mass at 1/2. Then / A/x(rfA) = 1/2, and
apply Theorem 30. ?
Note that if Y is a continuous semimartingale and В is standard Brownian
motion, then [У, В] is absolutely continuous as a consequence of the Kunita-
Watanabe inequality. Therefore, if / is C1 and un are partitions of [0, t], then
lim Y.f(Yu+u+1)(Bti+, -Bti)= f f(Ys)odBs,
with convergence in probability.
6 The Markov Nature of Solutions
One of the original motivations for the development of the stochastic integral
was to study continuous strong Markov processes (that is, diffusions), as
solutions of stochastic differential equations. Let В = {Bt)t>o be a standard
Brownian motion in Mn. K. Ito studied systems of differential equations of the
form
Xt =X0+ / f{s,Xs)dBs+ [ g{s,Xs)ds,
Jo Jo
and under appropriate hypotheses on the coefficients /, g he showed that a
unique continuous solution exists and that it is strong Markov.
Today we have semimartingale differentials, and it is therefore natural to
replace dB and ds with general semimartingales and to study any resulting
Markovian nature of the solution. If we insist that the solution itself be Markov
then the semimartingale differentials should have independent increments (see
Theorem 32); but if we need only to relate the solution to a Markov process,
then more general results are available.
For convenience we recall here the Markov property of a stochastic pro-
process which we have already treated in Chap. I. Assume as given a filtered
probability space (fi,.F, {J~t)t>o,P) satisfying the usual hypotheses.6
Definition. A process Z with values in Md and adapted to F = (.Ft)t>o is a
simple Markov process with respect to F if for each t > 0 the cr-fields J~t
and a{Zu;u > t} are conditionally independent given Zt.
Thus one can think of the Markov property as a weakening of the property
of independent increments. It is easy to see that the simple Markov property
is equivalent to the following. For и > t and for every / bounded, Borel
measurable,
E{f(Zu)\Ft}=E{f(Zu)\a{Zt}}. (*)
One thinks of this as "the best prediction of the future given the past and the
present is the same as the best prediction of the future given the present."
See Chap. I, Sect. 1 for a definition of the "usual hypotheses" (page 3).

292 V Stochastic Differential Equations
Using the equivalent relation (*), one can define a transition function
for a Markov process as follows, for s < t and / bounded, Borel measurable,
let
Note that if/(x) = 1a(x), the indicator function of a set A, then the preceding
equality reduces to
Identifying 1^ with A, we often write Ps<t(Zs,A) on the right side above.
When we speak of a Markov process without specifying the filtration of o-
algebras (^rt)t>o, we mean implicitly that T^ = o{Zs;s < t}, the natural
filtration generated by the process.
It often happens that the transition function satisfies the relationship
Ps,t = Pt-s
for t > s. In this case we say the Markov process is time homogeneous, and
the transition functions are a semigroup of operators, known as the transition
semigroup (Pt)t>o- In the time homogeneous case, the Markov property
becomes
A stronger requirement that is often satisfied is that the Markov property
hold for stopping times.
Definition. A time homogeneous simple Markov process is strong Markov
if for any stopping time T with P(T < oo) = 1, s > 0,
P(ZT+aeA\FT) = Pa(ZT,A)
or equivalently
E{f(ZT+s)\FT} = Ps(ZT,f),
for any bounded, Borel measurable function /.
The fact that we defined the strong Markov property only for time ho-
homogeneous processes is not much of a restriction, since if X is an M.d-valued
simple Markov process, then it is easy to see that the process Zt = (Xt,t) is
an Md+1-valued time homogeneous simple Markov process.
Examples of strong Markov processes (with respect to their natural fil-
trations of cr-algebras) are Brownian motion, the Poisson process, and indeed
any Levy process by Theorem 32 of Chap. I. The results of this section will
give many more examples as the solutions of stochastic differential equations.
Since we have defined strong Markov processes for time homogeneous pro-
processes only, it is convenient to take the coefficients of our equations to be
autonomous. We could let them be non-autonomous, however, and then with
an extra argument we can conclude that if X is the solution then the process
Yt = (Xt, t) is strong Markov.
We recall a definition from Sect. 3 of this chapter.

6 The Markov Nature of Solutions 293
Definition. A function / : M+ x Mn —> M is said to be Lipschitz if there
exists a finite constant к such that
(i) \f(t,x)-f(t,y)\ <k\x-y\,eachteR+, and
(ii) t •—> f(t, x) is right continuous with left limits, each x G M".
/ is said to be autonomous if f(t,x) = f(x).
In order to allow arbitrary initial conditions, we need (in general) a larger
probability space than the one on which Z is defined. We therefore define
ft = Mn x U
V = ey x P
where В denotes the Borel sets of Mn, and ey denotes the Dirac point mass
measure at y. For a point w = (y, w) ? Q, we further define
Xq(u) = y, when w = (y,w). (*)
Finally let J-t = (")«>* 3~и- A random variable Z defined on $7 is considered to
be extended automatically to $7 by the rule Z(w) = Z{uj), when To = {y,w).
We begin with a measurability result which is an easy consequence of
Sect. 3 of Chap. IV.
Theorem 31. Let Z^ be semimartingales A < j> < d), Hx a vector of adapted
processes in D for each x € Шп, and suppose (x,t,u>) н-» Щ{и>) is В® B+ ® J-
measurable.7 Let Fj be functional Lipschitz and for each x G M", Xх is the
unique solution of
\At ) = \Щ ) +
3 = 1
There exists a version of Xх such that (x,t,oj) ь-> Xх(u>) is В ® B+ ® J7
measurable, and for each x, Xх is a cadlag solution of the equation.
Proof. Let X°(x, t,ui) = Hx{u>) and define inductively
Xn+I(x,t,u>y = Hx + J2 I Fi(Xn(x,-,.))s-dZi.
i=i^
The integrands above are in L, hence by Theorem 63 in Chap. IV there exists
measurable, cadlag versions of the stochastic integrals. By Theorem 8 the
processes X" converge ucp to the solution X for each x. Then an application
of Theorem 62 of Chap. IV yields the result. ?
7 В denotes the Borel sets on Rn; B+ the Borel sets on

294 V Stochastic Differential Equations
We state and prove the next theorem for one equation. An analogous result
(with a perfectly analogous proof) holds for finite systems of equations.
Theorem 32. Let Z = (Z1,..., Zd) be a vector of independent Levy pro-
processes, Zo = 0, and let (/j) 1 < j < d, 1 < i < n, be Lipschitz functions. Let
Xq be as in (*) and let X be the solution of
J2 fj(s~,Xs_)dZi
(**)
P
Then X is a Markov process, under each P and X is strong Markov if the
f!j are autonomous.
Proof We treat only the case n = 1. Let T be an F stopping time, T < oo a.s.
Define QT = a{Z3T+u - Z3T; и > 0,1 < j < d}. Then QT is independent of TT
under P , since the ZJ are Levy processes, as a consequence of Theorem 32
of Chap. I. Choose a stopping time T < oo a.s. and let it be fixed. For и > 0
define inductively
Y°(x,T,u)=x,
d j-T+u
x + Y, /
Also, let X(x,T,u) denote the unique solution of
d /-Т+И
^2 /
d /-Т+И
^2 / fj(v~,X(x,T,v-))dZi,
i=i Jt
taking the jointly measurable version (cf., Theorem 31). By Theorem 8 we
know that X(x, T, u) is QT measurable. By approximating the stochastic
integral as a limit of sums, we see by induction that Yn(x,T, u) is QT
measurable as well. Under P we have X(Xo,T,u) = X(x,T,u) a.s., and
Yn(Xo,T,u) = Yn(x,T,u) Px-a.s., also. By uniqueness of solutions and us-
using Theorem 31, for all и > 0 a.s.
X(X0,0,T + u) = X(X(X0,0,T),T,u).
There is no problem with sets of probability zero, due to (for example) the
continuity of the flows. (See Theorem 37.) Writing Ex to denote expectation
on ft with respect to P , and using the independence of Тт and QT (as well
as of J~t and GT), we have for any bounded, Borel function h
Ex{h(X(X0,0, T + u))\TT) = E{h(X(x, 0, T +
= E{h(X(X(x,0,T),T,u))}l&

6 The Markov Nature of Solutions 295
where j(y) = E{h(X(y,T, «)}. The last equality follows from the elementary
fact that E{F(H, -)|W} = /(#), where f(h) = E{F(h, •)}, if-F is independent
of H and H is H measurable. This completes the proof, since the fact that
Ex{h(X(X0,0, T + u))\TT} is a function only of X(x,0, T) implies that
Ex{h(X(X0,0, T + v))\TT} = Ex{h(X(X0,0, T + и))\Х(Х0,0, Т)}. П
It is interesting to note that Theorem 32 remains true with Fisk-Straton-
ovich differentials. To see this we need a preliminary result.
Theorem 33. Let Z = (Zl,...,Zd) be a vector of independent Levy pro-
processes, Zo = 0. Then [Z\Zj]c =Oifi^j, and \Z\Z% = at, where
Proof First assume that the jumps of each Zl are bounded. Then the mo-
moments of Z% of all orders exist (Theorem 34 of Chap. I), and in particular
M\ = Z\ - E{Zi) is an L2 martingale for each i, with E{Z\~) = tE{Z\]. By
independence M%M.i is also a martingale and hence [Мг, ЛР] = 0 by Corol-
Corollary 2 of Theorem 27 of Chap. II. Therefore [Z\Zj]ct = [M\Mj)ct = 0 as
well.
Next consider A\ = [Zl, Zl]t = [M\M%]t. It is an immediate consequence
of approximation by sums (Theorem 22 of Chap. II) that A1 also has inde-
independent increments. Since
and the process J\ = J2o<s<t(^^lJ a^so dearly has independent incre-
increments, we deduce that [Мг,Мг]$ has independent increments as well. There-
Therefore [Мг,Мг]^ — ?{[Ml,M']4c} is a finite variation continuous martingale,
hence constant by Theorem 27 of Chap. II. Since [Ml, Мг]$ = 0, we deduce
that \1\г% = \М\М% = Е{[М\М1)$}. Since \M\M% is also a Levy
process we have E{[M%,Ml]t} = ?.Е{[М*,М*]],} (by the stationarity of the
increments), and also
E{ ? (AMiJ} = E{( ? AMtJ} = tE{ ? (AMIJ}
0<s<t 0<s<t 0<s<1
by the same reasoning. Therefore \Z\Z% = ЬЕ{[г*,г*]$} by subtraction.
If the jumps of Z are not bounded, let J\ = ^20<s<t AZll{\&.zi\>i}- Then
Zl = Zl — J% and Jl are independent Levy processes, and Jl is a quadratic
pure jump semimartingale. It therefore follows that [Z*,ZJ] = [Zl,Z^\ = 0
and that [Z\ Zl]c = [Z\ Z1}0, and the theorem is proved. ?
Theorem 34. Let Z = (Zl,...,Zci) be a vector of independent Levy pro-
processes, Zo = 0, and let (/]) 1 < j < d, 1 < i < n, be non-random F-S accept-
acceptable functions.8 Let X$ be as in Theorem 32 and X be the solution of
8 F-S acceptable functions are defined on page 278.

296 V Stochastic Differential Equations
d
3=1 Jo
Then X is a Markov process and X is strong Markov if the coefficients /j are
autonomous.
Proof. We treat only the case n = 1. By Theorem 33 [Z\ Zj\$ = 0 if i ф j and
\Z%,Z%\'i = a^, ai > 0. Therefore, by Theorem 22 the equation is equivalent
to
J2 f Ы*-,Х-№3. + \ Z> / ф ¦ fj)(s-,
j=1Jo * j=1 Jo ox
X.-)ds,
and since the process Yt = t is a Levy process we need only to invoke Theo-
Theorem 32 to complete the proof. ?
If the differentials are not Levy processes but only strong Markov pro-
processes which are semimartingales, then the solutions of equations such as (**)
in Theorem 32 need not be Markov processes. One does, however, have the
following result.
Theorem 35. Let Z be a strong Markov processes with values in K, Zq = 0,
such that Z is a semimartingale. Let / and g be Lipschitz functions. Let X$
be as in Theorem 32 and X be the solution of
I f(s-,X.-)dZ.+ f g(s,Xs-)ds.
Jo Jo
Then the vector process (X, Z) is Markov under P , each у € Ш, and strong
Markov if f and g are autonomous.
Proof. First recall that X is defined on $7 = К x Q and Z is automatically
extended to f2 as explained at the beginning of this section. The probability
Pv = ey x P is such that PV{Xq = y) = 1, each у е К. As in the proof
of Theorem 32 for a fixed stopping time T (T < oo a.s.) and x e M define
inductively for и > 0
Y°(x,T,u) =x
i-T+u
Yn+\x,T,u)=x+ / f(v-,Yn(x,T,v-))dZv
Jt
/
Jt
T+u
g(v-,Yn(x,T,v-))dv,
IT
and let X(x, T, u) denote the unique solution of
rT+u rT+u
X(x,T,u)=x+ f(v-,X(x,T,v-))dZv+ g{v-,X{x,T,v-))dv.
Jt Jt

6 The Markov Nature of Solutions 297
Next define QT = a{Zx+u — %т\ и > 0} for the same stopping time T. As in
the proof of Theorem 32 we see that Yn(x,T,u) is QT measurable for each
n > 1 and that X(x, T, u) is QT measurable as well. Next let h be Borel
measurable and bounded, and let G ? QT and bounded.
Observe that
E{h(XT)G\TT} = h{XT)E{G\TT}
= h(XT)E{G\ZT}
where j : R2 —> R is Borel. Since XT+U — X(XT,T,u) by the uniqueness
of the solution, the theorem now follows from the Monotone Class Theorem
(Theorem 8 of Chap. I). D
Theorem 35 can also be shown to be true with Stratonovich differentials,
but the proof is more complicated since the quadratic variation process is an
additive functional of Z, rather than a deterministic process (as is the case
when Z is a Levy process). For this type of result we refer the reader to
Cinlar-Jacod-Protter-Sharpe [34].
Each of Theorems 32, 34 and 35 can be interpreted with X having an
arbitrary initial distribution /i. Indeed for /j, a probability measure on (R, B),
define P" on fi by Р"(Л) = JRPx(A)[x(dx), for Л € В <g> Т. Then the conclu-
conclusions of the three theorems are trivially still valid for PM, and the distribution
of Xo is /i.
Traditionally the most important Markovian solutions of stochastic dif-
differential equations are diffusions. Suppose as given a space (fi, F, (Jrt)t>o, P)
satisfying the usual hypotheses.
Definition. An adapted process X with values in R™ is a diffusion9 if it has
continuous sample paths and if it satisfies the strong Markov property.
A restatement of Theorems 32 and 34 yields the following.
Theorem 36. Let В = (B1,...,Bd) be a standard Brownian motion on
M.d, Bo = 0, and let (/j) 1 < j < d, 1 < i < n, gl be autonomous Lipschitz
functions. Let Xq be as in Theorem 32 and let X be the solution of
*t = Xo + E / fAXs)dBi + f 9i{Xs)ds. (* * *)
J
E / fAXs)dBi + f
j=1jo Jo
ThenX is a diffusion. If the coefficients (fj)i<j<d, 1 < i < n, are non-random
F-S acceptable functions and if Y is the solution of
Ytl = Yj + Y, f f){Ys) о dBi + f gi{Ys)ds,
~[ Jo Jo
The definition of a diffusion is not yet standardized. We give a general definition.

298 V Stochastic Differential Equations
then Y is a diffusion.
In equation (***) of Theorem 36 the coefficients /j are called the diffusion
coefficients and the coefficients g% are called the drift coefficients.
Note that a diffusion need not be semimartingale, even though of course
the solutions of equations such as (* * *) are semimartingales. Indeed any
deterministic continuous function with paths of unbounded variation is a dif-
diffusion which is not a semimartingale. Another interesting example is provided
by Tanaka's formula. The process
rt
•t\= sign(B s)dBs + L°t
Jo
1Д
is a diffusion and it is a semimartingale, where В is standard Brownian motion
on R. Since the paths of L° are singular with respect to Lebesgue measure, and
since a semimartingale decomposition is unique for continuous processes (the
corollary of Theorem 31 of Chap. Ill), we see that \Bt\ cannot be represented
as a solution of (* * *). Another example is that of I-B^1/3 which is a time
homogeneous diffusion but not a semimartingale (Theorem 71 of Chap. IV).
An intuitive notion of a diffusion is to imagine a pollen grain floating down-
downstream in a river. The grain is subject to two forces: the current of the river
(drift), and the aggregate bombardment of the grain by the surrounding wa-
water molecules (diffusion). The coefficient f(t,x) then represents the sensitivity
of the particle at time t and place x to the diffusion forces. For example, if
part of the river water is warmer at certain times and places (due to sunlight
or industrial effluents, for example), then / might be larger. Analogously д
would be larger when the river was flowing faster due to a steeper incline.
We give three simple examples of diffusions.
Example. The stochastic exponential exp{Bt - \t} is a diffusion where
f(t,x) = x and g(t,x) = 0.
Example. Consider the simple system
Vt = V0+ f adBs + f aVsds,
Jo Jo
Xt = X0+ [ Vsds.
Jo
The process X can be used as a model of Brownian motion alternative to
Einstein's. It is called the Ornstein-Uhlenbeck Brownian motion, or simply
the Ornstein-Uhlenbeck process. Note that here the process X has paths
of finite variation and hence the process (Vt)t>o is a true velocity process for
X. Using integration by parts we can verify that
Vt = eat(V0
/¦
Jo

6 The Markov Nature of Solutions 299
is an explicit solution for V. Indeed
ft rt
e~atVt = V0+ Vs(-ae-as)ds + e~QWs
Jo Jo
f-t Pt rt
= Vo — a / V3e~a3ds + / e~asaVsds + / <
Jo Jo Jo
= Vo+
f e-a
Jo
and we are done. Since Vj) and the Brownian motion В are independent (by
construction), we see that when Vo has a Gaussian distribution then V is a
Gaussian process. If a is negative and we take a1 = —l/Ba), then V is а
stationary Gaussian process.
Example. Consider next the equation
Xt = / -ds + BU @<t< 1)
for S a standard Brownian motion.
For each to,O < to < 1, there is a solution which is a diffusion on [0, fo]. By
the uniqueness of solutions if t\ < t0 then the solution for [0, to] agrees with
the solution for [0, t\] on the interval [0, t\]. Thus we have a solution on [0,1).
If we can show that limt_>i Xt = 0, then the solution extends by continuity to
[0,1] and we will have constructed a diffusion X on [0,1] with Xo = X\ = 0,
known as the Brownian bridge.10
An application of integration by parts shows that the solution of the Brow-
Brownian bridge equation is given by
-^—dBs @<*<l).
Write Xt = f(t)Mt, where f(t) = A-t) and Mt = /„(l-s)^. To see that
limt_>i Xt = 0, we study Mt, making the change of variables t — u(l + u).
Then 0 < t < 1 corresponds to 0 < и < оо, and define
лг„ = м_^_, ди = t^l-, o<«<oo.
N is clearly a continuous Qu martingale, and moreover [N, N]u = u; hence N
is a standard Brownian motion by Levy's Theorem (Theorem 39 of Chap. II).
It is then easy to show, as a consequence of the Strong Law of Large Numbers,
that11
Nt
lim — =0 a.s.
t-»oo t
10 The Brownian bridge is also known as tied down Brownian motion, and alterna-
alternatively as pinned Brownian motion.
11 See, for example, Breiman [23, page 265].

300 V Stochastic Differential Equations
Let g(t,u>) = Nt(u>)/t so that Iimt_o5(*~1) w) = 0 a.s. Thus, limt_»o tNi/t = 0
a.s. We then have, replacing t with A — i)
t—> 1
_t) = lim(l - t)Ml/{2_t) = 0 a.s.,
and therefore limu_1(l - ы)М„ = 0 a.s., and hence limt_>i Xt = 0 a.s.
We now know that X is a continuous diffusion on [0,1], and that Xq =
Xi = 0 a.s. Also X is clearly a semimartingale on [0,1), but it is not obvious
that X is a semimartingale on [0,1]. One needs to show that the integral
Jo ?^ds < °° a-s- To see this calculate E{X?}, 0<t<l,
E{X2t) = f(tJE{M?} = f(tJE{[M,M]t}
= (l-t)
-t
By the Cauchy-Schwarz inequality
E{\Xt\}<E{X?}l/2 =
Therefore
Pl \Xs\^_ Г'ЕЦХ
ч^-г-г-
ds
s
whence Jo -j^ds < oo a.s. Therefore the solution X is a semimartingale on
[0,1].
Finally we remark that one can construct a similar Brownian bridge from
any value a to any value b, on an interval of arbitrary finite length т. Using
the interval [0, r] one has the equation
Bt, 0<t<r,
-dBs, 0<t<r,
with
solution
xt =
xt
L
= a +
= b.
Г Ь- Xg
Jo т — s

7 Plows of SDE: Continuity and Differentiability 301
7 Flows of Stochastic Differential Equations: Continuity
and Differentiability
Consider a stochastic differential equation of the form
F(X)s_dZs.
f
Jo
Obviously there is a dependence on the initial condition, and we can write the
solution in the form X(t, w, x), or Xх (u>). The study of the flow of a stochastic
differential equation is the study of the functions ф : x —» X(t,w,x) which can
be considered as mapping R™ —> R™ for (t,u>) fixed, or as mapping R" -» P",
where Vn denotes the space of cadlag functions from R+ to M™, equipped with
the topology of uniform convergence on compacts. It is important to distin-
distinguish between Vn and Dn. The former is a function space, and it is associated
in the literature with weak convergence results (see, e.g., Billingsley [17], or
Ethier-Kurtz [71], or Kurtz-Protter [136] for recent results in a semimartin-
gale context); the latter is the space of stochastic processes with cadlag paths,
and which are adapted to the underlying filtration.
Note that we have already encountered flows in Sect. 6 (cf., Theorem 31),
where we proved measurability of the solution with respect to a parameter
(which can be taken to be, of course, the initial condition). We will be inter-
interested in several properties of the flows: continuity, differentiability, injectivity,
and when the flows are diffeomorphisms of Mn.
We begin with continuity. We consider a general system of equations of
the form
XI = Щ + f F{Xx)s_dZs
Jo
(*)
where Xf and Hf are column vectors in R™, Z is a column vector of m
semimartingales with Zo = 0, and F is an n x m matrix with elements (F^).
For x fixed, for each у we have that Xt — Xf —Xf is a solution of the equation
Xt = Щ - Щ + f F(X)s-dZs, (**)
Jo
where ~F(Y) = F{XX + Y)- F(XX).
Theorem 37. Let Hx be processes in D™, and let x ь-> Hx : R™ -> Dn be
prelocally Lipschitz continuous in S_p, some p > n. Let F be an n x m matrix
of functional Lipschitz operators (F^), l<«<n, 1 < a < m.12 Then there
exists a function X(t,w,x) on R+ x fi x R™ such that
(i) for each x the process Хх(ш) = X(t, to, x) is a solution of (*), and
(ii) for almost all u>, the flow x к-> X(-,u>,x) from R" into T>n is continuous
in the topology of uniform convergence on compacts.
See page 250 for the definition of functional Lipschitz.

302 V Stochastic Differential Equations
Proof. We recall the method of proof used to show the existence and unique-
uniqueness of a solution (Theorem 7). By stopping at a fixed time to, we can
assume the Lipschitz process is just a random variable К which is finite
a.s. Then by conditioning13 we can assume without loss of generality that
this Lipschitz constant is non-random, and we call it с < оо. By replac-
replacing Щ with X? + f* F@)s^dZs, and then by replacing F with G given by
G(Y)t = F(Y)t — F@)t, we can further assume without loss of generality that
F@) = 0. Then for /3 = Cp(c), by Theorem 5 we can find an arbitrarily large
stopping time T such that ZT~ e <S(/3), and Hx is Lipschitz continuous in
S_p on [0,T). Then by Lemma 2 (preceding Theorem 7) we have that for the
solution X of (**)
\\XT~\\gP < Cp(c,Z) ||(Я* - Н»)т-\\&, for any p > 2,
and some (finite) constant Cp(c, Z). Choose p > n, and we have
E{sup Щ - ХУ\*} < Cp(c, Z)K\\x - y||p, (* * *)
s<T
due to the Lipschitz hypothesis on x *-> Hx. By Kolmogorov's Lemma (The-
(Theorem 72 of Chap. IV) we have the result on R™ x [0, T). However since T was
arbitrarily large, the result holds as well on R™ x Q x R+. ?
For the remainder of this section we will consider less general equations.
Indeed, the following will be our basic hypotheses, which we will supplement
as needed.
Hypothesis (Hi). Za are given semimartingales with Zft = 0, 1 < a < m.
Hypothesis (H2). fla : R" -> R are given functions, 1 <i <n, l<a<m,
and f(x) denotes the n x m matrix (/„(x)).
We will study the system of equations
Jo
which we also write
ft
n
Xt = x+ f(Xs_)dZs
Jo
where it is understood that Xt and x are column vectors in R™, /(Xs_) is an
n x m matrix, and Z is a column vector of m semimartingales.
To study the differentiability of the flow we will need a more general theo-
theorem on existence and uniqueness of solutions. Indeed, we wish to replace our
customary (global) Lipschitz condition with a local Lipschitz condition.
See the proofs of Theorems 7, 8, or 15 for this argument.

7 Flows of SDE: Continuity and Differentiability 303
Definition. A function / : R™ —» R is said to be locally Lipschitz if there
exists an increasing sequence of open sets Ak such that (Jfc Ak = R™ and / is
Lipschitz with a constant K^ on each Ak.
For example, if / has continuous first partial derivatives, then it is locally
Lipschitz, while if its continuous first partials are bounded, then it is Lipschitz.
If /, g, are both Lipschitz, then their product fg is locally Lipschitz. These
coefficients arise naturally in the study of Fisk-Stratonovich equations (see
Sect. 5).
Theorem 38. Let Z be as in (HI) and let the functions (/?) in (H2) be
locally Lipschitz. Then there exists a function ((x, u>) : R™ x fi —> [0, oo] such
that for each x ((x, •) is a stopping time, and there exists a unique solution of
;-)dZs (®®)
up to C(ar, •) with limsupj^^^.) ]|Xt|| = oo a.s. on {? < oo}. Moreover
x >—> ?(ar, ui) is lower semi-continuous, strictly positive, and the flow of X
is continuous on [0,?(z, •)).
Remark. Before proving the theorem we comment that for each x fixed
the stopping time T(u>) = (,(x,u>) is called an explosion time. Thus Theo-
Theorem 38 assures the existence and uniqueness of a solution up to an explosion
time; and at that time, the solution does indeed explode in the sense that
limsup^y ||Xt|| = -boo on {T < oo}. Note however that if the coefficients
(/?) in (H2) are (globally) Lipschitz, then a.s. ( = oo for all x (Theorem 7).
Proof. Let Ai be open sets increasing to R™ such that there exist (he), a
sequence of C°° functions with compact support mapping Rn to [0,1] such
that hi = 1 on A%. For each I we let Xt(t, u>, x) denote the solution (continuous
in x) of the equation
Xt =
/ ge(X..)dZa
Jo
where the matrix ge(x) is defined by h^(x)f(x).
Define stopping times, for x fixed,
Se(w,x) - inf{t > 0 : Xt(t,w,x) ? Ae or Xe(t-,w,x) <? Ae}.
By Theorem 37 the flow is uniformly continuous on compact sets, hence the
functions Si are lower semi-continuous. Then on [0,?^) we have that JQ =
Xe+i by the uniqueness of the solutions, since they both satisfy equation
). We wish to show that the relation
Хе(-,ш,х) = Xt+i(;<jj,x) on[0,Se(u),x)) (L)
holds for all ieE" simultaneously. Choose an x and let

304 V Stochastic Differential Equations
Ax = {w: Хе(-,ш,х) = Xe+i(-,uj,x) on [0,S*(w,x))}.
Then Р(ЛХ) = 1. We set A = UzeQ" ^ where Q™ denotes the rationale in
Rn. Then P(A) = 1 as well, and without loss of generality we take A — fi.
Next for arbitrary у e R", let yn -» y, with у„ е Q". Then
Se(w,y) < liminf Se(uj,yn),
n—>oo
and therefore the relation (L) holds for у as well, by the previously established
continuity.
Observe that Sf(uj,x) < Sg+i(u>,x), and let C(x,w) — supf Sg(u>,x),
for each x. Then ? is lower semi-continuous because the Sp are, and it is
strictly positive. Further, we have shown that there exists a unique func-
function X(-,u,x) on [0, ?(x,w)) which is a solution of (<E><?>), and it is equal
to X^(-,w,x) on [0,Si(uj,x)), each ? > 1. Indeed, on [0,Se(uJ,x)) we have
X(-,w,x) = Xt(-,w,x), and since X?(-,w,x) e Л^, we have ht(Xe(-,w,x)) = 1,
wherefore X% is a solution of (<8><g>) on [0,5^).
By our construction we have that X(Se(u>, x), u>, x) belongs to A\. Indeed,
letting Se(w, x) = s, then X(s—,w, x) = Xt(s—,w, x) = и ? Ag, for some value
u. Since и e Ag we have he (u) = 1, and thus X and JQ have the same jump at
s and we can conclude X(s,u>,x) = Xe(s,w,x). But Xt(s,uj,x) or Xt(s—,uj,x)
must be in A\ by right continuity and the definition of Sg; therefore X(s, w, x)
or X(s-,w,x) is in Л^. This shows that limsupt^ \\Xt\\ = ooon {С < оо}. П
It is tempting to conclude that if there are no explosions at a finite time for
each initial condition x a.s. (null set depending on x), then also C(#r) = oo
a.s. (with the same null set for all initial values x). Unfortunately this is not
true as the next example shows.
Example. Let В be a complex Brownian motion. That is, let B1 and B2
be two independent Brownian motions on R and let Bt = ВI + iB%, where
i2 — — 1. Consider the stochastic differential equation
Jo
Z2.dB,
where the initial value z is complex. This equation has a closed form solution
given by
Indeed, if we set f(x) = z(l + zx) and Zt = f(Bt), then f(Bt) = -Z?,
and since Bt = B\ + iB%, we have by Ito's formula
Zt = z- f Z2sdBs + \ f f"{Bs){d[B\B%+i2d\B2,B2}s);
Jo l Jo
since d[B1,B1]s = d[B2, B2]s = ds, we see that Zt = z - /„* Z2dBs.

7 Flows of SDE: Continuity and Differentiability 305
For z fixed we know that P{3t : Bt — —1/z) = 0, since Brownian motion
in R2 a.s. does not hit a specified point. Therefore Z does not have explosions
in a finite time for each fixed initial condition Zo = z. On the other hand
for any ujq and to fixed we have BtQ(w0) = zq for some zq G C, and thus for
the initial value z — —1/zo we have an explosion at the chosen finite time tg.
Thus each trajectory has initial conditions such that it will explode at a finite
time, and P{u> : 3 z with C(u>, z) < oo} = 1.
We next turn our attention to the differentiability of the flows. To this end
we consider the system of n + n2 equations (assuming that the coefficients (/?)
are at least C1)
"' ft
a=l J°
m n rt
e*=l j = l'
A < i < n) where D denotes an n x n matrix-valued process and 8\. = 1 if
i = к and 0 otherwise (Kronecker's delta). A convenient convention, sometimes
called the Einstein convention, is to leave the summations implicit. Thus the
system of equations (D) can be alternatively written as
I ra{
We will use the Einstein convention when there is no ambiguity. Note that in
equations (D) if X is already known, then the second system is linear in D.
Also note that the coefficients for the system (D) are not globally Lipschitz,
but if the first partials of the (/?) are locally Lipschitz, then so also are the
coefficients of (D).
Theorem 39. Let Z be as in (HI) and let the functions (/?) in (H2) have
locally Lipschitz first partial derivatives. Then for almost all u> there exists a
function X(t,u>,x) which is continuously differentiable in the open set {x :
?(ar, u>) > t}, where ? is the explosion time (cf., Theorem 38). If (/?) are
globally Lipschitz then C, = oo. Let Dk(t,u),x) = -^-X{t,u),x). Then for each
x the process (X(¦, to, x), D(-, u>, x)) is identically cadlag, and it is the solution
of equations (D) on [0, ?(z, •)).
Proof. We will give the proof in several steps. In Step 1 we will reduce the
problem to one where the coefficients are globally Lipschitz. We then resolve
the first system (for X) of (D), and in Step 2 we will show that, given X, there
exists a "nice" solution D of the second system of equations, which depends

306 V Stochastic Differential Equations
continuously on x. In Step 3 we will show that D\ is the partial derivative in
Xk of Xх in the distributional sense.14 Then since it is continuous (in x), we
can conclude that it is the true partial derivative.
Step 1. Choose a constant N. Then the open set {x : ((x,lj) > N} is a
countable union of closed balls. Therefore it suffices to show that if В is one
of these balls, then on the set Г = {u> : Vx ? B, ?(ar, u>) > N} the function
x t—* X(t,u>,x) is continuously differentiable on B. However by Theorem 38
we know that for each w e Г, the image of X as x runs through В is compact
in Vn with 0 <t < N, hence it is contained in a ball of radius R in R™, for R
sufficiently large.
We fix the radius R and we denote by К the ball of radius R of R™ centered
at 0. Let
A = {w : for x e В and 0 < t < N, X(t, ш, x) ? K}.
We then condition on A. That is, we replace P by Рл, where Рл(А) =
Р(А\Л) = Рр?д^ ¦ Then Рл <С Р, so Z is still a semimartingale with re-
respect to Рл (Theorem 2 of Chap. II). This allows us to make, without loss of
generality, the following simplifying assumption: if x ? В then ?(z, w) > N,
and X(t,w,x) eK,0<t<N.
Next let h : M™ —> R be C°° with compact support and such that h(x) = 1
if x e К and replace / with fh. Let Z be implicitly stopped at the (constant
stopping) time N. (That is, ZN replaces Z.) With these assumptions and
letting Рл replace P, we can therefore assume—without loss of generality—
that the coefficients in the first equation in (D) are globally Lipschitz and
bounded.
Step 2. In this step we assume that the simplifying assumptions of Step 1
hold. We may also assume by Theorem 5 that Z ? <S(/3) fc>r a /3 which will be
specified later. If we were to proceed to calculate formally the derivative with
respect to Xk of X1, we would get
dX\ _ ^ д
dxk
Therefore our best candidate for the partial derivative with respect to Xfc is
the solution of the system
m rt n
*4)
? / E ^
a=iJo 3=1 axi
14 These derivatives are also known as derivatives in the generalized function sense.

7 Flows of SDE: Continuity and Differentiability 307
and let D be the matrix (Dlk). Naturally Xs = X(s,ui,x), and we can make
explicit this dependence on x by rewriting the above equation as
Ukt -
where the summations over a and j are implicit (Einstein convention). We
now show that Dk = (D\,..., D%) is continuous in x. Fix x,y ? Rn and let
Vs(u>) =Dk(s,w,x) -Dk(s,Lj,y). Then
Jo oxj
where У/» = ЕГ=1/0* l^(x«-)dZ"- Note that ЬУ SteP 1 we know that
|^-(X^_) is bounded; therefore since Za e <S(/3), the Yj are in 5(c/3) for a
constant c. If C is small enough, by Lemma 2 (preceding Theorem 7) we have
that for each p > 2 there exists a constant CP(Z) such that
\\g,<Cp{Z)\\H\\g,.
However we can also estimate ||ЯЦ^р. If we let
then Я4г = ЕГ=1 /о JasdZf' an<^ therefore by Emery's inequality (Theorem 3)
we have that
which in turn implies
% < Cp(Z)\\J\\gr.
We turn our attention to estimating || J||sp. Consider first the terms
дР дР
dx~{ s~> ~ dx

308 V Stochastic Differential Equations
By the simplifying assumptions of Step 1, the functions -^ are Lipschitz in
K, and Xх takes its values in K. Therefore Theorem 37 applies, and as we
saw in its proof (inequality (* * *)) we have that
<K\\Xx-Xy\\s^<K(p,Z)\\x-y\\.
Next consider the terms DJkx. We have seen that these terms are solutions of
the system of equations
f
Jo
and therefore they can be written as solutions of the exponential system
with Yg = /„* %&(XZ_)dZZ. As before, by Lemma 2,
\\Dix\\g,P<C2p{Z).
Recalling the definition of J and using the Cauchy-Schwarz inequality gives
<C2p(Z)\\x-yl
which in turn combined with previous estimates yields
||V||r <C(p,Z)\\x-y\\.
Since V was defined to be Vs(ui) = Dk(s,ui,x) — Dk(s,u>,y), we have shown
that (with p > n)
E \sup \\Dk(s,u>,x) - Dk(s,uj,y)\\A < C(p,Z)P\\x - y\\P,
{s<N J
and therefore by Kolmogorov's Lemma (Theorem 72 of Chap. IV) we have
the continuity in x of Dk(t,ui,x).
Step 3. In this step we first show that Dk(t,u>,x), the solution of equations
(*4) (and hence also the solution of the n2 equations of the second line of (D))
is the partial derivative of X in the variable xk in the sense of distributions
(i.e., generalized functions). Since in Step 2 we established the continuity of
.Dfc in x, we can conclude that Dk is the true partial derivative.

7 Flows of SDE: Continuity and Differentiability 309
Let us first note that with the continuity established in Step 2, by increas-
increasing the compact ball K, we can assume further that Dk(s,ui,x) € К also, for
s < N and all x e B.
We now make use of Cauchy's method of approximating solutions of dif-
differential equations, established for stochastic differential equations in The-
Theorem 16 and its corollary. Note that by our simplifying assumptions, Y =
(X,D) takes its values in a compact set, and therefore the coefficients are
(globally) Lipschitz. The process Y is the solution of a stochastic differential
equation, which we write in vector and matrix form as
Yt=V
Г' f(Y.-)dZ..
Jo
Let ar be a sequence of partitions tending to the identity, and with the no-
notation of Theorem 16 let Y(a) = (X(a),D(a)) denote the solution of the
equation of the form
f f{Y°+)°dZs.
f
Jo
For each (ay) the equations (D) become difference equations, and thus trivially
dX(ar)
dxk
= Dk{ar).
The proof of the theorem will be finished if we can find a subsequence rq
such that limg_>ooX(crr.?) = X and lim^oo Dk(<?rq) = Dk, in the sense of
distributions, considered as functions of x.
We now enlarge our space Q, exactly as in Sect. 6 (immediately preceding
Theorem 31):
U = R2n
x п
u>t
where B2n denotes the Borel sets of R2n. Let A be normalized Lebesgue mea-
measure of K. Finally define
P = A x P.
We can as in the proof of Step 2 assume that Z e S@) for /? small enough
and then,
lim (X(<rr), D(ar) = (X, D) in 52,
J >OO —
by Theorem 16. Therefore there exists a subsequence rq such that
oo
M = J2^P\\(X(rq),D(rq)) - (X,D)\\ e Ll(dP).
9 = 1

310 V Stochastic Differential Equations
The function M = M(w,x) is in L1(X x P), and therefore for P-almost all ui
the function x t—> M(ui,x) € L1(dX). For u> not in the exceptional set, and t
fixed it follows that
limo(X(rq),D(rq)) = (X,D)
A a.e. Further, it is bounded by the function M(w, ¦) + \\(X(t,w, -),D(t,u>, -))||
which is integrable by hypothesis. This gives convergence in the distributional
sense, and the proof is complete. ?
We state the following corollary to Theorem 39 as a theorem.
Theorem 40. Let Z be as in (HI) and let the functions (/?) in (H2) have
locally Lipschitz derivatives up to order N, for some N, 0 < N < oo. Then
there exists a solution X(t,w,x) to
l = Xi + J2 ra
which is N times continuously differentiable in the open set {x : <^(x, w) > t},
where ? is the explosion time of the solution. If the coefficients (/?) are globally
Lipschitz, then С = oo.
Proof If N = 0, then Theorem 40 is exactly Theorem 38. If N = 1, then
Theorem 40 is Theorem 39. If N > 1, then the coefficients of equations (?>)
have locally Lipschitz derivatives of order N — 1 at least. Induction yields
(X,D)eCN-\ whence X e CN. О
Note that the coefficients (fla) in Theorem 40 are locally Lipschitz of order
N if, for example, they have N + 1 continuous partial derivatives; that is, if
/*, G CW+1(R"), for each г and a, then (/?) are locally Lipschitz of order N.
8 Flows as Diffeomorphisms: The Continuous Case
In this section we will study a system of differential equations of the form
X\ = Xi + V f Fi{X)s-dZf, 1 < i < n,
(*)
where the semimartingales Za are assumed to have continuous paths with
Zo = 0. The continuity assumption leads to pleasing results. In Sect. 10
we consider the general case where the semimartingale differentials can have
jumps. The flow of an equation such as (*) is considered to be an K"-valued
function ip : Rn —» R™ given by ip(x) = X(t,w,x), for each (t,w). We first
consider the possible injectivity of ip, of which there are two forms.

8 Flows as Diffeomorphisms: The Continuous Case 311
Definition. The flow ip of equation (*) is said to be weakly injective if for
each fixed x, у е R™, x ф у,
P{w :3t: X(t,LJ,x) = X(t,w,y)} = 0.
Definition. The flow ip of equation (*) is said to be strongly injective
(or, simply, injective) if for almost all ui the function ц> : x —> X(t,ui,x) is
injective for all t.
For convenience we recall here a definition from Sect. 3 of this chapter.
Definition. An operator F from Ю™ into D is said to be process Lipschitz
if for any X, Y € Ю™ the following two conditions are satisfied.
(i) For any stopping time T, XT" = YT" implies F(X)T~ = F(Y)T~.
(ii) There exists an adapted process К е L such that
Actually, process Lipschitz is only slightly more general than random Lip-
Lipschitz. The norm symbols in the above definition denote Euclidean norm, and
not sup norm. Note that if F is process Lipschitz then F is also functional
Lipschitz and all the theorems we have proven for functional Lipschitz coeffi-
coefficients hold as well for process Lipschitz coefficients. If / is a function which
is Lipschitz (as defined at the beginning of Sect. 3) then / induces a process
Lipschitz operator. Finally, observe that by Theorem 37 we know that the
flow of equation (*) is continuous from R™ into R™ or from R" into T>n a.s.,
where T>n has the topology of uniform convergence on compacts.
Theorem 41. Let Za be continuous semimariingales, 1 < a < m, H a
vector of adapted cadlag processes, and F an nxm matrix of process Lipschitz
operators. Then the flow of the solution of
is weakly injective.15
Proof. Let x, у G R™, x ф у. Let Xх, Xy denote the solutions of the above
equation with initial conditions x, у respectively. We let и = x — у and U =
Xх - XV. We must show P{w : 3t : Ut(u>) = 0} = 0. Set V = F(XX)- -
F(Xy)_. Then V G L and \V\ < K\U\. Further, the processes U and V are
related by
Ut = u+ [ VsdZs.
Let T = inf{t > 0 : Ut = 0}; the aim is to show P(T = oo) = 1. Since U is
continuous the stopping time T is the limit of a sequence of increasing stopping
We are using the Einstein convention on sums.

312 V Stochastic Differential Equations
times Sk strictly less than T. Therefore the process 1[о,т) = lim^oo l[o,sfc] is
predictable.
We use Ito's formula (Theorem 32 of Chap. II) on [О, Г) for the function
f(x) =log||a;||. Note that
df 1 d\\x\\ 1 xt xt
г)т llrll г)т llrll llrll llrll2'
ихг \\x\\ ихг уху \\x\\ \\x\\
d2f 1 Ixj d\\x\\ _ 1 2x±
32f 2xiXj
\\x\\A '
Therefore on [0,T),
+ 5 E/ йр*-, </•!. - E/o
Since <Я7* = ^a V^dZ", the foregoing equals
log Ht^H-tog ||«
* 1
-E
i,j,a,0
All the integrands on the right side are predictable and since ||V|| <
they are moreover bounded by К and K2 in absolute value. However on
{T < 00} the left side of the equation, log ||J7t|| - log ||и||, tends to -00 as t
increases to T; the right side is a well-defined non-exploding semimartingale
on all of [0,00). Therefore P(T < 00) = 0, and the proof is complete. D
In the study of strong injectivity the stochastic exponential of a semi-
semimartingale (introduced in Theorem 37 of Chap. II) plays an important role.
Recall that if Z is a continuous semimartingale, then XqS(Z) denotes the
(unique) solution of the equation
Xt = Xq + / X.dZs,
/
Jo
and ?(Z)t = exp{Zt - \[Z,Z\t}. In particular, P(\nis<tS{Z)s > 0) = 1.
Theorem 42. For iel", let Hx be in Dfc such that they are locally bounded
uniformly in x. Assume further that there exists a sequence of stopping times
i increasing to oo a.s. such that \\(H^_ — й"^)т'||^'- < K\\x — y\\, each

8 Flows as Diffeomorphisms: The Continuous Case 313
(- > 1> for a constant К and for some r > n. Let Z = (Z1,... ,Zk) be к
semimartingales. Then the functions
ft
т t—> / TJxrl7
JO
x ь-» [Hx ¦ Z, Hx ¦ Z]t
have versions which are continuous as functions from R" into T>, with T>
having the topology of uniform convergence on compacts.
Proof. By Theorem 5 there exists an arbitrarily large stopping time T such
that ZT~ G H°°. Thus without loss of generality we can assume that Z e H°°,
and that Hx is bounded by some constant K, uniformly in x. Further we
assume \\H1 - Hl\\sr < K\\x - y\\. Then
up || /* Hx_dZs - f H»_dZ,\\r} < CE{snp \\HX_ - Щ_\\г}\\г\\гн~
t Jo Jo t =
where we have used Emery's inequality (Theorem 3). The result for /Hf_dZs
now follows from Kolmogorov's Lemma (Theorem 72 of Chap. IV). For the
second result we have
|| [Hx_ -Z,HX-Z\- [Hi -Z,Hy_- Z] ||r
)+Ht(H:i
<2к\\г\\%г\\нх-ну\\§:,
and the result follows. D
Theorem 43. Let F be a matrix of process Lipschitz operators and Xх the
solution of (*) with initial condition x, for continuous semimartingales Za,
1 < a < m. Fix x, у ? R". For r ? R there exist for every x, у G R™ with
x ф у (uniformly) locally bounded predictable processes Ha(x,y), Ja'P(x,y),
which depend on r, such that
where
Ar
(x,
V)t =
ft
Jo
ft
Jo

314 V Stochastic Differential Equations
Proof. Fix x, у e Rn and let U = Xх - X*, V = F{XX)- - F(X»)_. Ito's
formula applies since U is never zero by weak injectivity (Theorem 36). Using
the Einstein convention,
r4UlUi +5)\\Us\\T-2}d[U\U%.
Let (•, •) denote Euclidean inner product on R™. It suffices to take
H?(x,y)=r\\Us\\-2(Us,Vsa),
(where Va is the a-th column of V); and
One checks that these choices work by observing that dU\ —
Finally the above allows us to conclude that
Л
Jo
and the result follows. ?
Before giving a key corollary to Theorem 43, we need a lemma. Let H00
be the space of continuous semimartingales X with Xo = 0 such that X has
a (unique) decomposition
X = N + A
where N is a continuous local martingale, A is a continuous process of finite
variation, Nq = Aq = 0, and such that [NjN]^ and Jo°° \dAs\ are in L°°.
Further, let us define
coo
\dAs\\\L°°-
\\Х\\ЙОО = \\[N,N}U? +
Jo
Lemma. For every p,a < oo, there exists a constant C(p,a) < oo such that
if \\Х\\Й„ < a, then \\S(X)\\iP < C(p,a).
Proof. Let X = N + A be the (unique) decomposition of X. Then
\\PSP = E{supexp{p{Xt - ]:[X,X]t)}}
- t 2
< E{epX'} (recall that X* = sup \Xt\)
t
< E{exp{pN* + pa}}
= epaE{epN'},

8 Flows as Diffeomorphisms: The Continuous Case 315
since \At\ < a, a.s. We therefore need to prove only an exponential maximal
inequality for continuous martingales. By Theorem 42 of Chap. II, since N
is a continuous martingale, it is a time change of a Brownian motion. That
is, Nt = B[jv,jv]t) where В is a Brownian motion defined with a different
filtration. Therefore since [N, N}oo < a2, we have
and hence E{epN'} < E{exp{pB*2}}. Using the reflection principle (Theo-
(Theorem 33 of Chap. I) we have
2 2
E{exp{pB*a2}} < 2E{exp{pBa2}} = ^|
D
Note that in the course of the proof of the lemma we obtained C{p, a) =
2
Corollary. Let — oo < r < oo and p < oo, and let (Лг(х, y)t)t>o be as given
in Theorem 43. Then ?(Ar(x, y)) is locally in ^p, uniformly in x, y.
Proof. We need to show that there exists a sequence of stopping times Тц
increasing to oo a.s., and constants C( < oo, such that ||5(Лг(а;, y))Tt \\s_p < Ci
for all x, у е К™, x фу. ~
By stopping, we may assume that Za and {Za,Z@] are in H_ and that
\Ha\ and \Ja^\ < b for all (x,y), 1 < a, C < m. Therefore \\Лг(х^у)\\йоо < С
— OO
for a constant C, since if X € Я and if К is bounded, predictable, then
К ¦ X e Я°° and \\K ¦ XWjja* < \\К\\з°°\\Х\\йоо, as can be proved exactly
analogously to Emery's inequalities (Theorem 3). The result now follows by
the preceding lemma. ?
Comment. A similar result in the right continuous case is proved by a dif-
different method in the proof of Theorem 62 in Sect. 10.
Theorem 44. Let Za be continuous semimartingales, 1 < a < m, and F an
n x m matrix of process Lipschitz operators. Then the flow of the solution of
Xt=x+ f F(X)s-dZs
Jo
is strongly injective on K™.
Proof. It suffices to show that for any compact set Ccl", for each N, there
exists an event of probability zero outside of which for every x,y € С with
Ы\\Х(8,и,х)-Х(8,и,у)\\>0.
s<N

316 V Stochastic Differential Equations
Let x, y, s have rational coordinates. By Theorem 43 a.s.
The left side of the equation is continuous (Theorem 37). As for the right side,
?(Лг(х, у)) will be continuous if we can show that the processes H"(x, y) and
Jf >/3(ж, у), given in Theorem 43, verify the hypotheses of Theorem 42. To
this end, let В be any relatively compact subset of R™ x R™ \ {(x,x)} (e.g.,
В = Bxx B2 where Bi,B2 are open balls in K" with disjoint closures). Then
|jar — y\\r is bounded on В for any real number r. Without loss we take r = 1
here. Let U(x,y) = Xх -X», V(x,y) = F(X*)_ -F(X»)_, and let Va(x,y)
be the a-th column of V. Then for (x,y) and (x',yr) in В we have
Ha(x,y)-Ha(x',y') (**)
= (\\U(x,y)\\-2 - \\U(x',y')\\-2)(U(x,y),Va(x,y))
+ \\U(x', y')\\-2(U(x, y) - U(x', y'), Va(x, y))
+ \\U(x',y')\\-2(U(x',y>),Va(x,y) - Va(x',y')).
The first term on the right side of (**) above is dominated in absolute value
by
\\\U(x,y)\\ - \\U(x',y')\\\(\\U(x,y)\\ + \\U(x',y')\\\) уШуа(х v)n
<K\\U(x,y)-U(x',y')\\(\\U(x,y)\\ + \\U(x',y')\\)\\U(x',y')\\-2,
where we are assuming (by stopping), that F has a Lipschitz constant K.
Since U(x, y) — U(x', y') = U(x, x') — U(y, y'), the above is less than
K{\\U{x,x')\\+\\U{y,y')\\){\\U{x,y)\\
= K{\\x - x'\\S{Ax{x,x')) + \\y - j
{\\x - у\\?(Лг(х,у)) + \\x' - у'ЩЛ^х^у'Шх' - у'\\-2?{Л_2{х',у')
<К1\\(х,у)-(х',у')\\(?(Л1(х,х'))+?(Л1(у,у')))(?(М^У))
By the lemma following Theorem 43, and by Holder's and Minkowski's
inequalities we may, for any p < oo, find stopping times Ti increasing to oo a.s.
such that the last term above is dominated in 5P norm by K( || (x, y) — (x', y') \\
for a constant Ki corresponding to Ti. We get analogous estimates for the
second and third terms on the right side of (**) by similar (indeed, slightly
simpler) arguments. Therefore Ha satisfies the hypotheses of Theorem 42, for
(x,y) € B. The same is true for Ja^, and therefore Theorem 42 shows that
Ar and [yl^ylr] are continuous in (x,y) on B. (Actually we are using a local
version of Theorem 42 with (x, y) ? В С R2n instead of all of R2n; this is not
a problem since Theorem 42 extends to the case x ? W open in R™, because

8 Flows as Diffeomorphisms: The Continuous Case 317
Kolmogorov's Lemma does—recall that continuity is a local property.) Finally
since Ar and [Лг, Лг] are continuous in (x, y) G В we deduce that ?(Лг(х, y))
is continuous in {(x, y) G M2n : x ф у}.
We have shown that both sides of
\\X(s,w,x)-X(s,w,y)\\r = \\x-y\\r?(Ar(x,y))s
can be taken jointly continuous. Therefore except for a set of probability zero
the equality holds for all (x, y, s)et"xI"xR|. The result follows because
?(Ar(x, y))t is defined for all t finite and it is never zero. D
Theorem 45. Let Za be continuous semimartingales, 1 < a < m, and let F
be an n x m matrix of process Lipschitz operators. Let X be the solution of
(*). Then for each N < oo and almost all ш
lim inf \\X(s,u),x)\\ = oo.
¦ Ill ^ »r II \ ' ' /II
||x||-»oo s<N
Proof By Theorem 43 the equality
is valid for all r G R.
For x ф 0 let Yx = \\XX - Х°\\~1. (Note that Yx is well-defined by
Theorem 41.) Then
(x,0)) (***)
\yx -Yy\ < ц^0101
Define Y°° = 0. The mapping x i-> x||x||~2 inspires a distance (I on Rn \ {0}
by d(x,y) = |ff|j||||. Indeed,
У ll2 (\\x-y\\'2
11 № h\r VN
By Holder's inequality we have that
\\?(Л1(х,у))?(Л.1(х,0)?(Л-1(у,0))\\ш-
and therefore by the corollary to Theorem 43 we can find a sequence of stop-
stopping times (Te)e>i increasing to oo a.s. such that there exist constants Ce with
(using (* * *))
т
Next set

318 V Stochastic Differential Equations
0 < ||x|| < oo,
||x|| = 0.
Then ||(УХ - Yy)Te\\rsr < Crt\\x - y\\r on Rn, and by Kolmogorov's Lemma
(Theorem 72 of Chap. IV), there exists a jointly continuous version of
(t,x) i-> Ytx, on Rn. Therefore Нтцхц_>0Угх exists and equals 0. Since
(У1) = ||ХЖ"ЖИ - Х°\\, we have the result. ?
Theorem 46. Let Za be continuous semimartingales, 1 < a < m, and F be
an n x m matrix of process Lipschitz operators. Let X be the solution of (*).
Let ip : Rn —> Rn be the flow ip(x) = X(t,w,x). Then for almost allw one has
that for all t the function <p is surjective and moreover it is a homeomorphism
from Rn to Rn.
Proof. As noted preceding Theorem 41, the flow ip is continuous from Rn to
Vn, topologized by uniform convergence on compacts; hence for a.a. lj it is
continuous from Rn to Rn for all t.
The flow tp is injective a.s. for all t by Theorem 44.
Next observe that the image of Rn under ip, denoted y(Rn), is closed.
Indeed, let y>(Rn) denote its closure and let у € y(Rn). Let (xk) denote a
sequence such that lim^o, ^(ij) = y. By Theorem 45, limsup^^^ ||х*,|| <
oo, and hence the sequence (xk) has a limit point x e Rn. Continuity implies
ip(x) = y, and we conclude that ip(Rn) = y(Rn); that is, ip(Rn) is closed.
Then, as we have seen, the set {xk} is bounded. If Xk does not converge to
x, there must exist a limit point z ф x. But then ip(z) = у = <р(х), and this
violates the injectivity, already established. Therefore ip~ * is continuous.
Since ip is a homeomorphism from Rn to </?(Rn), the subspace fp(M.n) of
Rn is homeomorphic to a manifold of dimension n in Rn; therefore by the
theorem of the invariance of the domain (see, e.g., Greenberg [84, page 82]),
the space y(Rn) is open in Rn. But </?(Rn) is also closed and non-empty. There
is only one such set in Rn that is open and closed and non-empty and it is
the entire space Rn. We conclude that y>(Rn) = Rn. ?
Comment. The proof of Theorem 46 can be simplified as follows: extend ip
to the Alexandrov compactification R" = Rn U {oo} of Rn to Tp by
I oo, x = oo.
Then Ip is continuous on R" by Theorem 45, and obviously it is still injective.
Since R™ is compact, Tp is a homeomorphism of R" onto 7p(R%). However R"
is topologically the sphere Sn, and thus it is not homeomorphic to any proper
subset (this is a consequence of the Jordan-Brouwer Separation Theorem (e.g.,
Greenberg [84, page 79]). Hence ^(R?) = R™.

8 Flows as Diffeomorphisms: The Continuous Case 319
We next turn our attention to determining when the flow is a diffeomor-
phism of Rn. Recall that a diffeomorphism of Rn is a bijection (one to one
and onto) which is C°° and which has an inverse that is also C°°. Clearly the
hypotheses on the coefficients need to be the intersection of those of Sect. 7
and process Lipschitz.
First we introduce a useful concept, that of right stochastic exponentials,
which arises naturally in this context. For given n, let Z be an n x n matrix
of given semimartingales. If X is a solution of
X* = 1+1 Xs-dZs,
f Xs-c
Jo
where X is an n x n matrix of semimartingales and / is the identity matrix,
then X = ?{Z), the (matrix-valued) exponential of Z. Since the space of n x n
matrices is not commutative, it is also possible to consider right stochastic
integrals, denoted
(Z :H)t= f (dZs)Hs,
Jo
where Z is an n x n matrix of semimartingales and H is an n x n matrix of
(integrable) predictable processes. If ' denotes matrix transpose, then
(Z : Я) = (H' ¦ Z'Y,
and therefore right stochastic integrals can be defined in terms of stochastic
integrals. Elementary results concerning right stochastic integrals are collected
in the next theorem. Note that / Y-dZ and [Y, Z] denote nxn matrix-valued
processes here.
Theorem 47. Let Y, Z be given nxn matrices of semimartingales, H an
nxn matrix of locally bounded predictable processes. Then,
(i) YtZt - Y0Z0 = /„* Ys_dZs + J*(dYs)Zs_ + [У, Z]t;
(ii) [H-Y,Z] =H-[Y,Z); and
(Hi) [Y,Z:H} = [Y,Z]:H.
Moreover ifF is an nxn matrix of functional Lipschitz operators, then there
exists a unique nxn matrix of 3-valued processes which is the solution of
/'
./о
(dZs)F(X)s_.
Proof The first three identities are easily proved by calculating the entries
of the matrices and using the results of Chap. II. Similarly the existence and
uniqueness result for the stochastic integral equation is a simple consequence
of Theorem 7. ?
Theorem 47 allows the definition of the right stochastic exponential.

320 V Stochastic Differential Equations
Definition. The right stochastic exponential of ал п х п matrix of semi-
martingales Z, denoted ?H(Z), is the (unique) matrix-valued solution of the
equation
Xt = I+ f (dZe)Xe..
Jo
We illustrate the relation between left and right stochastic exponentials
in the continuous case. The general case is considered in Sect. 10 (see Theo-
Theorem 63). Note that ?R(Z) = ?{Z')'.
Theorem 48. Let Z be an n x n matrix of continuous semimartingales with
Zo = 0. Then ?{Z) and SR{-Z + [Z, Z]) are inverses; that is, S(Z)?R(-Z +
[Z,Z]) = I.
Proof. Let U = ?{Z) and V = ?R(-Z + [Z,Z]). Since U0VQ = I, it suffices
to show that d(UtVt) = 0, all t > 0. Note that
dV = (-dZ + d[Z,Z])V,
(dU)V = (UdZ)V, and
d[U,V] = Ud[Z,V] = -Ud[Z,Z}V.
Using Theorem 47 and the preceding,
d(UV) = UdV + (dU)V + d{U, V]
= U(-dZ + d[Z, Z])V + UdZV - Ud[Z, Z)V
= o,
and we are done. D
The next theorem is a special case of Theorem 40 (of Sect. 7), but we state
it here as a separate theorem for ease of reference.
Theorem 49. Let (Z1,..., Zm) be continuous semimartingales with Z^ = 0,
1 < i < m, and let (fa), 1 < i < n, 1 < a < m, be functions mapping Шп to
R, with locally Lipschitz partial derivatives up to order N, 1 < N < oo, and
bounded first derivatives. Then there exists a solution X(t,u),x) to
J0
fia{Xe)dZf,
such that its flow <p : x —> X(x, t,uj) is N times continuously differentiable on
Шп. Moreover the first partial derivatives satisfy the linear equation
a=lj-1
where S-j, is Kronecker's delta.

9 General Stochastic Exponentials and Linear Equations 321
Observe that since the first partial derivatives are bounded, the coefficients
are globally Lipschitz and it is not necessary to introduce an explosion time.
Also, the value N = oo is included in the statement. The explicit equation for
the partial derivatives comes from Theorem 39.
Let D denote the n x n matrix-valued process
Dt = (Dit)i<i<n,i<fc<n- (®)
The process D is the right stochastic exponential ?R(Y), where Y is defined
by
Combining Theorems 48 and 49 and the above observation we have the im-
important following result.
Theorem 50. With the hypotheses and notation of Theorem 49, the matrix
Dt is non-singular for allt> 0 and ieR", a.s.
Theorem 51. Let (Z1,..., Zm) be continuous semimartingales and let (fa),
1 < i < n, 1 < a < m, be functions mapping Шп to Ш, with partial derivatives
of all orders, and bounded first partials. Then the flow of the solution of
a=l
fa(Xs)dZ», 1 < * < n,
is a diffeomorphism from Шп to Rn.
Proof. Let (p denote the flow of X. Since (/„)i<i<n,i<a<m have bounded first
partials, they are globally Lipschitz, and hence there are no finite explosions.
Moreover since they are C°°, the flow is C°° on Rn by Theorem 49. The
coefficients (fa) are trivially process Lipschitz, hence by Theorem 46 the flow
ip is a homeomorphism; in particular it is a bijection of Rn. Finally, the matrix
Dt (defined in (®) preceding Theorem 50) is non-singular by Theorem 50, thus
ip is C°° by the Inverse Function Theorem. Since <p~l is also C°°, we conclude
ip is a diffeomorphism of Rn. ?
9 General Stochastic Exponentials and Linear Equations
Let 2bea given continuous semimartingale with Zq = 0 and let ?{Z)t denote
the unique solution of the stochastic exponential equation
Xt = 1 + / XsdZs. (*)
Jo
Then Xt = ?{Z)t = exp{Zt - \[Z,Z]t} (cf., Theorem 37 of Chap. II). It is
of course unusual to have a closed form solution of a stochastic differential

322 V Stochastic Differential Equations
equation, and it is therefore especially nice to be able to give an explicit
solution of the stochastic exponential equation when it also has an exogenous
driving term. That is, we want to consider equations of the form
=Ht+ f
Jo
Xt=Ht+ f Xs_dZs, (**)
Jo
where H G D (cadlag and adapted), and Z is a continuous semimartingale. A
unique solution of (**) exists by Theorem 7. It is written ?jj{Z).
Theorem 52. Let H be a semimartingale and let Z be a continuous semi-
semimartingale with Zo = 0. Then the solution ?h(Z) of equation (**) is given
by
eH(Z)t = ?(Z)t{H0 + ( ?(г);Ч(н3 - [н, z\s)}.
Proof. We use the method of "variation of constants." Assume the solution is
of the form Xt = CtUt, where Ut = ?(Z)t, the normal stochastic exponential.
The process С is cadlag while U is continuous. Using integration by parts,
dXt = Ct-dUt + UtdCt + d[C, U]t
= Ct-UtdZt + UtdCt + Utd[C, Z]t
If X is the solution of (**), then equating the above with (**) yields
dHt + Xt.dZt = Xt_dZt + Utd{Ct + [C, Z]t)
or
dHt = Utd{Ct + [C,Z]t}.
Since U is an exponential it is never zero and 1/U is locally bounded. Therefore
Calculating the quadratic covariation of each side with Z and noting that
[[C, Z],Z]= 0, we conclude
Therefore equation (* * *) becomes
dH = dC+
and Ct = Jl U~ld{H8 - [Я, Z)s). Recall that Ut = ?{Z)t and Xt = CtUt, and
the theorem is proved. ?

9 General Stochastic Exponentials and Linear Equations 323
Since ?(Z)^1 — l/?(Z)t appears in the formula for ?jj{Z), it is worthwhile
to note that (for Z a continuous semimartingale)
/ 1 \=dZ-d[Z,Z]
\?{Z)) ?{Z)
and also
A more complicated formula for ?h{Z) exists when Z is not continuous
(see Yoeurp-Yor [236]). The next theorem generalizes Theorem 52 to the case
where H is not necessarily a semimartingale.
Theorem 53. Let H be cadlag, adapted (i.e., H G Ш), and let Z be a con-
continuous semimartingale with Zq = 0. Let Xt — ?jj(Z)t be the solution of
,t
Xt = Hf + I Xs-dZs.
Jo
Then Xt = ?H(Z)t is given by
Xt=Ht+ ?(Z)t [ ?{Z)-\Hs_dZs - Hs_d[Z, Z]s).
Jo
Proof. Let Yt = Xt - Ht. Then Y satisfies .
[
= f Hs_dZs + [ Ys^
Jo Jo
Yt=
= Kt [
Jo
where К is the semimartingale Я_ • Z. By Theorem 52,
Yt = ?(Z)t{K0 + f ?{г)-Ч{Ка - [K,Z].)}
Jo+
and since KQ = 0 and [K, Z)t = f* H..d[Z, Z]s,
Yt = ?(Z)t f ?{Z)-\Hs_dZs - Hs^d[Z,Z]s),
Jo
from which the result follows. ?
Theorem 54 uses the formula of Theorem 52 to give a pretty result on the
comparison of solutions of stochastic differential equations.

324 V Stochastic Differential Equations
Lemma. Suppose that F is functional Lipschitz such that if Xt{oS) = 0, then
F(X)t-(w) > 0 for continuous processes X. Let С be a continuous increasing
process and let X be the solution of
Xt = x0 + j F(X)s_dCs,
Jo+
with xQ > 0. Then P{31 > 0 : Xt < 0} = 0.
Proof. Let Г = inf {t > 0 : Xt = 0}. Since XQ > 0 and X is continuous, Xs > 0
for all s < T on {T < oo}. The hypotheses then imply that F(X)T- > 0 on
{T < сю}, which is a contradiction. ?
Comment. In the previous lemma if one allows xq = 0, then it is necessary
to add the hypothesis that С be strictly increasing at 0. One then obtains the
same conclusion.
Theorem 54 (Comparison Theorem). Let (Za)i<a<:m be continuous
semimartingales with Zft — 0, and let Fa be process Lipschitz. Let A be a
continuous, adapted process with increasing paths, strictly increasing att = O.
Let G and H be process Lipschitz functional^ such that G(X)t_ > H{X)t_
for any continuous semimartingale X. Finally, let X and Y be the unique
solutions of
Xt = x0 + f G(X)s_dAs + f F(X)s_dZs,
Jo+ Jo
H(Y)s_dAs+ f F(Y)s_dZs
o+ Jo
where x0 > y0 and F and Z are written in vector notation. Then P{4t > 0 :
Xt < Yt} = 0.
Proof. Let
ft
Nt
= j {F(X)S- - F(Y)S^}(XS - Ys)-4{x^Yi]dZs, and
Jo
Ct=xo-yo+ [ {G(X)S_ - H(Y)s_}dAs.
Jo+
Then Ut = Ct + /0* Us-dNs, and by Theorem 52
Ut = ?(N)t{(x0 -Ы+/
Next set

9 General Stochastic Exponentials and Linear Equations 325
vt =
and define the operator К on continuous processes W by
K(W) = G(W?(N) + Y)- H(Y).
Note that since G and H are process Lipschitz, if Wt = 0 then G(W?(N) +
Y)t- = G(Y)t... Therefore К has the property that Wt(w) = 0 implies that
K(W)t- > 0. Note further that K(V) = G(U + Y)~ H(Y) = G(X) - H(Y).
Next observe that
= x0 - yo + / ?{NO1dCs
Jo+
?(N);1{G(X)s--H(Y)s.}dAs
= xo-yo+ f K(V)s-?(N):xdAs
Jo+
K(V)s-dDs,
о
where Dt = J0,?(N)~ldAs. Then D is a continuous, adapted, increasing
process which is strictly increasing at zero. Since V satisfies an equation of
the type given in the lemma, we conclude that a.s. Vt > 0 for all t. Since
?{N)^1 is strictly positive (and finite) for all t > 0, we conclude Ut > 0 for
all t > 0, hence Xt > Yt for all t > 0. ?
Comment. If Xq > г/о (i-e-> ^o = 2/0 is n°t allowed), then the hypothesis that
A is strictly increasing at 0 can be dropped.
The theory of flows can be used to generalize the formula of Theorem 52.
In particular, the homeomorphism property is used to prove Theorem 55.
Consider the system of linear equations given by
Xt=Ht+yl A{_Xs_dZi (*4)
where H = (Нг), 1 < i < n is a vector of n semimartingales, X takes values
in Rn, and Л-3 is an n x n matrix of adapted, cadlag processes. The processes
Z-', 1 < j <m, are given, continuous semimartingales which are zero at zero.
Define the operators Fj on Dn by
F(X)t = A\Xt
where A> is the nxn matrix specified above. The operators Fj are essentially
process Lipschitz. (The Lipschitz processes can be taken to be \\АЦ\ which

326 V Stochastic Differential Equations
is cadlag, not caglad, but this is unimportant since one takes F(X)t- in the
equation.)
Before examining equation (*4), consider the simpler system A < i,k <
n),
where
xi-
0, гфк.
Letting I denote the nxn identity matrix and writing the preceding in matrix
notation yields
/ M-Us-dZi, (*5)
E /
j=i Jo
where U takes its values in the space of n x n matrices of adapted processes
inD.
Theorem 55. Let A^, 1 < j < m, be n x n matrices of cadlag, adapted
processes, and let U be the solution of (*5). Let Xf be Hie solution of (*4)
where Ht = x, x еШп. Then X* = Utx and for almost all w, for all t and x
the matrix Ut(cj) is invertible.
Proof. Note that Ut is an n x n matrix for each (t,uj) and x G Rn, so that
UfX is in Rn. If X* = Utx, then since the coefficients are process Lipschitz we
can apply Theorem 46 (which says that the flow is a homeomorphism of Шп)
to obtain the invertibility of Ut(co).
Note that U is also a right stochastic exponential. Indeed, U = ?R(V),
where Vj = Jo YlT=i Ai-dZJs7 and therefore the invertibility also follows from
Theorem 48.
Thus we need to show only that Xf =¦ Utx. Since Utx solves (*4) with
Ht = x, we have Utx = Xf a.s. for each x. Note that a.s. the function
x >—> U(uj)x is continuous from Rn into the subspace of T>n consisting of
continuous functions; in particular (t,x) i—* Ut(cj)x is continuous. Also as
shown in the proof of Theorem 46, (x,t) i—> Xf is continuous in x and right
continuous in t. Since Utx = Xf a.s. for each fixed x and t, the continuity
permits the removal of the dependence of the exceptional set on x and t. ?
Let U^1 denote the nxn matrix-valued process with continuous trajec-
trajectories a.s. denned by (U~l)t(co) =
Recall equation (*4)
m „t
3=1 Jo
И)

9 General Stochastic Exponentials and Linear Equations 327
where H is a column vector of n semimartingales and Zq = 0. Let [H, Z3}
denote the column vector of n components, the ith one of which is [Hl, Z3].
Theorem 56. Let И be a column vector of n semimartingales, Z] A < j <
m) be continuous semimartingales with Zq = 0, and let A], 1 < j < m be
nxn matrices of processes in ID. Let U be the solution of equation (*5). Then
the solution XH of (*A) is given by
XtH = UtH0 + Ut / U;\dHs - J2 A{-d[H, Z%)
Jo+ i=1
Proof. Write XH as the matrix product UY. Recall that U~l exists by Theo-
Theorem 48, hence Y = U~1XH is a semimartingale, that we need to find explicitly.
Using matrix notation throughout, we have
d(UY) = dH + J2 AiX
Integration by parts yields (recall that U is continuous)
m
(dU)Y- + U(dY) + d[U, Y] = dH + ^ Aj_U-Y-dZj,
by replacing X with UY on the right side above. However U satisfies (*5) and
therefore
m
{dU)Y_ = J2A-u-Y-dZJi
3 = 1
and combining this with the preceding gives
U(dY)+d[U,Y] =dH,
or equivalently
dY = U-1dH-U-1d[U,Y]. (*6)
Taking the quadratic covariation of the preceding equation with Z, we have
since \U~1d\U,Y},Z3] = 0, 1 < j < m. However since U satisfies (*5),
d[U,Y\ -
3=1 3=1
3 = 1

328 V Stochastic Differential Equations
since U equals ?/_. Substitute the above expression for d[U,Y] into (*6) and
we obtain
m
dY = U~\dH -Y,A-d[H,Zj}),
and since XH = UY, the theorem is proved. D
10 Flows as Diffeomorphisms: The General Case
In this section we study the same equations as in Sect. 8, namely
X\ = Xi + JT f Fi(X).-dZf, 1 < i < n, (*)
except that the semimartingales {Za)\<a<m are no longer assumed to he con-
continuous. For simplicity we still assume that Zo = 0. In the general case it is
not always true that the flows of solutions are diffeomorphisms of Rn, as the
following example shows.
Example. Consider the exponential equation in R,
f*
Xt — x + / Xs-dZs.
Jo
Let Z be a semimartingale, Zq = 0, such that Z has a jump of size —1 at a
stopping time Г, Т > 0 a.s. Then all trajectories, starting at any initial value
x, become zero at T and stay there after T, as is trivially seen by the closed
form of the solution with initial condition x:
Xt=x exp{Zt--[Z,Z]ct} TT (l + AZs)e-AZ\
2 A A
0<s<t
Therefore, injectivity of the flow fails, and the flow cannot be a diffeomorphism
We examine both the injectivity of the flow and when it is a diffeomorphism
of Rn. Recall the hypotheses of Sect. 7, to which we add one, denoted (H3).
Hypothesis (HI). Za are given semimartingales with Zff = 0, 1 < a < m.
Hypothesis (H2). /? : Rn —> R are given functions, I <i < n, 1 < a < m,
and f(x) denotes the n x m matrix (/„(a:)).
The system of equations
* =xi + E f ш
(*)

10 Flows as Diffeomorphisms: The General Case 329
may also be written
[
Jo
[ f(Xs-)dZs
Jo
where Xt and x are column vectors in Rn, /(Xs_) is an n x m matrix, and Z
is a column vector of m semimartingales.
Hypothesis (H3). / is C°° and has bounded derivatives of all orders.
Note that by Theorem 40 of Sect. 7, under (H3) the flow is C°°. The key
to studying the injectivity (and diffeomorphism properties) is an analysis of
the jumps of the semimartingale driving terms.
Choose an e > 0, the actual size of which is yet to be determined. For
(Za)i<a<m we can find stopping times 0 = To < T\ < T2 < ¦ ¦ ¦ tending a.s.
to oo such that
have an H°° norm less than e (cf., Theorem 5). Note that by Theorem 1,
[Za'j, Z'^'ijoo < e as well, hence the jumps of each Za>i are smaller than e.
Therefore all of the "large" jumps of Za^ occur only at the times (Tj), j > 1.
Let X3t (x) denote the solution of (*) driven by the semimartingales Za>i.
Outside of the interval (Tj-i,Tj) the solution is
Next define the linkage operators
using vector and matrix notation. We have the following obvious result.
Theorem 57. The solution X of (*) is equal to, for Tn <t < Tn+1,
Xt(x)=X?+\xn+),
where
xo+ = x
xi- =Xfx_(a:), x1+ = H1(x1-)
X2+ = H2(X2~)
xn- = X?n_(z(n_i)+), xn+ = Hn{xn-).
Theorem 58. The flow ip : x —> Xt(x,uf) of the solution X of (*) is a
diffeomorphism if the collections of functions
x i—у X3t (x,lj) x i—> H3 (x, w)
are diffeomorphisms.

330 V Stochastic Differential Equations
Proof. By Theorem 57, the solution (*) can be constructed by composition of
functions of the types given in the theorem. Since the composition of diffeo-
morphisms is a diffeomorphism, the theorem is proved. D
We begin by studying the functions x —> X3T.{x,w) and x —> X™+1(x,lj).
Note that by our construction and choice of the times Tj, we need only to
consider the case where Z = Z? has a norm in H°° smaller than e.
The following classical result, due to Hadamard, underlies our analysis.
Theorem 59 (Hadamard's Theorem). Let g : Rn -> Rn be C°°. Suppose
(i) Нт^ц^оо ||5(a;)|| = oo, and
(ii) the Jacobian matrix g'(x) is an isomorphism o/Rn for all x.
Then g is a diffeomorphism o/Rn.
Proof. By the Inverse Function Theorem the function g is a local diffeomor-
diffeomorphism, and hence it suffices to show it is a bijection of Rn.
To show that g is onto (i.e., a surjection), first note that <?(Rn) is open and
non-empty. It thus suffices to show that <?(Rn) is a closed subset of Rn, since
Rn itself is the only nonempty subset of Rn that is open and closed. Let (?j)j>i
be a sequence of points in Rn such that lirn^oo g(xi) = у € Rn. We will show
that у e <7(Rn). Let Xi — tiVi, where U > 0 and ||vj|| = 1. By choosing a
subsequence if necessary we may assume that Vi converges to и € Sn, the
unit sphere, as i tends to oo. Next observe that the sequence (ij)«>i must be
bounded by condition (i) in the theorem: for if not, then tt = \\xi\\ tends to oo
along a subsequence and then ||<7(?ifc)|| tends to oo by (i), which contradicts
that limj^oo g(xi) = y. Since (?»)»>! is bounded we may assume lim^oo tt =
to E Rn again by taking a subsequence if necessary. Then limj—oo xt = tov,
and by the continuity of g we have у — lim^oo g(Xi) = g(tov).
To show g is injective (i.e., one-to-one), we first note that g is a local
homeomorphism, and moreover g is finite-to-one. Indeed, if there exists an
infinite sequence (a;n)n>i such that g(xn) = yo, all n, for some t/o, then by
condition (i) the sequence must be bounded in norm and therefore have a
cluster point. By taking a subsequence if necessary we can assume that xn
tends to x (the cluster point), where g(xn) = yo, all n. By the continuity of g
we have g{x) = г/о as well. This then violates the condition that g is a local
homeomorphism, and we conclude that g is finite-to-one.
Since g is a finite-to-one surjective homeomorphism, it is a covering map.16
However since Rn is simply connected the only covering space of Rn is Rn (the
fundamental group of Rn is trivial). Therefore the fibers g~1(x) for x € Rn
each consist of one point, and g is injective. D
The next step is to show that the functions x н-> Xj,.(x,ui) and x i—>
X™+1(x,u>) of Theorem 58 satisfy the two conditions of Theorem 59 and are
16 For the algebraic topology used here, the reader can consult, for example,
Munkries [183, Chapter 8].

10 Flows as Diffeomorphisms: The General Case 331
thus diffeomorphisms. This is done in Theorems 62 and 64. First we give a
result on weak injectivity which is closely related to Theorem 41.
Theorem 60. Let Za be semimartingales, 1 < a < m with Zft — 0, and let F
be an n x m matrix of process Lipschitz operators with non-random Lipschitz
constant K. Let Нг e D, 1 < i < n (cadlag, adapted). //?™=1 ||-?а||/г» < е,
for e > 0 sufficiently small, then the flow of the solution of
Xt=x+Ht+ f
Jo
is weakly injective.
17
Proof. Let x, у & M.n, and let Xх, Xy denote the solutions of the above equa-
equation with initial conditions x, y, respectively. Let и = x - y, U = Xх - Xy,
and V = F(XX)_ - F{Xy)-. Then V e L and |V| < K\U-\. Also,
Ut = u+ f VsdZs.
Jo
Therefore AUS = J2a V?&Zf and moreover (using the Einstein convention
to leave the summations implicit)
<\\\Us-\\
if e is small enough. Consequently \\US\\ > |||C/S_||. Define T = infjt > 0 :
Ut- = 0}. Then C/t_ ^Oon [0,T) and the above implies Ut ф 0 on [0,T) as
well. Using Ito's formula for f(x) = log \\x\\, as in the proof of Theorem 41 we
have
log ||?/t||-log|H (**)
yi,ayi,f3
л jji yi,a i ft yi,ayi,f3
~ / IIГ7 II2 s 9 / ИГ/ Ц2 dlZ '
Jo \\Us-\\ ? Jo \\Us-\\
ft jji jji y
-Jo \\u.-
ft jji jji yi,ayj,P
0<s<t "Us
For s fixed, let
17 We are using vector and matrix notation, and the Einstein convention on sums.
The Einstein convention is used throughout this section.

332 V Stochastic Differential Equations
Js = {log\\U.\\ -log||C/s_|| - -p=
\\us-\\
so that the last sum on the right side of equation (**) can be written
Eo<s<t Js- ВУ Taylor's Theorem
d2f
2 —J—(x)\
OXOX
where f(x) = log||a;||, and Is denotes the segment with extremities Us and
?/,_. Since
1
and since ||AC/S|| < ^||C/S_||, we deduce
which in turn implies
\Js\<C\\Us-f\\AZs
Js" \\ut.p-
Therefore
0<s<t 0<s<t а
on [О, Г). Returning to (**), as t increases to T, the left side tends to — oo on
{T < oo} and the right side remains finite. Therefore P(T < oo) = 0, and U
and U- never vanish, which proves the theorem. D
Theorem 61. Let (^a)i<a<m be semirnartingales, Zff = 0, F an n x m
matrix of process Lipschitz operators with a non-random Lipschitz constant,
and Hl G D, 1 < г < п. /f E™=1 Ц^Цд00 < e for e >0 sufficiently small,
then for r ? Ш there exist uniformly locally bounded predictable processes
Ha(x,y) and Ka<P(x,y), which depend onr, such that
where Xх is the solution of
Xt = x + Ht+ f F(X)s_dZs.
Jo
The semimartingale AT is given by

10 Flows as Diffeomorphisms: The General Case 333
Ar(x,y)t = f H?(x,y)dZ? + f
Jo Jo
where Jt = Ylo<s<t ^s' aru^ w^ere ^s is an adapted process such that \AS\ <
Cr(AZ«J.
Proof. Fix x, у e Rn, x ф y, and let U = Xх - Xy, V = F{XX)- - F(Xy)-.
By Theorem 60 U is never zero. As in the proof of Theorem 43, by Ito's
formula,
\\Uy\\r = \\x - y\\r + f r\\Us-\\r
Jo
+ \ [\{(г-2)\\и^\
& Jo
0<s<t
Let Ls denote the summands of the last sum on the right side of the above
equation. If g(x) = \\x\\r, then
\LS\<C\\AUS\\
where Is is the segment with boundary Us and C/s_. However,
да л ¦ n
II = r\(r — 2)l|a;||'— xx¦ + 5*||o;llr~ I
which is less than Cr||a;||r~2. However as in the proof of Theorem 60, we
have ||Af/s|| < |||?78_||, which implies (for all r positive or negative) that
||я||г < Сг||^8-1Г for all x between Us- and Us. Hence (the constant С
changes)
<C\\Us4r\\AZs\\2.
Therefore, let As = ||C/s_||~rI/s, and we have
\A.\< E
0<s<t 0<s<t
an absolutely convergent series with a bound independent of (x,y). To com-
complete the proof it suffices to take
and

334 V Stochastic Differential Equations
as in the proof of Theorem 43. Note that l{c/s_^o} is indistinguishable from
the zero process by weak injectivity (Theorem 60). These choices for Ha and
Ka'$ are easily seen to work by observing that Щ = Z)™=i /o K*'Q?^?> ancl
the preceding allows us to conclude that
\\Ut\\r = \\x-y\\r+ f \\Us_\\4Ar(x,y)s,
Jo
and the result follows. D
Theorem 62. Let (Za)i<a<m be semimartingales, Z§ = 0, F an n x m
matrix of process Lipschitz operators with a non-random Lipschitz constant,
and W e D, 1 < i < n. Let X = X(t,w, x) be the solution of
[
Jo
F(X)s_dZs. (*)
If 51Г=1 Il-Z^lltf00 < ? for ? > 0 sufficiently small, then for each N < oo and
almost all ш
lim inf ||X(s,o;,a;)|| = oo.
||ж||—юо s<N
Proof. The proof is essentially the same as that of Theorem 45, so we only
sketch it. For x ф 0 let Yx = \\XX - X0^1, which is well-defined by weak
injectivity (Theorem 60). Then
\yx\ = WxW
\YX-Yy\ < ||ХХ
by Theorem 61, where Лг(х, у) is as defined in Theorem 61. Set Y°° = 0.
Since H^H/foe < e each Za has jumps bounded by e, and the process Jt
defined in Theorem 61 also has jumps bounded by Cre2. Therefore we can
stop the processes Лг(х,у) at an appropriately chosen sequence of stopping
times (Tg)t>\ increasing to oo a.s. such that each Лг(х, у) & S(e) for a given e,
and for each ?, uniformly in (x, y). However if Z is a semimartingale in S(e),
then since ?(Ar(x,y)) satisfies the equation Ut = 1 + /0 Us^dAr(x,y)s, by
Lemma 2 of Sect. 3 of this chapter we have
\\S(Ar(x,y))\\sp <C(p,z)<<x,

10 Flows as Diffeomorphisms: The General Case 335
where C(p, z) is a constant depending on p and z = ||Лг(а;, у)\\н°° < ке: the
bound for ?, provided of course that e is sufficiently small. We conclude that
for these Tg there exist constants Ce such that
wherep > n, and where d is the distance on Rn\{0} given by d(x, y) = K,.^),,
Set
0 < ||ж|| < oo,
Y°° = 0,
Then ||(Yx - Yv)T'\\%, < C*\\x - y\\P on Rn, and by Kolmogorov's Lemma
(Theorem 72 of Chap. IV) we conclude that Итцх||_>о Yx exists and it is zero.
Since (У*) = ||XxHx|1 - X°\\, the result follows. D
If (p is the flow of the solution of (*), Theorem 62 shows that
lira \\(p(x)\\ = +oo,
\\x\\-* oo
and the first condition in Hadamard's Theorem (Theorem 59) is satisfied.
Theorem 63 allows us to determine when the second condition in Hadamard's
Theorem is also satisfied (see Theorem 64), but it has an independent interest.
First, however, some preliminaries are needed.
For given n, let Z be an n x n matrix of given semimartingales. Recall that
X = ?{Z) denotes the (matrix-valued) exponential of Z, and that ?R(Z) de-
denotes the (matrix-valued) right stochastic exponential of Z, which was defined
in Sect. 8, following Theorem 47.
Recall that in Theorem 48 we showed that if Z is an n x n matrix of
continuous semimartingales with Zq = 0, then ?{Z)?R{—Z + [Z, Z]) = /, or
equivalently ?{-Z + [Z, Z])?R{Z) — I. The general case is more delicate.
Theorem 63. Let Z be an n x n matrix of semimartingales with Zq = 0.
Suppose that Wt = -Zt + [Z,Z]ct + E0<s<t(/ + Д^ГЧД^J is a well-
defined semimartingale. Then
?(W)t?R(Z)t = I
for all t > 0.
Proof Let U = ?(W), V = ?R(Z), and Jt = Eo<s<t(/ +
Then,
dU = U-(-dZ + d[Z, Z)c + dJ),
dV = (dZ)V-,
and therefore

336 V Stochastic Differential Equations
= U-(-dZ + d[Z, Z]c
d[U,V] = U-d[W,
= -U-d[Z, Z]V- + U-d[J, Z]V-,
since d[Z, [Z, Z]c] = 0. By Theorem 47
d(UV) = U_dV + (dU)V- + d[U, V];
using the above calculations several terms cancel, yielding
d(UV) = U_d[Z,Z}cV_ + U-(dJ)V. - U-d[Z,Z]V- + U.d[J,Z]V-.
Since [Z, Z]t = [Z, Z\4 + E0<s<t(A^J, and
0<s<t 0<s<t
the preceding becomes
d(UV) - -U_
Since
(AZSJ = ((AZSJ + (AZSK)(I + AZs)-\
the above equation implies d(UV)t = 0, all t > 0. However UoVo = I, and
therefore UtVt = I, all t > 0. П
Corollary. Let Z be a square matrix of semimartingales. If ||Z||#oo < e for
г > 0 sufficiently small, then ?R(Z)t is invertible for all t > 0.
Proof. If || Z\\h°° < ? then the jumps of Z are bounded by e > 0 and therefore
the process W of Theorem 63 is a well-defined semimartingale. D
Theorem 64. Let (^a)i<a<m be semimartingales with Zo = 0, and let f be
a matrix of coefficients satisfying Hypotheses (H2) and (H3). Let X be the
unique solution of
Xt = x+ f f(X.-)dZe. (*)
Jo
The Jacobian matrix Dlk{t,u),x) = -^-Х1^,и),х) is invertible for each t > 0
provided \\Z\\h°° < ? for sufficiently small e > 0.

10 Flows as Diffeomorphisms: The General Case 337
Proof. By Theorem 39 (in Sect. 7) the Jacobian matrix D satisfies the right
stochastic exponential equation
and the matrix semimartingale differential -g^(Xs-)dZ? satisfies the hy-
hypotheses of the corollary of Theorem 63, whence the result. ?
Before stating the principal result of this section, we need to define two
subsets of Km; recall that under Hypotheses (HI), (H2), and (H3), that Z =
{Za)i<a<m is a given m-tuple of semimartingales and that f(x) = {fa(x)) is
an n x m matrix of C°° functions. Let
P = {z e Km : H(x) = x + f(x)z is a diffeomorphism of K™}
X ={z?Rm : H{x) = x + f(x)z is injective in K™}.
Clearly PCI.
Theorem 65. Let Z and f be as given in Hypotheses (HI), (H2), and (H3),
and let X be the solution of
Xt=x+ f f(X,-)dZa. (*)
Jo
The flow of X is a.s. a diffeomorphism of K™ (resp. trajectories of X from
different initial points a.s. never meet) for all t if and only if all the jumps of
Z belong to P (resp. all the jumps of Z belong to X).
Proof. Recall the processes Х^,.(х,ш) and X™+ (x,u>) defined in Theorem 57,
and the linkage operator W (x) = x + f(x)AZx:i, defined immediately preced-
preceding Theorem 57. By hypothesis the linkage operators H^(x) are clearly diffeo-
diffeomorphisms of K™ (resp. injective), and by Theorems 62 and 64, Hadamard's
conditions are satisfied (Theorem 59), and therefore the functions x н->
Xj,.(x,(jj) and x н-> X™+1(x,w) are diffeomorphisms of K™ if e > 0 is taken
small enough in the definition of the stopping times (Tj)j>i, which it is always
possible to do. Therefore by Theorem 58 the flow ip : x —> Xt(x,w) is a.s. a
diffeomorphism of K™ for each t > 0.
The necessity is perhaps the more surprising part of the theorem. First
observe that by Hadamard's Theorem (Theorem 59) the set P contains a
neighborhood of the origin. Indeed, if z is small enough and x is large enough
then ||/(x)z|| < ||ж||/2 since / is Lipschitz, which implies that ||Д"(х)|| > ||х||/2
and thus condition (i) of Hadamard's Theorem is satisfied. On the other hand
H'(x) — I + f'(x)Z is invertible for all x for ]|z|| small enough because f'(x)
is bounded; therefore condition (ii) of Hadamard's Theorem is satisfied. Since
f(x) is C°° (Hypothesis (H3)), we conclude that P contains a neighborhood
of the origin.

338 V Stochastic Differential Equations
To prove necessity, set
A = {u : 3 5 > 0 with AZs{u) G Vе},
Г2 = {uj : 3 5 > 0 with AZs{uj) e Iе}.
Suppose -P(A) > 0. Since V contains a neighborhood of the origin, there
exists an e > 0 such that all the jumps of Z less than e are in V. We also take
? so small that all the functions x н-> X^ (x) are diffeomorphisms as soon as
the linkage operators Hk are, all к < i.
Since the jumps of Z smaller than e are in V, the jumps of Z that are in
Vе must take place at the times Tj. Let
A> = {u : AZTi e P, all i < j, and AZTj e X>c}.
Since -P(-Ti) > 0, there must exist a j such that P(Aj) > 0. Then for w G Aj,
x t-> X7v_(a:,a;) is a diffeomorphism, but H^(x,uj) is not a diffeomorphism.
Let u)q G Л^ and io = Tj(u)o). Then а; н-> Xt0(x,u)o) is not a diffeomorphism,
and therefore
P{ui : 3 t such that Z —> Xt(x,w) is not a diffeomorphism} > 0,
and we are done. The proof of the necessity of the jumps belonging to I to
have injectivity is analogous. D
Corollary. Let Z and / be as given in Hypotheses (HI), (H2), and (H3), and
let X be the solution of
Xt = x+ f{Xs_)dZs.
Jo
Then different trajectories of X can meet only at the jumps of Z.
Proof. We saw in the proof of Theorem 65 that two trajectories can intersect
only at the times Tj that slice the semimartingales Za into pieces of H°°
norm less than e. If the Za do not jump at Tj0 for some jo, however, and
paths of X intersect there, one need only slightly alter the construction of Tj0
(cf., the proof of Theorem 5, where the times Tj were constructed), so that
Tj0 is not included in another sequence that ?-slices {Za)i<a<m, to achieve
a contradiction. (Note that if, however, (Za)i<a<m has a large jump at Tj0,
then it cannot be altered.) D
11 Eclectic Useful Results on Stochastic Differential
Equations
We begin this collection of mostly technical results concerning stochastic dif-
differential equations with some useful moment estimates. And we begin the
moment estimates with preliminary estimates for stochastic integrals. The
first result is trivial, and the second is almost as simple.

11 Eclectic Useful Results on Stochastic Differential Equations 339
Lemma. For any predictable (matrix-valued) process H and for any p > 1,
and for 0 < t < 1, we have
-E{sup
s<t
f Hsds } < f E{\Hs\p}ds.
Jo Jo
Proof. Since 0 < t < 1, we have that ds on [0,1] is a probability measure,
and since f(x) = xp is convex for p > 1, we can use Jensen's inequality. The
result then follows from Fubini's Theorem. ?
Lemma. For any predictable (matrix-valued) process H and multidimen-
multidimensional Brownian motion B, and for any p > 2, and for 0 < t < 1, there exists
a constant cp depending only on p such that
up / HsdBs }<cp f E{\Hs\p}ds.
<t [Jo Jo
E{sup / HsdBs
s<t \Jo
Proof. The proof for the case p = 2 is simply Doob's inequality; we give it for
the one dimensional case.
?{sup( ГHsdBsJ} < 4E{\H-B,H-B]t} - 4E{ f H2sds} = 4 / E{H2s}ds,
s<t Jo Jo Jo
where the last equality is by Fubini's Theorem. For the case p > 2, we use
Burkholder's inequality (see Theorem 48 of Chap. IV,
?{sup
s<t
f HsdBs } < cpE{{H -B,H- Bft'2} <cp f E{\Hs\p}ds
Jo Jo
where the last inequality follows from Jensen's inequality (recall that p > 2
so that f(x) = xpl2 is a convex function, and since 0 < t < 1 we have used
that ds on [0,1] is a probability measure) and Fubini's Theorem. ?
Theorem 66. Let Z be a d-dimensional Levy process with E{\\ Zt \\lp} < со
forO < t < 1, H a predictable (matrix-valued) process and p > 2. Then then-
exists a finite constant Kv such that for 0 < t < 1 we have
If HsdZs }<KP f E{\Hs\p}ds.
\Jo Jo
E{sup
s<t \JO
Proof We give the proof for the one dimensional case.18 Since the Levy pro-
process Z can be decomposed Zt — bt + aBt + Mt, where M is a purely dis-
discontinuous martingale (that is, M is orthogonal in the L2 sense to the stable
subspace generated by continuous L2 martingales), it is enough to prove the
inequality separately for Zt = bt, and Zt = crBt, and Zt = Mt. The preceding
two lemmas prove the first two cases, so we give the proof of only the third.
18 An alternative (and simpler) proof using random measures can be found in [105].

340 V Stochastic Differential Equations
In the computations below, the constants Cp and the functions Kp(-) vary
from line to line. Choose the rational number к such that 2k < p < 2k+l,
Applying the Burkholder-Gundy inequalities19 for p > 2 we have
E{
f
Jo
HsdMs
} < (CP)pE{
Set
/ \Hs\2d[M,M]t
Jo
= / \x\2vM{dx) < со.
Since [M, M] is also a Levy process, we have that [M, M]t — olm^ is also a
martingale. Therefore the inequality above becomes
E{
f
Jo
} <CPE{
\Hs\2d([M,M]s-aMs)
p/2
f\Hs\4s
Jo
p/2
}¦
We apply a Burkholder-Gundy inequality again to the first term on the right
side above to obtain
E{
H8dMs
¦K,
,(«)[/
Hs\4s}.
We continue recursively to get
E{\ HsdMs
s<t
)E{ f
\Hs\4s}.
Next we use the fact that, for any sequence a such that ||a||;« is finite,
\\a\\p < ЦоЦ/, for 1 < q < 2. As 1 < p2~k < 2 we get
s<t
s<t
s<t
whence
19 The Davis inequality is for the (important) case p = 1 only.

11 Eclectic Useful Results on Stochastic Differential Equations 341
s<t
s<t
Note that J2s<t \AMS\P is an increasing, adapted, cadlag process, and its
(non-random) compensator is At = t j \x\pi/M(dx), which is finite by hypoth-
hypothesis. Since \H\P is a predictable process,
Jo
r\p - As
r<s
is a martingale with zero expectation. Therefore we have
E{\ J* HsdMs\p} <{CP J \x\puM{dx) + Kp(t) f^(J i
¦E{f\Hs\pds}
Jo
It remains to show that, for any 1 < г < к,
(J \xf uM{dx)f2~l < (J \x\2yM{dx)Y'2 + J \x\piyM(dx).
Let Am := / \x\2VM{dx), so that
liM{dx) :- -—\x\2vM{dx)
лм
is a probability measure. Denote 2* by q. One has to show
*Mq(J N9~ Wc*r))p/e < \%2 + XmJ \x\p-2m(dx).
If
the preceding inequality is obvious. On the other hand, if
then it is sufficient to prove that
X^9~\J\x\9-2fiM{dx)r/9 <J\x\p-2m(dx).
But the bound on Am and Jensen's inequality give the result.
?

342 V Stochastic Differential Equations
Theorem 66 can be used to prove many different results concerning solu-
solutions of stochastic differential equations driven by Levy processes. We give two
examples. (These types of results are true more extensively (for example in n
dimensions) with similar proofs; see for example [105].) Our first example is a
moment estimate. As a convenience, we continue to work on the time interval
[0,1].
Theorem 67. Let a be continuously differentiable with a bounded derivative,
and let Z be a Levy process with Levy measure v such that J,x,>1 \x\pis(dx) <
oo, which is equivalent to E{\Zt\p} < oo for p > 2 and for all t, 0 < t < 1.
Let Xх denote the unique solution of the stochastic differential equation
dXt = a{Xt_)dZu Xo = x. (*)
Then we have
E{suP\Xx\p}<K(l + \x\p)
s
where К is of course a positive constant.
Before proving this theorem we need to recall Gronwall's inequality.20
Theorem 68 (Gronwall's Inequality). Let a be a function from K+ to
itself, and suppose
a(s) < c + к a(r)dr < oo
Jo
forO < 5 < t. Thena(t) < cekt. Moreover if с = 0 thena vanishes identically.
The proof is simple and usually taught in elementary differential equations
courses.
Proof of Theorem 67. This is a consequence of Theorem 66 and Gronwall's
inequality. ?
The next corollary follows easily from Theorem 67.
Theorem 69. Let a and Z be as in Theorem, 67. Let g be continuous and for
q > 0 define
|| g \\q= mi{a > 0 : \g(x)\ < a(l + \x\*)}.
Forq = 0 we let \\ g ||o be the L°° norm. If Pt denotes the transition semigroup
of the Markov process Xх of equation (*), then we have for q > 0
|| Ptg \\q< К || g ||g .
20 Extensions of this classic version of Gronwall's inequality axe in the exercises for
this chapter. See also [182].

11 Eclectic Useful Results on Stochastic Differential Equations 343
We now turn to our second example, which consists of moment estimates
on the flows of the solution Xх of equation (*). We first establish some nota-
notation. For a Levy process Z with Levy measure v and with finite p-th moment
for p > 2, we decompose it as Zt = bt + cBt + Mt where В is a standard Brow-
nian motion and M is a martingale independent from В and 1? orthogonal
to all continuous martingales. We then define
which is finite since by assumption i?{|Zt|p} < oo and p > 2. Here we consider
the multidimensional case: both Z and the solution Xх of equation (*) are
assumed to be d-dimensional, where d > I.21 We let X* ' represent a A;-th
derivative of the flow of Xх.
Theorem 70. For к > 1 assume that a is differentiable к times and that
all partial derivatives of a of order greater than or equal to one are bounded.
Assume also that Z has moments of order kp with p > 2. With the nota-
notation defined right before this theorem, we have that there exists a constant
K(k,p,a,T]p) such that for 0 < s <1,
W\P} < K.
Proof. We recall from Theorem 39 that Xх together with XX)^ constitute
the unique solution of the system of equations
as long as к > 2, and with a slightly simpler formulation if At = 1. (Note
that this is the same equation (D) as on page 305.) We can write the second
equation above in a slightly more abstract way
f
i=2Jo
if к > 2; and if к = 1 the equation is the simpler
21 One could also allow the dimensions of Z and Xх to be different, but we do not
address that case here.

344 V Stochastic Differential Equations
f
Jo
We next observe that the components of Fk^x^,... ,x^k г+1)) are sums of
terms of the form
6*5
(**)
, and an "empty" product
k-i+1 ocj
3 = 1 r=l
and where x^'1 is the Z-th component of a;
equals 1.
We thus want now to prove that
E{sup
for all i — 2,..., N and for some constant К = К(к,р, а, щр) (in the remain-
remainder of the proof К = К(к,р,сг,г)кр) varies from line to line). And of course,
it is enough to prove that if G is any monomial as in (**), then
We will use Theorem 66 and the fact that W is bounded. Note that Theo-
Theorem 66 implies that
s<t
[SHsdZs\P'}<K(p,r,p) f E{\Hs\P'}ds
Jo JO
for 2 < p' < p, where the constant K{p,r)p) depends only on p, r)p, and the
dimensions of H and Z. For this we see that the left side of the previous
equation is smaller than
k+l-i
k+l-i
К Е{
by Holder's inequality, since ^ jatj = k. The recurrence assumption yields
that each expectation above is smaller than some constant K(p, k, a, T]kp), and
we obtain the required inequality. D
We now turn to the positivity of solutions of stochastic differential equa-
equations. This issue arises often in applications (for example in mathematical
finance) when one wants to model a dynamic phenomenon where for reasons
relating to the application, the solution should always remain positive. We
begin with a very simple result, already in widespread use.

11 Eclectic Useful Results on Stochastic Differential Equations 345
Theorem 71. Let Z be a continuous semimartingale, a a continuous func-
function, and suppose there exists a (path-by-path) unique, non-exploding solution
to the equation
Xt=X0+ I a(Xs)XsdZ
f e{Xs)Xs
Jo
with Xo > 0 almost surely. Let T = ini{t > 0 : Xt = 0}. Then P(T < oo) = 0.
In words, if the coefficient of the equation is of the form xa(x) with cr(x)
continuous on [0, oo), then the solution X stays strictly positive in finite time
if it begins with a strictly positive initial condition. We also remark that in
the applied literature the equation is often written in the form
Note that if a(x) is any function, one can then write a(x) = x(p*p-), but
we then need the new function ^^p- to be continuous, well-defined for all
nonnegative x, and we also need for a unique solution to exist that has no
explosions. This of course amounts to a restriction on the function a. Since
xa(x) being Lipschitz continuous is a sufficient condition to have a unique
and non-exploding solution, we can require in turn for a to be both bounded
and Lipschitz, which of course implies that xcr(x) is itself Lipschitz.
Proof of Theorem 71. Define Tn = mi{t > 0 : Xt - \jn or Xt = Xo V n},
and note that P(Tn > 0) = 1 because P(Xq > 0) = 1 and Z is continuous.
Using Ito's formula up to time Tn we have
ln(\XTn\) = ln(X0)+ a(Xs)dZs-- a(XsJd[Z,Z}s,
Jo z Jo
and since a is assumed continuous, it is bounded on the compact set [1/n, n],
say by a constant с Since the stopping times Tn | T we see that on the event
{T < oo} the left side of the above equation tends to oo while the right side
remains finite, a contradiction. Therefore P(T < oo) = 0. ?
Theorem 71 has an analogue when Z is no longer assumed to be continuous.
Theorem 72. Let Z be a semimartingale and let a be a bounded, continuous
function such that there exists a unique non-exploding solution of
Xt = Xq +
Jo
with Xq > 0 almost surely. If in addition we have \AZS\ < ,Ут,е almost surely
for all s > 0 for some г with 0 < г < 1, then the solution X stays positive
a.s. for all t > 0.

346 V Stochastic Differential Equations
Proof. || a 11oo refers to the L°° norm of a. The proof is similar to the proof
of Theorem 71. Using Ito's formula gives
- \ f
* Jo
s<t
and let us consider the last term, the summation. We have
|Xs|-ln|Xs_| -a(Xs_)AZs}
s<t
= -У
where |i?s| < 1 by the Mean Value Theorem, applied и>-Ъу-ш, which implies
the above is
s<t
which is a convergent series for each finite time t. The only ways X can be
zero or negative is to cross 0 while continuous, which cannot happen by an
argument analogous to the proof of Theorem 71, or to jump to 0 or across into
negative values. Assume X has not yet reached (—oo, 0] at time 5—. Then we
have
\AXS\ < Xs.\a(Xs_)AZs\ < Xa.(l - e) < Xs_,
which implies that X cannot cross into (—oo,0] by jumping. Hence we have
that X must be positive on [0, oo). ?
We now turn to something completely different. In Sect. 6, Theorem 32,
we showed that an equation of the form
d Л
with Z a vector of independent Levy processes, had a (unique) solution which
was strong Markov. We now show what amounts to a converse, which is per-
perhaps surprising. That is, if one wants a solution of a stochastic differential
equation to be time homogeneous Markov, one essentially must have that the
driving semimartingale is a Levy process!

Bibliographic Notes 347
Theorem 73. Let (fi, T, F, P) be a filtered probability space with Z a cadlag
semimartingale defined on the space. Let f be a Borel function which is never
0 and is such that for every x el the equation
dXt = f{Xt-)dZt, Xo = x
has unique (strong) solution, which we denote Xх. If each of the processes Xх
is a time homogeneous Markov process with the same transition semigroup,
then Z is a Levy process.
For the proof of this theorem, we assume the reader is familiar with the
abstract theory of Markov processes, as can be found (for example) in [215].
Proof. Let fi' be the function space of cadlag functions defined on M+ with
Xt(w) = w(t), the projection operator. Let ?' be the canonical filtration and
let F't) be the shift operators. If P'x denotes the law of Xх, then we have that
(Sl',F',6't,X',P?) is a Markov process in the Dynkin (or Blumenthal-Getoor)
sense. Since / is never zero, we can write
f
Jo
f
Jo
We can therefore define on $1', relative to each law Px, the stochastic integral
z't= f f(X's_)-4x's,
Jo
and we have that the law of Z' under Px is the law of the process Z — Zo.
However Z' is also an additive functional, and hence the Markov property
implies
Z[)\T'} = E'x{g(Z's) о o't\Ft}
where we have also used that the law of Z' under Px is the law of the process
Z — Zo. This in turn implies that Z't+S — Z't is P'x independent of T't, and
hence also independent from Z'r for all r <t, and it also implies that the law
of Z'tJrS — Z't is the same as the law of Zs — Zq. Using the identity of the laws of
Z' under Px with that of the process Z — Zq once again, the result follows. ?
Bibliographic Notes
The extension of the i?p norm from martingales to semimartingales was im-
implicit in Protter [197] and first formally proposed by Emery [64]. A compre-
comprehensive account of this important norm for semimartingales can be found in

348 V Stochastic Differential Equations
Dellacherie-Meyer [46]. Emery's inequalities (Theorem 3) were first established
in Emery [64], and later extended by Meyer [174].
Existence and uniqueness of solutions of stochastic differential equations
driven by general semimartingales was first established by Doleans-Dade [51]
and Protter [198], building on the result for continuous semimartingales in
Protter [197]. Before this Kazamaki [123] published a preliminary result, and
of course the literature on stochastic differential equations driven by Brownian
motion and Lebesgue measure, as well as Poisson processes, was extensive. See,
for example, the book of Gihman-Skorohod [81] in this regard. These results
were improved and simplified by Doleans-Dade-Meyer [54] and Emery [65];
our approach is inspired by Emery [65]. Metiver-Pellaumail [161] have an
alternative approach. See also Metivier [158]. Other treatments can be found
in Doleans-Dade [52] and Jacod [103]. The approach of L. Schwartz [214] is
nicely presented in [44].
The stability theory is due to Protter [199], Emery [65], and also to
Metivier-Pellaumail [162]. The semimartingale topology is due to Emery [66]
and Metivier-Pellaumail [162], while a pedagogic treatment is in Dellacherie-
Meyer [46].
The generalization of Fisk-Stratonovich integrals to semimartingales is
due to Meyer [171]. The treatment here of Fisk-Stratonovich differential
equations is new. The idea of quadratic variation is due to Wiener [230].
Theorem 18, which is a random Ito's formula, appears in this form for
the first time. It has an antecedent in Doss-Lenglart [58], and for a very
general version (containing some quite interesting consequences), see Sznit-
man [222]. Theorem 19 generalizes a result of Meyer [171], and Theorem 22
extends a result of Doss-Lenglart [58]. Theorem 24 and its corollary is from
Ito [99]. Theorem 25 is inspired by the work of Doss [57] (see also Ikeda-
Watanabe [92] and Sussman [221]). The treatment of approximations of the
Fisk-Stratonovich integrals was inspired by Yor [238]. For an interesting ap-
application see Rootzen [211]. For more examples of solutions of stochastic dif-
differential equations with formulas, see [128].
The results of Sect. 6 are taken from Protter [196] and Cinlar-Jacod-
Protter-Sharpe [34]. A comprehensive pedagogic treatment when the Markov
solutions are diffusions can be found in either Stroock-Varadhan [220] or
Rogers-Williams [209, 210].
Work on flows of stochastic differential equations goes back to 1961 and the
work of Blagovescenskii-Freidlin [18] who considered the Brownian case. For
recent work on flows of stochastic differential equations, see Kunita [131, 133],
Ikeda-Watanabe [92] and the references therein. There are also flows results for
the Brownian case in Gihman-Skorohod [81], but they are L2 rather than al-
almost sure results. Much of our treatment is inspired by the work of Meyer [177]
and that of Uppman [224, 225] for the continuous case, however results are
taken from other articles as well. For example, the example following The-
Theorem 38 is due to Leandre [140], while the proof of Theorem 41, the non-
confluence of solutions in the continuous case, is due to Emery [68]; an alter-

Exercises for Chapter V 349
native proof is in Uppman [225]. For the general (right continuous) case, we
follow the work of Leandre [141]. A similar result was obtained by Fujiwara-
Kunita [78].
Our presentation on moment estimates follows [205] and [105], although it
dates back to [16]. A perhaps simpler treatment using random measures can
be found in [105]. See also [106]. The results on when solutions of stochastic
differential equations are always positive is well known, at least in the Brown-
ian case, but a reference is not easily found in the literature. The general case
may be new. Theorem 73 is from [108].
Exercises for Chapter V
Exercise 1. Let X be a solution of the stochastic differential equation
dXt = X?dBt + Xfdt
with initial condition Xo ф 0. Show that X is defined on [0, T), where T =
inf{t > 0 : Bt = 1/-Xo}. Moreover show that X is the unique solution on
[0,T). Finally show that X explodes at T.
Exercise 2. Let cr(x) = \f\x~\- Note that a is Lipschitz continuous everywhere
except at x = 0. Let X be a solution of the equation
dXt = a(Xt)dBt + ^dt
with initial condition Xo. Find an explicit solution of this equation and de-
determine where it is defined.
Exercise 3. Let X be the unique solution of the equation
dXt = a(Xt)dBt + b(Xt)dt
with Xo given, and a and b both Lipschitz continuous. Let the generator of
the (time homogeneous) Markov process X be given by
Af(x)= lim */(*)-/(*>
J v ; t-»o,t>o t
where Ptf(x) equals the transition function22 Ps(Xo,f) when Xo = x almost
surely. A is called the infinitesimal generator of X. Use Ito's formula to
show that if / has two bounded continuous derivatives (written / e C2), then
the limit exists and A is denned. In this case / is said to be in the domain of
the generator A.
This notation is defined in Sect. 5 of Chap. I on page 35.

350 Exercises for Chapter V
Exercise 4. (Continuation of Exercise 3.) With the notation and terminology
of Exercise 3, show that the infinitesimal generator of X is a linear partial
differential operator for functions / G C\ given by
Exercise 5. Let / 6 C|, let a and b be Lipschitz continuous, let X be the
unique solution of the stochastic integral equation
Xt=X0+ f a(Xs)dBs + f b(Xs)ds, (*)
Jo Jo
and let A denote its infinitesimal generator. If Xo = x, where x ? M, prove
Dynkin's expectation formula for Markov processes in this case:
E{f(Xt) - /(Xo) - f Af(Xs)ds} = 0. (**)
Jo
Exercise 6. As a continuation of Exercise 5, show that
(Ptf)(y) ~ f(y) - [\PsAf)(y)ds = 0
Jo
for у e M, and that limsj0 PsAf = Af for (for example) all / in C°° with
compact support.
Exercise 7. As in Exercise 5, extend Dynkin's expectation formula by show-
showing that it still holds if one replaces in equation (**) the integration upper
limit t with a bounded stopping time T, and use this to verify Dynkin's for-
formula in this case. That is, show that if / e C\ and a and b are both Lipschitz,
and X is a solution of equation (*) of Exercise 5, then
fT
f(Xr) — f(Xo) — / Af(Xs)ds is a local martingale.
Jo
Exercise 8. Let Zbea Levy process with Levy measure v and let X satisfy
the equation
Xt=x+ [ f(Xe.)dZe
Jo
where / is Lipschitz continuous. The process X is taken to be d-dimensional,
and Z is n-dimensional, so / takes its values in Rd x Mn. We use (column)
vector and matrix notation. The initial value is some given x ? M.d. Let Px
denote the corresponding law of X starting at x. Show that if g € C°° with
compact support on Rd, then

Exercises for Chapter V 351
Ag(x) =\ ? (^
J
u{dy) {g(x + f(x)y) - g(x) - Vg(x)f(x)),
where Vy is a row vector and b is the drift coefficient of Z, * denotes transpose,
and A is of course the infinitesimal generator of X.
Exercise 9. In the setting of Exercise 8 show that
Co) - / Ag(X,
Jo
g{Xt) - g(X0) — I Ag(Xs)ds is a local martingale
Jo
for each law Px. Show further that it is a local martingale for initial laws
of the form P^, where P^ is given by: for Л с П measurable, Р^(Л) =
JRd Px(A)^(dx) for any probability law /i on Md.
Exercise 10. Let Z be an arbitrary semimartingale, and let To = 0 and
(Tn)n>i denote the increasing sequence of stopping times of the jump times of
Z when the corresponding jump size is larger than or equal to 1 in magnitude.
Define en = sign(AZ<rn) with во = 1- For a semimartingale Y define Un(Y)t
to be the solution of the exponential equation
for Tn < t < Tn+i. Show that the (unique) solution of the equation
*\Xt.\dZ.
о
is given by Xt = ?„>о X?l[TntTn+l)(t), where X? = XTnUn(enZ)t, for Tn <
t < Tn+i.
Exercise 11. Show that if M is an L2 martingale with AMS > — 1, and
(M, M)oo is bounded, then ?(MHO > 0 a.s. and it is square integrable.
(See [147].)
Exercise 12. Let M be a martingale which is of integrable variation. Suppose
also that the compensator of the process X^s<t I^-Wsl is bounded. Show that
the local martingale ?{M) is also of integrable variation.
Exercise 13. Let Z be a semimartingale with decomposition Z = L + V
where ? is a local martingale with ALS ф -1 for all finite s, and V has paths
of finite variation on compacts. Let
Show that ?{Z) = ?{L)?{U). (See [157].)

352 Exercises for Chapter V
* Exercise 14 (Memin's criterion for exponential martingales). Let
M be a local martingale, let Jt = Y2a<t Д-^Л{|ДМа|>А}! an<l let ^t — Jt~ Jt
and Nt — Mt - Lt. (Both L and N are of course also local martingales.) Let
At = [M,M]c+
0<s<t 0<s<t
and assume that the compensator of A is bounded. Show that this implies
that ?{M) is a uniformly integrable martingale. {Note: This is related to
Kazamaki's and Novikov's criteria, but without the assumption of path con-
continuity. Also see [156].)
Exercise 15 (Gronwall's inequality). . Let Cj be a cadlag increasing pro-
process and suppose 0 < At < а + /0 As~dCs for t > 0. Show that At < aeCi,
each t > 0. (Note: The classical Gronwall inequality is usually stated with
Ct =t. Also see [85].)
The next five exercises outline the Metivier-Pellaumail method for showing
the existence and uniqueness of solutions of stochastic differential equations.
* Exercise 16. Let M be an L2 martingale. Show that for any predictable
stopping time T, E{(E{AMT|J"T-}J} < ?{(M,M)T_}. Show also that for
any totally inaccessible stopping time T, we have Е{(Е{^Мт\Тт-}J} <
E{(M,M)t}- Conclude that for an arbitrary stopping time T we have
Е{{Е{&Мт\?т-}?} < Е{{М,М)т-}.
Exercise 17 (Metivier-Pellaumail inequality). Let M be an L2 mar-
martingale and let M*_ = sups<t \MS\. Show that for any stopping time T we
have
(Hint: Use Exercise 16).
Exercise 18. Let M be a locally square integrable local martingale. Show
that for any stopping time T we have
E{sup(Ms - M0J} < 4?'{(M,M)T_ + [M, M]T_}.
s<T
Exercise 19 (extended Gronwall's inequality). Let A and С be two
increasing processes with A adapted and E{AOO} < oo, and С such that
0 < Coo < к everywhere for some constant k. Suppose that for every stopping
time T and for some positive constant a we have
T-
Then ElAoo} < aek. (Hint: Use Lebesgue's change of time lemma, together
with Gronwall's inequality of Exercise 15. Also see [110, page 576].)

Exercises for Chapter V 353
Exercise 20. Use Exercises 16 through 19 as needed to show, via a Picard
iteration method, that a unique solution exists to the stochastic differential
equation
dXt = f(X,-)dZa
where / is Lipschitz continuous and Z is an arbitrary semimartingale.
Exercise 21. Show that there are an infinite number of solutions (on the
same probability space) to the equation
= 3 / xhs + 3 / Xi dBs, Xo = 0,
Jo Jo
where Б is a standard Brownian motion. (See [121, page 293].)
*Exercise 22. Let a satisfy a weakening of the Lipschitz condition of the
form \o{y) — &(x)\ < k(\x — y\) for all i, j ? I, with J^ ,K$du = oo and к
increasing with к@) = 0. (Such a condition is a variant of a standard condition
for uniqueness of solutions in the theory of ordinary differential equations; see
for example [35, page 60]. Let b be Lipschitz continuous. Show that there
exists a unique solution of the equation
dXt = b(Xt)dt + a{Xt)dBt
where В is standard Brownian motion. (See [232].)
Exercise 23. Let a and b be Lipschitz continuous and let X be the unique
solution of equation (*) of Exercise 5. Show that 2xb(x) + cr(xJ < с + k\x\2,
where с and k are positive constants. Use Ito's formula on X2 to show the
following moment estimate: E{X2} < (E{X^} + ct)ekt. (See [135].)
*Exercise 24 (Euler method of approximation). Let a and b be con-
continuous and such that a unique path-by-path solution X of equation (*) of
Exercise 5 exists. Let (?fc)fc>i be i.i.d. with mean zero and variance r2. Let
Xo be independent of (?&)*:>i an<i ^ Xk be defined inductively by
Ь(Хк)
n
Define Xn(t) = X[nt] where [nt] denotes the integer part of nt. Let Bn(t) =
7Й ЕЙ &> Vn(t) = If1, and finally observe that
Xn(t)=X{0)+ f a(Xn(s-))dBn(s) + f b{Xn{s-))dVn(s).
Jo Jo
By Donsker's Theorem we know that (Bn,Vn) converges weakly (in distri-
distribution) to (tB, V), where В is a standard Brownian motion and V(t) = t.
Show that Bn is a martingale, and that Xn converges weakly to X. (See [136]
and [137].)

354 Exercises for Chapter V
*Exercise 25. Let Zbea Levy process which is a square integrable martin-
martingale. Let / be continuously differentiable with a bounded derivative and let
Xх be the unique solution of
Xx = x + f f{Xx_)dZs.
Jo
/о
Show that Xх is also a square integrable martingale.
*Exercise 26. In the framework of Exercise 25, let X'tx be the solution of
ft
Y x _ I , / fi/ ух \ у x jy
Jo
Show that X x is also a square integrable martingale.
**Exercise 27. In the framework of Exercises 25 and 26, for each measurable
function g with at most linear growth, set
Ptg(x) = E{g(Xx)} and Qt9(x) = E{g(Xf)X'tx}.
Assume that / is infinitely differentiable, and that g is twice differentiable,
bounded and both of its first two derivatives are bounded. Show that the
function (t,x) i-> Ptg(x) is twice differentiable in x and once differentiable in
t, that all the partial derivatives are continuous in (t,x), and further that
д " ' ~ Qtg'{x).

VI
Expansion of Filtrations
1 Introduction
By an expansion of the filtration, we mean that we enlarge the filtration
(•^t)t>o to get another filtration (Ht)t>o such that the new filtration satisfies
the usual hypotheses and Tt С Ht, each t > 0. There are three questions we
wish to address: A) when does a specific, given semimartingale remain a semi-
martingale in the enlarged filtration; B) when do all semimartingales remain
semimartingales in the enlarged filtration; C) what is a new decomposition of
the semimartingale for the new filtration.
The subject of the expansion of filtrations began with a seminal paper
of K. Ito in 1976 (published in [100] in 1978), when he showed that if В
is a standard Brownian motion, then one can expand the natural filtration
(Jrt)t>o of В by adding the ст-algebra generated by the random variable B\
to all Tt of the filtration, including of course To- He showed that В remains
a semimartingale for the expanded filtration, he calculated its decomposition
explicitly, and he showed that one has the intuitive formula
f HsdBs = f BxHsdBa
Jo Jo
where the integral on the left is computed with the original filtration, and
the integral on the right is computed using the expanded filtration. Obviously
such a result is of interest only for 0 < t < 1. We will establish this formula
more generally for Levy processes in Sect. 2.
The second advance for the theory of the expansion of filtrations was the
1978 paper of M. Barlow [7] where he considered the problem that if L is a
positive random variable, and one expands the filtration in a minimal way to
make L a stopping time, what conditions ensure that semimartingales remain
semimartingales for the expanded filtration? This type of question is called
progressive expansion and it is the topic of Sect. 3.

356 VI Expansion of Filtrations
2 Initial Expansions
Throughout this section we assume given an underlying filtered probability
space (ft,T, (Tt)t><a-,P) which satisfies the usual hypotheses. As in previous
chapters, for convenience we denote the filtration [Tt)t><a by the symbol F.
The most elementary result on the expansion of filtrations is due to Jacod
and was established in Chap. II (Theorem 5). We recall it here.
Theorem 1 (Jacod's Countable Expansion). Let A be a collection of
events in T such that if Aa, Ар е A then Aa П Ар = ф, а ф C. Let 7it be the
filtration generated by Tt and A. Then every ((^)г>о,Р) semimartingale is
an ((Ht)t>o,P) semimartingale also.
We also record a trivial observation as a second elementary theorem.
Theorem 2. Let X be a semimartingale with decomposition X = M + A
and let Q be a a-algebra independent of the local martingale term M. Let Ш
denote the filtration obtained by expanding F with the one a-algebra Q (that
is, Ht = JiV Q, each t > 0 and Ht = Ht+). Then X is апШ semimartingale
with the same decomposition.
Proof. Since the local martingale M remains a local martingale under H, the
theorem follows. D
We now turn to Levy processes and an extension of Ito's first theorem. Let
Z be a given Levy process on our underlying space, and define H = (Ht)t>o
to be the smallest filtration satisfying the usual hypotheses, such that Z\ is
Ho measurable and Tt С Ht for all t > 0.
Theorem 3 (Ito's Theorem for Levy Processes). The Levy process Z
is an Ш semimartingale. If moreover E{\Zt\} < oo, all t > 0, then the process
f
t/\\ 17 17
CIS
1 -s
is an Ш martingale on [0, oo).
Proof. We begin by assuming E{Z?} < oo, each t > 0. Without loss of gener-
generality we can further assume E{Zt} = 0. Since Z has independent increments,
we know Z is an F martingale. Let 0<s<t<lbe rationale with s = j/n
and t = k/n. We set
Then Zx - Zs = YSZj Yi and Zt-Za = Ylilj Yi- The random variables
are i.i.d. and integrable. Therefore

2 Initial Expansions 357
fc-l n-l , . n-1
E{Zt — ZS\Z\ — Zs] —
The independence of the increments of Z yields E{Zt — ZS\HS} — E{Zt —
ZS\ZX - Zs}; therefore E{Zt - ZS\HS) = j5f (^i - Zs) for all rationals, 0 <
s < t < 1. Since Z is an F martingale, the random variables (Zt)o<t<i are
uniformly integrable, and since the paths of Z are right continuous, we deduce
E{Zt - Zs\ns} = jE^{Zi - Za) for all reals, 0 < s < t < 1. By Fubini's
Theorem for conditional expectations the above gives
E{Mt - MS\HS} = E{Zt - ZS\HS} - I --^E{Zl - Zu\Hs}du
Ju
= 0.
There is a potential problem at t = 1 because of the possibility of an explo-
explosion. Indeed this is typical of initial enlargements. However if we can show
E{f0 \Z^ ds} < oo this will suffice to rule out explosions. By the sta-
tionarity and independence of the increments of Z we have E{\Z\ — Zs\} <
E{(Zi — ZsJ}% < a(l — s)i for some constant a and for all s, 0 < s < 1.
Therefore E{f^ |z^lfa|rfs} < af* ::^^ds < oo. Note that if t > 1 then
Tt — Wt, and it follows that M is a martingale. Since Zt = Mt + JQ '~^? ds
we have that Z is a semimartingale.
Next suppose only that -E{|Zt|} < oo, t > 0 instead of Z being in L2 as
we assumed earlier. We define
jl ^~* Л 7 1 A T^ \~* Л 7 1
0<s<t 0<s<t
(Since Z has cadlag paths a.s. for each u> there are at most a finite number of
jumps bigger than a fixed size on each compact time set; hence each Jl is finite
a.s.) By the results of Chap. I on Levy processes we have that Yt = Zt — J^ + J2
is a Levy process with bounded jumps, and hence it is square integrable.
Additionally, the processes Y, J1, and J2 are jointly independent.
Combining these facts with the preceding proof yields
Nt=Yt-
— s
is an H martingale, where H is the smallest right continuous filtration ob-
obtained by expanding F with (Yx,Jl,J2). Note that H С Н'. The same argu-
argument, plus the observation that since J* does not jump at time t = 1 and is

358 VI Expansion of Filtrations
therefore constant in a (random) neighborhood of 1, which in turn yields that
Jo ' \Za ds < oo a.s., shows that (Ji~~f0 \ZSS ds)t>o is a local martingale for
f I 117*7*1
H . Moreover it is a martingale for H as soon as E{f0 \ZS ds} < oo. But
the function t i—> E{ Jt*} = atf for all i by the stationarity of the increments.
Hence
/•1 I Ji Ji /¦! ji ji f
E{ \J^Lds} = m ±_Lids}l = l
Jo i ~~s Jo l ~ s Jo
1 1 - s J
11 _ „
is
ds
о 1~s
— Bj <C 00.
Since У, J1, and J2 are all independent, we conclude that M is an H' martin-
martingale. Since M is adapted to H, by Strieker's Theorem it is also an H martingale,
and thus Z is an H semimartingale.
Finally we drop all integrability assumptions on Z. We let
Jt = У~] ^^s^-{\az3\>i} and also Xt = Zt — j\.
0<s<t
Then X is also a Levy process, and since X has bounded jumps it is in Lp
for all p > 1, and in particular ?{X2} < oo, each t > 0. Let Ш(Х\) denote
F expanded by the adjunction of the random variable Xx. Let К = H(Xi) V
M(Jl), the filtration generated by H(Xi) and H(J11). Then H(Zi) с К. But
X is a semimartingale on Ш(Х\), and since J1 is independent of X we have
by Theorem 2 that X is а (К, Р) semimartingale. Therefore by Strieker's
Theorem, X is an (H, P) semimartingale; and since J\ is a finite variation
process adapted to H we have that Z is an (H, P) semimartingale as well. ?
The most important example of the above theorem is that of Brownian
motion, which was Ito's original formula. In this case let H = H(J3i), and we
have as a special case the H decomposition of Brownian motion:
d / d f ^1 — Bs f Bi — Bs f Bi —
Bt = (Bt- / — ds) + / — ds = Pt+ —
Jo 1-s Jo 1 - s Jo 1 -
B B f B — Bs
s
Note that the martingale /3 in the decomposition is continuous and has
[j3, /3]t — t, which by Levy's theorem gives that /3 is also a Brownian mo-
motion. A simple calculation gives Ito's original formula, for a process H which
is F predictable:
( [ B\~Bs
HsdBs = Si / HsdPs+Вг [ Н3Щ—
Jo Jo L ~~
= / B1Hsdp3+ f ВгНэЦ^
Jo Jo l ~
= [ BxHadBa
Jo
^ds
s

2 Initial Expansions 359
where since the random variable B\ is Ho measurable, it can be moved inside
the stochastic integral. We can extend this theorem with a simple iteration;
we omit the fairly obvious proof.
Corollary. Let Z be a given Levy process with respect to a filtration F, and
let 0 = to < t\ < ¦ ¦ ¦ < tn < oo. Let H denote the smallest filtration satis-
satisfying the usual hypotheses containing F and such that the random variables
Ztl,...,Ztn are all Ho measurable. Then Z is an H semimartingale. If we
have a countable sequence 0 = to < t\ < ¦ • ¦ < tn < ¦ ¦ ¦, we let r = supn tn,
with H the corresponding filtration. Then Z is an H semimartingale on [0, r).
We next give a general criterion (Theorem 5) to have a local martingale
remain a semimartingale in an expanded filtration. (Note that a finite varia-
variation process automatically remains one in the larger filtration, so the whole
issue is what happens to the local martingales.) We then combine this theo-
theorem with a lemma due to Jeulin to show how one can expand the Brownian
filtration. Before we begin let us recall that a process X is locally integrable
if there exist a sequence of stopping times (Tn)n>i increasing to сю a.s. such
that Е{\Хтп1{тп>0}\} < °° f°r eacn n. Of course, if Xq = 0 this reduces to
the condition -Е{|-Хт„|} < сю for each n.
Theorem 4. Let M be an? local martingale and suppose M is a semimartin-
semimartingale in an expanded filtration Ш. Then M is a special semimartingale in H.
Proof. First recall that any local martingale is a special semimartingale. In
particular the process Mt* = sups<t \MS\ is locally integrable (see Theorem 33
of Chap. Ill), and this of course remains locally integrable in the expanded
filtration H, since stopping times remain stopping times in an expanded fil-
filtration. Since M is an H semimartingale by hypothesis, it is special because
Mt* is locally integrable (see Theorem 34 of Chap. III). ?
Theorem 5. Let M be an? local martingale, and let H be predictable such
that f0 Hgd[M,M]s is locally integrable. Suppose Ш is an expansion of? such
that M is an Ш semimartingale. Then M is a special semimartingale in Ш
and let M = N + A denote its canonical decomposition. The stochastic inte-
integral process (Jo HsdMs)t>o is an Ш semimartingale if and only if the process
(f0 HsdAs)t>o exists as a path-by-path Lebesgue-Stieltjes integral a.s.
Proof. First assume that -E{/0°° H%d[M, M]s} < сю, which implies that Я • M
(where H ¦ M denotes the stochastic integral process (Jo HsdMs)t>o) is a
square integrable martingale, and hence by Theorem 4 it is a special semi-
semimartingale in the H filtration. Let M = N + A be the canonical H de-
decomposition of M, and it follows that H ¦ M = H • N + H ¦ A is the
canonical H decomposition of H ¦ M. By the lemma following Theorem 23
of Chap. IV we have E{f™ Hfd[N,N]s} < E{f™ H*d[M,M]s} < сю and
E{$™H2sd[A,A]s} < ?{/0°°H2sd[M,M\s) < сю. This allows us to conclude
that H is (W2,M) integrable, calculated in the H filtration.

360 VI Expansion of Filtrations
To remove the assumption E{f?° H%d[M, M]s} < oo, we only need to
recall that H ¦ M is assumed to be locally square integrable, and thus take M
stopped at a stopping time Tn that makes H ¦ M square integrable, and we
are reduced to the previous case. ?
Theorem 6 (Jeulin's Lemma). Let R be a positive, measurable stochas-
stochastic process. Suppose for almost all s, Rs is independent of Ts; and the law
(or distribution) of Rs is fj, which is independent of s, with //({0}) = 0 and
Jo°° xfi(dx) < oo. If a is a positive predictable process with fQ asds < oo a.s.
for each t, then the two sets below are equal almost surely:
POO POO
{ / Rsasds < oo} = { / asds < сю} a.s.
Jo Jo
Proof. We first show that {/0°° Rsasds < сю} С {/0°° asds < сю} a.s. Let A
be an event with P(A) > 0, and let J = 1a, Jt = E{J\!Ft}, the cadlag version
of the martingale. Let j = inft Jt. Then j > 0 on {J = 1}. We have a.s.
POO
E{lARt\Tt}= E{lAl{Rt>u}\Tt}du. (*)
Jo
Consider next
E{lAl{Rt>u}\Ft} = E{(lA - l{Rt<u))+\Ft) > (E{(lA - 1{Дг<и})|^})+
by Jensen's inequality, and this is equal to (E{lA\!Ft} - м@> и])+> where fj, is
the law of R4. Continuing with (*) we have
> Г(Е{1А\Ъ} - M@, u])+du = Ф{Е{1А\Ъ}) =
Jo
where Ф(х) = J^°(x — (j,@, u])+du. Note that Ф is increasing and continuous
on [0,1], and Ф > 0 on @,1] because //({0}) = 0.
Choose An = {/0°° Rsasds < n} for the event A in the foregoing. Then
POO /»OO
00 > nP(An) > E{lAn / Rsasds) = E{ / lAnRsasds}
Jo Jo
= E{ E{lAnRs\Fs}asds}>E{ 0(Je)aads},
Jo Jo
which implies that /0°° Ф(Js)asds < 00 a.s. But
POO /»0O
/ Ф(Js)asds > Ф{э) I asds,
Jo Jo
POO
OO >
and therefore Jo°° asds < 00 a.s. on {J = 1}. That is, Jo°° asds < 00 a.s. on
An. Letting n tend to 00 through the integers gives {/0°° Rsasds < 00} С

2 Initial Expansions 361
{/o°° asds < 00} a.s. We next show the inverse inclusion. Let Tn = inf{? > 0 :
/„ asds > n}. Then Tn is a stopping time, and n > E{f0 " asds}. Moreover
E{ f
Jo
Rsasds} = L
Jo
= E{ f " E{E{Rs\fs}\FsATJasds} = E{ f " E{Rs}asds}
Jo Jo
= aE{ / asds} <
Jo
an < сю
where a is the expectation of Rs, which is finite and constant by hypothesis.
Therefore Jo " Rsasds < сю a.s. Let ш be in {Jo asds < сю}. Then there exists
an n (depending on us) such that Tn(u>) = сю. Therefore we have the inclusion
{/0°° Rsasds < сю} Э {/0°° asds < сю} a.s. and the proof is complete. ?
We are now ready to study the Brownian case. The next theorem gives
the main result.
Theorem 7. Let M be a local martingale defined on the standard space of
canonical Brownian motion. Let H be the minimal expanded filtration con-
containing B\ and satisfying the usual hypotheses. Then M is апШ semimartin-
gale if and only if the integral Jo ,*_s\d\M,B]s\ < сю a.s. In this case
^t — /0 B\Zfs d[M,B]s is an Ш local martingale.
Proof. By the Martingale Representation Theorem we have that every F lo-
local martingale M has a representation Mt = Mo + /0 HsdBs, where H is
predictable and f0 H^ds < сю a.s., each t > 0. By Theorem 5 we know
that M is an H semimartingale if and only if J* \HS\ \Zf ds is finite
a.s., 0 < t < 1. We take as = J^L, and Rs = l{s<1} l^~_^"l. Then
/o \Ha\^B\Zf'^ds = /0 asRsds, which is finite only if /0 asds < сю a.s. by
Jeulin's Lemma. Thus it is finite only if JQ -L=d=ds < сю a.s. But
,J3]S= /" -jl==d[\H\-B,B
Jo
and this completes the proof. ?
As an example, let 1/2 < a < 1, and define

362 VI Expansion of Filtrations
Then H is trivially predictable and also /0 H^ds < oo. However Jo Hs i—rfs
is divergent. Therefore M — H ¦ В is an F local martingale which is not an H
semimartingale, by Theorem 7, where of course Ш = M(Bi). Thus we conclude
that not all F local martingales (and hence a fortiori not all semimartingales)
remain semimartingales in the H filtration.
We now turn to a general criterion that allows the expansion of filtration
such that all semimartingales remain semimartingales in the expanded filtra-
filtration. It is due to Jacod, and it is Theorem 10. The idea is surprisingly simple:
recall that for a cadlag adapted process X to be a semimartingale, if Hn is a
sequence of simple predictable processes tending uniformly in (t, us) to zero,
then we must have also that the stochastic integrals Hn ¦ X tend to zero in
probability. If we expand the filtration by adding a a-algebra generated by
a random variable L to the F filtration at time 0 (that is, cr{L} is added to
fo), then we obtain more simple predictable processes, and it is harder for
X to stay a semimartingale. We will find a simple condition on the random
variable L which ensures that this condition is not violated. This approach is
inherently simpler than trying to show there is a new decomposition in the
expanded filtration.
We assume that L is an (E, ?)-valued random variable, where E is a stan-
standard Borel space1 and ? are its Borel sets, and we let M(L) denote the smallest
filtration satisfying the usual hypotheses and containing both L and the orig-
original filtration F. When there is no possibility of confusion, we will write H
in place of H(L). Note that if Y e Н$ = Tt V a{L}, then Y can be written
Y(lo) = G(lo,L(lo)), where (ui,x) >—> G(co,x) is an Tt®? measurable function.
We next recall two standard theorems from elementary probability theory.
Theorem 8. Let Xn be a sequence of real-valued random variables. Then Xn
converges to 0 in probability if and only if limn^oo E{mm(l, |Xn|)} = 0.
A proof of Theorem 8 can be found in textbooks on probability (see for
example [109]).
We write 1Л \Xn\ for min(l, \Xn\). Also, given a random variable L, we let
Qt(u>,dx) denote the regular conditional distribution of L with respect
to Tt, each t > 0. That is, for any Л € ? fixed, Qt(-,A) is a version of
E{l{L€A}\^t}, and for any fixed w, Qt(u>, dx) is a probability on ?. A second
standard elementary result is the following.
Theorem 9. Let L be a random variable with values in a standard Borel
space. Then there exists a regular conditional distribution Qt(co,dx) which is
a version of
For a proof of Theorem 9 the reader can see, for example, Breiman [23,
page 79].
1 (E, ?¦) is a standard Borel space if there is a set Г € Б, where Б are the Borel
subsets of R, and an injective mapping ф : E —> Г such that ф is ? measurable and
ф~г is Б measurable. Note that (Rn, Bn) are standard Borel spaces, 1 < n < oo.

2 Initial Expansions 363
Theorem 10 (Jacod's Criterion). Let L be a random variable with values
in a standard Borel space (E,?), and let Qt(co,dx) denote the regular condi-
conditional distribution of L given Tt, each t > 0. Suppose that for each t there
exists a positive a-finite measure r\t on (E,?) such that Qt(u>,dx) <gc rjt(dx)
a.s. Then every F semimartingale X is also an M(L) semimartingale.
Proof. Without loss of generality we assume Qt(u>,dx) <?i rjt{dx) surely. Then
by Doob's Theorem on the disintegration of measures there exists an ? ® Ft
measurable function qt(x,co) such that Qt(co,dx) = qt(x,co)r]t(dx). Moreover
since E{JEQt(;dx)} = E{E{l{Y€E}\Ft}} = P(Y € E) = 1, we have
E{ f Qt(;dx)} = E{[ qt(x,w)Tit(dx)} = [ E{qt{x,u)}rit(dx) = 1.
Je Je Je
Hence for almost all x (under r]t{dx)), we have qt{x, •) € Lx{dP).
Let X be an F semimartingale, and suppose that X is not an H(L) semi-
semimartingale. Then there must exist а и > 0 and an e > 0, and a sequence Hn
of simple predictable processes for the H filtration, tending uniformly to 0 but
such that infn E{1 A \Hn ¦ X\} > s. Let us suppose that tn < u, and
n-l
i=0
with Jf € Tu Vct{L}. Hence Jf has the form д^{ш,Ь{ш)), where (u>,x) t->
gi(w,x) is Tti ®? measurable. Since Hn is tending uniformly to 0, we can
take without loss \Hn\ < 1/n, and thus we can also assume that |<^| < 1/n.
We write
n-l
i=0
and therefore (х,ш) ^ Щ'х(ш) and (х,ш) ^ (Яп-Х • X)t(w) are each ? <g> Tu
measurable, 0 < t < u. Moreover one has clearly Hn-X = Hn<L-X. Combining
the preceding, we have
A \Hn -XU\} = E{[(IA \Hn'x ¦ XU\)QU(-,dx)}
Je
A \Hn'x ¦ Xu\)qu(-, dx)Vu(dx)}
= f E{{1 A \Hn<* ¦ Xu\)qu(-,
Je
where we have used Fubini's Theorem to obtain the last equality. However the
function hn(x) = E{{1 A \Hn'x ¦ Xu\)qu(-,x)} < E{qu(-,x)} € Ll{d-qu), and
since hn is non-negative, we have

364 VI Expansion of Filtrations
lim
im E{\ A \Hn ¦ Xu\) = lim / E{{1 A \Hn ¦ Xu\)qu(; xft^dx)
—изо n—изо Jv?
= / lim E{{1 A\Hn ¦ X^q^-
JEn->oo
by Lebesgue's Dominated Convergence Theorem. However qu{-,x) € Ll(dP)
for a.a. x (under (dr]u)), and if we define dR = cqu(-, x)dP to be another proba-
probability, then convergence in P-probability implies convergence in Д-probability,
since R<.P. Therefore lim^^ ER{A A \Hn>x ¦ Xu\)} = 0 as well, which im-
implies
0 = lim -ER{A A \Hn'x ¦ Xu\)}
п—юо с
= EP{(lA\Hn>x-Xu\)qu(;x)}
for a.a. x (under (dr]u)) such that qu(-,%) & Ll{dP). Therefore the limit of
the integrand in (*) is zero for a.a. x (under (dj]u)), and we conclude
lim ?{AЛ|Яп-Хи|)}=0,
n—»oo
which is a contradiction. Hence X must be a semimartingale for the filtration
H°, where H° = Tt V a{L}. Let X — M + A be a decomposition of X under
H°. Since H° need not be right continuous, the local martingale M need
not be right continuous. However if we define Mt = Mt if t is rational; and
liniux^ueQ Mu if t is not rational; then Mt is a right continuous martingale
for the filtration H where Tit — f]u>t ^u- Letting At = Xt —Mt, we have that
Xt = Mt+At is an H decomposition of X, and thus X is an H semimartingale.
?
A simple but useful refinement of Jacod's Theorem is the following where
we are able to replace the family of measures щ be a single measure 77.
Theorem 11. Let L be a random variable with values in a standard Borel
space (E,?), and let Qt(u>,dx) denote the regular conditional distribution of L
given Tt, each t > 0. Then there exists for each t a positive a-finite measure
rjt on (E, 8) such that Qt(u),dx) <C T]t(dx) a.s. if and only if there exists one
positive a-finite measure rj(dx) such that Qt(co,dx) <gc rj(dx) for all со, each
t > 0. In this case r\ can be taken to be the distribution of L.
Proof. It suffices to show that the existence of r\t for each t > 0 implies
the existence of 77 with the right properties; we will show that the dis-
distribution measure of L is such an 77. As in the proof of Theorem 10 let
(x,lo) н-> qt(x,u>) be ?¦ ® Tt measurable such that Qt(i0,dx) = qt(x,io)r]t(dx).
Let at(x) = E{qt(x,u>)}, and define
I 0, otherwise.

2 Initial Expansions 365
Note that at{x) = 0 implies qt(x, •) = 0 a.s. Hence, qt(x,ui) = rt(x,co)at(x)
a.s.; whence rt(:r,u>)at(:r)?7t(d:r) is also a version of Qt(u>, dx).
Let r/ be the law of L. Then for every positive ? measurable function g we
have
Jg{x)r,{dx) = E{g(L)} = E{Jg(x)Qt(-,dx)}
g{x)qt{x,-)r)t{dx)}
i
= f g(x)E{qt(x,-)}r]t(dx)
Je
= / g(x)at(x)r}t(dx)
Je
from which we conclude that at(x)r]t(dx) = r](dx). Hence, Qt(co,dx) =
rt(u>,x)r](dx), and the theorem is proved. D
We are now able to re-prove some of the previous theorems, which can be
seen as corollaries of Theorem 11.
Corollary 1 (Independence). Let L be independent of the filtration F.
Then every F semimartingale is also an M(L) semimartingale.
Proof. Since L is independent of !Ft, E{g(L)\Jrt\ = E{g(L)} for any bounded,
Borel function g. Therefore
E{g(L)\Ft) = [ Qt(u,,dx)g(x) = [ f](dx)g(x) = E{g(L)},
Je Je
from which we deduce Qt(w, dx) = r]{dx), and in particular Qt(w, dx) < rj(dx)
a.s., and the result follows from Theorem 11. ?
Corollary 2 (Countably-valued random variables). Let L be a random
variable taking on only a countable number of values. Then every F semi-
semimartingale is also an M(L) semimartingale.
Proof. Let L take on the values ai, a%, аз,.... The distribution of L is given
by r](dx) = YliLi P{L — ai)?ai(dx), where eai(dx) denotes the point mass at
O!j. With the notation of Theorem 11, we have that the regular conditional
distribution of L given Tt, denoted Qt(w,dx), has density with respect to r\
given by
The result now follows from Theorem 11. D

366 VI Expansion of Filtrations
Corollary 3 (Jacod's Countable Expansion). Let Л = (A1;A2,...) be
a sequence of events such that А* П A,: = 0, i ф j, all in T, and such that
USi Ai = fl. Let H be the filtration generated by F and Л, and satisfying
the usual hypotheses. Then every F semimartingale is an H semimartingale.
Proof. Define L = Y^li 2~*1л<- Then И = M{L) and we need only to apply
the preceding corollary. ?
Next we consider several examples.
Example (Ito's example). We first consider the original example of Ito,
where in the standard Brownian case we expand the natural filtration F with
a{Bi}. We let H denote M(Bi). We have
= E{g{Bl -Bt + Bt)\Tt) = Jg(x
where r]t(dx) is the law of B\ - Bt and where we have used that Bi — Bt is in-
independent of Tt- Note that T]t(dx) is a Gaussian distribution with mean 0 and
variance A — t) and thus has a density with respect to Lebesgue measure. Since
Lebesgue measure is translation invariant this implies that Qt(u>,dx) <§C dx
a.s., each t < 1. However at time 1 we have E{g(B\) \!F\} = g{B\), which yields
Qi(u>, dx) — e{Bi(ui)}{dx), which is a.s. singular with respect to Lebesgue mea-
measure. We conclude that any F semimartingale is also an H(J3i) semimartingale,
for 0 < t < 1, but not necessarily including 1. This agrees with Theorem 7
which implies that there exist local martingales in F which are not semimartin-
galesinll(Si).
Our next example shows how Jacod's criterion can be used to show a
somewhat general, yet specific result on the expansion of filtrations.
Example (Gaussian expansions). Let F again be the standard Brownian
filtration satisfying the usual hypotheses, with В a standard Brownian motion.
Let V = Jo°° g(s)dBs, where Jo°° g(sJds < сю, д a deterministic function. Let
a = inf {t > 0 : Jt°° g(sJds = 0}. If h is bounded Borel, then as in the previous
example
E{h(V)\Ft} = E{h( f g(s)dBs + f g(s)dBs)\Tt}
Jo Jt
= j h(f g{s)dBs + x)r]t{dx),
where rjt is the law of the Gaussian random variable /t°° g(s)dBs. If a = сю,
then r]t is non-degenerate for each t, and r\t of course has a density with
respect to Lebesgue measure. Since Lebesgue measure is translation invariant,
we conclude that the regular conditional distribution of Qt(u, dx) of V given
Tt also has a density, because

2 Initial Expansions 367
Qt(w,h) = E{h{V)\Tt} = Jh(J g(s)dBs+x)r,t(dx).
Hence by Theorem 10 we conclude that every F semimartingale is an M(V)
semimartingale.
Example (expansion via the end of a stochastic differential equa-
equation). Let В be a standard Brownian motion and let X be the unique solution
of the stochastic differential equation
pt pt
Xt=X0+ / cr{Xs)dBs + / b{Xs)ds
Jo Jo
where a and b are Lipschitz. In addition, assume a and b are chosen so that
for h Borel and bounded,
E{h{Xx)\Tt} = Jh(x)w(l-t,Xt,x)dx
where тгA — t,u,x) is a deterministic function.2 Thus Qt(u>,dx) — тгA —
t,Xt(oj),x)dx, and Qt(ui,dx) is a.s. absolutely continuous with respect to
Lebesgue measure if t < 1. Hence if we expand the Brownian filtration F
by initially adding X\, we have by Theorem 10 that every F semimartingale
is an H(.Xi) semimartingale, for 0 < t < 1.
The mirror of initial expansions is that of filtration shrinkage. This has
not been studied to any serious extent. We include one result (Theorem 12
below), which can be thought of as a strengthening of Strieker's Theorem,
from Chap. II. Recall that if X is a semimartingale for a filtration H, then it
is also a semimartingale for any subfiltration G, provided X is adapted to G,
by Strieker's Theorem. But what if a subfiltration F is so small that X is not
adapted to it? This is the problem we address. We will deal with the optional
projection Z of X onto F.
Definition. Let H = (Ht)t>o be a bounded measurable process. It can be
shown that there exists a unique optional process °H, also bounded, such that
for any stopping time T one has
E{HT1{T<OO}} = E{°HT1{T<OO}}.
The process °H is called the optional projection of H.
We remark that °Ht — Ht a.s. for each fixed time t, but the null set
depends on t. Therefore were we simply to write as a process (E{Ht\J-t})t>o-,
instead of °H, it would not be uniquely determined almost surely. This is why
we use the optional projection. The uniqueness follows from Meyer's section
theorems, which are not treated in this book, so we ask the reader to accept
it on faith.
2 Sufficient conditions are known for this to be true. These conditions involve Xo
having a nice density, and requirements on the differentiability of the coefficients.
See, for example, [190] and [193].

368 VI Expansion of Filtrations
Definition. Let H = (#t)t>0 be a bounded measurable process. It can be
shown that there exists a predictable process PH, also bounded, such that for
any predictable stopping time T one has
Е{Нт1{т<оо}} = Е{рНт1{т«х>}}-
The process PH is called the predictable projection of H.
It follows that for the optional projection, for each stopping time T we
have
°#T = E{HT\FT} a.s. on {T < 00}
whereas for the predictable projection we have that
PHT = E{HT\rT-} a.s. on {T < 00}
for any predictable stopping time T. For fixed times t > 0 we have then
of course °Ht = E{Ht\Tt) a.s., and one may wonder why we don't simply
use the "process" {E{Ht\Ft})t>o instead of the more complicated object °H.
The reason is that this is defined only almost surely for each t, and we have
an uncountable number of null sets; therefore it does not uniquely define a
process. (Related to this observation, note in contrast that in some cases
(E{Ht\Ft})t>0 might exist even when °H does not). We begin our treatment
with two simple lemmas.
Lemma. Let F С G, and let X be a G martingale but not necessarily adapted
to F. Let Z denote the optional projection of X onto F. Then Z is an F
martingale.
Proof. Let s < t. Then
Zs = E{XS\TS} = E{E{Xt\gs}\Ts} = E{E{Xt\gt}\Ts} = E{Zt\Ts}
and the result follows. D
The Azema martingale, a projection of Brownian motion onto a subfil-
tration to which it is not adapted, is an example of the above lemma. The
projection of an increasing process, however, is a submartingale but not an
increasing process.
Lemma. Let fcG, and let X be a G supermartingale but not necessarily
adapted to F. Let Z denote the optional projection of X onto F. Then Z is
an F supermartingale.
Proof. Let s < t. Then
Zs = E{XS\TS} > E{E{Xt\gs}\Ts} - E{Xt\Ts} = E{E{Xt\Ft}\Fs)
= E{Zt\Ts),
and the result follows. ?

3 Progressive Expansions 369
The limitation of the two preceding lemmas is the need to require integra-
bility of the random variables Xt for each t > 0. We can weaken this condition
by a localization procedure.
Definition. We say that a G semimartingale X starting at 0 is an F spe-
special, G semimartingale if there is a sequence (Tn)n>i of F stopping times
increasing a.s. to сю, and such that the stopped processes XT™ can be written
in the form XT" = Mn + An where Mn is a G martingale with Mg = 0
and where An has integrable variation over each [0, t], each t > 0, and with
Ao=0.
Theorem 12 (Filtration Shrinkage). Let G be a given filtration and let F
be a subfiltration of Q. Let X be an? special, G semimartingale. Then the F
optional projection of X, called Z, exists, and it is a special semimartingale
for the F filtration.
Proof. Without loss of generality we can assume Tn < n for each n > 1. We
set To = 0 and let Xn = XT" -XT~-\ and TV" = MT--MT—1 with iV° = 0.
For each n there are two increasing processes Cn and Dn, each starting at 0,
with Xn — Nn + Cn — Dn, and moreover we can choose this decomposition
such that the following holds:
?{sup \Nt\} < oo,
t
and where t < Tn_i implies Ct™ = D™ = JVf = 0, and t > Tn implies
ct ~ стп = Dt ~ DTn = Nt - NTn = °- The integrability condition implies
that the F optional projections of Cn, Dn, and Nn all exist and have cadlag
versions. By the previous two lemmas the optional projection of iVn is an F
martingale, and those of Cn and Dn are F submartingales. Therefore letting
°Xn, °Nn, °Cn, and °Dn denote the respective F optional projections of Xn,
Nn, Cn, and Dn, we have that °Xn =° Nn+°Cn-°Dn exists and is a special
F semimartingale.
Since Tn-i and Tn are F stopping times, we have that also the F optional
projections °iVn, °Cn, and °Dn and hence °Xn are all null over the stochastic
interval [0,Tn_i] and constant over (Tn,oo). Then J2n>i °^" is a c&d\hg
version of °X = Z and thus Z is a special F semimartingale. D
3 Progressive Expansions
We consider the case where we add a random variable gradually to a filtration
in order to create a minimal expanded filtration allowing it to be a stopping
time. Note that if the initial filtration F = {Tt)t>o is given by Tt = {0,^}
for all t, then G = (Q)t>o given by Qt = a{L Л s;s < i] is the smallest
expansion of F making L a stopping time. Note that Qt = cr{L Л t} as well.
Let L be a strictly positive random variable. Let Л = {(t,w) : t < L(oj)}.

370 VI Expansion of Filtrations
Then L = sup{t : (t,w) s Л}. In this sense every positive random variable
is the end of a random set. Instead however let us begin with a random set
Л с М+ x Q and define L to be the end of the set A. That is,
L(<jj) — sup{i : (t,ui) e Л}
where we use the (unusual) convention that sup@) = 0-, where {0—} is an
extra isolated point added to the non-negative reals [0, oo] and which can be
thought of as 0— < 0. We also define J- = -^o- The purpose of {0-} is to
distinguish between the events {w : Л(и>) = 0} and {w : Л(и>) — {0}}, each of
which could potentially be added to the expanded filtration.
The smallest filtration expanding F and making the random variable L a
stopping time is G° defined by Q® = Tt V a{L Л t}; but G° is not necessarily
right continuous. Thus the smallest expanded filtration making L a stopping
time and satisfying the usual hypotheses is G given by Qt = f)u>tGu- Nev-
Nevertheless, it turns out that the mathematics is more elegant if we consider
expansions slightly more rich than the minimal ones. In order to distinguish
progressive expansions from initial ones, we will change the notation for the
expanded filtrations. Beginning with our usual filtered probability space sat-
satisfying the usual hypotheses (tt, T, F, P)m where of course F denotes the fil-
filtration (^")t>0) and a random variable L which is the end of a random set, we
define the expanded filtration to be FL and is given by
Г e FtL <=> {Г e T and 3 Ft e Tt : Г П {L > t} = Гг П {L > t}}.
This filtration is easily seen to satisfy the usual hypotheses, and also it makes
L into a stopping time. Thus GcF1. There are two useful key properties the
filtration FL enjoys.
Lemma. If Я is a predictable process for FL then there exists a process
J which is predictable for F such that H — J on [0,L]. Moreover, if T is
any stopping time for FL then there exists an F stopping time 5 such that
S AL = T AL a.s.
Proof. Let Г be an event in T^. Then events of the form (t, oo) x Г form a
generating set for V(?L), the predictable sets for FL. Let Hs — 1{D>Оо)хг}(^)
and then take J to be Js = l(t,oo)xrt(s)- The first result follows by an appli-
application of the Monotone Class Theorem. For the stopping time T, note that it
suffices to take H = l[o,T], and let J be the F predictable process guaranteed
by the first half of this lemma, and take 5 = inf {t: Jt = 0}. ?
We next define a measure ць on [0, сю] х п by
HL(J) = E{JLl{L>0_}}
for any positive, measurable process J. For such a measure ць there exists an
increasing process 1{*>l} which is null at 0— but which can jump at both 0

3 Progressive Expansions 371
and +00. We will denote AL = (A^L>o, the (predictable) compensator
of l{t>L} for the filtration F. Therefore if J is an F predictable bounded
process we have
E{JLl{L>0_}} = E{ [ JsdALs).
J[0,oo]
We now define what will prove to be a process fundamental to our analy-
analysis. The process Z defined below was first used in this type of analysis by
J. Azema [3]. Recall that if H is a (bounded, or integrable) ?L process, then
its optional projection °H onto the filtration F exists. We define
Note that l{L>t} 1S decreasing, hence by the lemma preceding Theorem 12
we have that Z is an F supermartingale. We next prove a needed technical
result.
Theorem 13. The set {t : 0 < t < 00, Zt- = 0} is contained in the set (L, 00]
and is negligible for the measure dAL.
Proof. Let T(w) = inf{i > 0 : Zt{w) = 0, or Zt-(w) = 0 for t > 0}. Then
it is a classic result for supermartingales that for almost all u> the function
t 1—» Zt(oj) is null on \T(oj), 00]. (This result is often referred to as "a non-
negative supermartingale sticks at zero.") Thus we can write {Z = 0} as the
stochastic interval [T, 00], and on [0, T) we have Z > 0, Z~ > 0. We have
E{A^A^} = P(T <L)= E{ZTl{T<oo}} = 0, hence dAL is carried by [0,T].
Note that since dl{L>ty is carried by the graph of L, written [L], we have
L <T, and hence we have Z > 0, Z- > 0 on [0, L). Next observe that the set
{Z- = 0} is predictable, hence 0 = E{l{Zt_=o}^l{L>t}} = E{l{Zt=OydAf}
and hence {Z- = 0} is negligible for dAL. Note that this further implies that
P{ZL- > 0) = 1, and again that {Z- = 0} С {L, 00]. ?
We can now give a description of martingales for the filtration FL, as
long as we restrict our attention to processes stopped at the time L. What
happens after L is more delicate. For an integrable process J we let p J denote
its predictable projection.
Theorem 14. Let Y be a random variable with E{\Y\} < 00. A right con-
continuous version of the martingale Yt = E{Y\J-^} is given by the formula
At
Moreover the left continuous version Y- is given by
1

372 VI Expansion of Filtrations
Proof. Let OL denote the optional cr-algebra on Ш+ x П, corresponding to
the filtration FL. On [0,L), OL coincides with the trace of О on [0,1,). (By
О we mean of course the optional cr-algebra on R+ x Q corresponding to the
underlying nitration F.) Moreover on [L,oo), OL coincides with the trace of
the cr-algebra B(M+)®^roo on [L, oo). The analogous description of VL holds,
with [0, L) replaced by @, L], and with [L, oo) replaced with (L, oo). It is then
simple to check that the formulas give the bona fide conditional expectations,
and also the right continuity is easily checked on [0, L) and [L, oo) separately.
The second statement follows since Z- > 0 on @, L] by Theorem 13. D
We now make a simplifying assumption for the rest of this paragraph. This
assumption is often satisfied in the cases of interesting examples, and it allows
us to avoid having to introduce the dual optional projection of the measure
Simplifying assumption to hold for the rest of this paragraph. We
assume L avoids all F stopping times. That is, P(L = T) = 0 for all F
stopping times T.
Definition. The martingale ML given by MtL = A\ + Zt is called the fun-
fundamental L martingale.
Note that it is trivial to check that ML is in fact a martingale, since AL
is the compensator of 1 — Z. Note also that M^ = A^, since Z^ = 0. Last,
note that it is easy to check that ML is a square integrable martingale.
Theorem 15. Let X be a square integrable martingale for the F filtration.
Then (XtAi)t>o is a semimartingale for the filtration ?L. Moreover X4Al —
Jo Y^d{X,ML)s is a martingale in the ?L filtration.
Proof. Let С be the (non-adapted) increasing process Ct = 1{j>l}- Since С
has only one jump at time L we have E{Xl} = E{f?° XsdCs}. Since X is a
martingale it jumps only at stopping times, hence it does not jump at L, and
using that AL is predictable and hence natural we get
/¦OO /-OO
E{XL} = E{ Xs-dCs} = E{ Xs-dALs} = ^
Jo Jo
M^} - E{[X, MLU} = E{(X,
Suppose that Я is a predictable process for FL, and J is a predictable process
for F which vanishes on {Z_ = 0} and is such that J = H on @,L]. We
are assured such a process J exists by the lemma preceding Theorem 13.
Suppose first that H has the simple form H = /ilDiOO) for bounded h ? J-^.
If j is an Tt random variable equal to h on {t < L}, then we can take
J = jl(t,oo) and we obtain H ¦ Xoo = h(Xb — Xt)l{t<Ly. In this way we can
define stochastic integrals for non-adapted simple processes. We have then
E{(H ¦ X)oo} = E{(J ¦ X)l}, and using our previous calculation, since J ¦ X
is another square integrable martingale, we get

3 Progressive Expansions 373
/>OO
E{(H ¦ X)oo} = E{(J- X, ML)OO} = E{ / Jsd(X, ML)S}.
Jo
Since (X, ML) is F predictable, we can replace H by g°'L] because it has
the same predictable projection on the support of d(X, ML). This yields
/>oo pL ту
E{ HsdXs} = E{ ^d(X,ML)s}. (**)
J0 Jo ^s-
Last if we take the bounded ?L predictable process H to be a stochastic
interval [0,T Л L], where T is an FL stopping time, we obtain E{Xtal -
/0TAL ^zd(X,ML)s} = 0, which implies by Theorem 21 of Chap. I that
Xt/\l - /о Л Y^d{X,ML)s is a martingale. П
We do not need the assumption that X is a square integrable martingale,
which we made for convenience. In fact the conclusion of the theorem holds
even if X is only assumed to be a local martingale. We get our main result as
a corollary to Theorem 15.
Corollary. Let X be a semimartingale for the F filtration. Then (XtAi,)t>0
is a semimartingale for the filtration ?L.
Proof. If X is a semimartingale then it has a decomposition X = M + D.
The local martingale term M can be decomposed into X = V + N, where
V and N are both local martingales, but V has paths of bounded variation
on compact time sets, and N has bounded jumps. (This is the Fundamental
Theorem of Local Martingales, Theorem 25 of Chap. III.) Clearly V and D
remain finite variation processes in the expanded filtration ?L, and since M
has bounded jumps it is locally bounded, hence locally square integrable, and
since every F stopping time remains a stopping time for the FL filtration, the
corollary follows from Theorem 15. ?
We need to add a restriction on the random variable L in order to study
the evolution of semimartingales in the expanded filtration after the time L.
Definition. A random variable L is called honest if for every t < oo there
exists an Tt measurable random variable Lt such that L = Lt on {L < t}.
Note that in particular if L is honest then it is J-qq measurable. Also, any
stopping time is honest, since then we can take L = L Л t which is of course
Tt measurable by the stopping time property.
Example. Let X be a bounded cadlag adapted processes, and let Xt+* =
sups<tX+ and X+* = supsX+. Then L = inf{s : X+* = X+*}, is honest
because on the set {L < t) one has L — inf{s : Xs = X^*}, which is Tt
measurable.
Theorem 16. L is an honest time if and only if there exists an optional set
Л С [0, oo] x п such that L(uj) — sup{i < oo : (t,w) G A}.

374 VI Expansion of Filtrations
This is often described verbally by saying "L is honest if it is the end of
an optional set."
Proof. The end of an optional set is always an honest random variable. Indeed,
on {L < t}, the random variable L coincides with the end of the set Лп ([О, t] x
fi), which is Tt measurable.
For the converse we suppose L is honest. Let (Lt)t>o be an F adapted
process such that L = Lt on {L < t}. Since we can replace Lt with Lt At
we can assume without loss of generality that Lt < t. There is also no loss of
generality to assume that Lt is increasing with t j since we can further replace
Lt with sups<4Z,s. Last, it is also no loss, now that it is increasing, to take
it right continuous. We thus have that the process (Lt)t>o is optional for the
filtration F. Last, L is now the end of the optional set {(t,u>) : Lt(w) = t}. D
When L is honest we can give a simple and elegant description of the
filtration TL.
Theorem 17. Let L be an honest time. Define
gt = {r.r= (An{L >t})U(Bn{L <t}) for some A,B eft}
Then G = (Gt)t>o constitutes a filtration satisfying the usual hypotheses.
Moreover L is a G stopping time. A process U is predictable for G if and
only if it has a representation of the form
where H and К are F predictable processes.
Proof. Let s<t and take Я e Qs, of the form (An{L> s}) U (В П {L < s})
with А, В G Ts. We will show that H G Gt, which shows that the collection G
is filtering to the right.3 Since L is an honest time, there must exist D e Tt
such that {L < s} = D П {L < t}. Therefore
H П {L < t} = [(А П Dc) U(Bfl D)} n{L< t},
with [(А П Dc) U (J5 П D)\ G Tt. The fact that each Qt is a cr-algebra, and also
that <G is right continuous, we leave to the reader. Note that {L < t} e Gt
which implies that L is a G stopping time, as we observed at the start of the
proof of Theorem 16. For the last part of the theorem, let U = (Ut)t>o be a
cadlag process adapted to the <G filtration. Let Ht and Kt be Tt measurable
random variables such that
Ut = #t on {L > t} and Ut = Kt on {L < t}.
Next set Ht = lim infs/t Hs;Kt = liminfsy4 Js for all t > 0. Suppose that s
is a rational number. Take t = 0 and we have the equality
That is, if s < t, then Qs С Gt-

3 Progressive Expansions 375
U = Н1[0,ц + Kl{LiOo].
One can extend this result to predictable processes by the Monotone Class
Theorem. D
Theorem 18. Let X be a square integrable martingale for the F filtration.
Then X is a semimartingale for the G filtration if L is an honest time. More-
Moreover X has a G decomposition
etl\L i
JO %s-
+{ / —d(X, ML)S - l{t>L} / —d(X, ML)S}.
Jo "s- Jl l ~ ^s-
Before beginning the proof of the theorem, we establish a lemma we will
need in the proof. It is a small extension of the local behavior of the stochastic
integral established in Chap. IV.
Lemma (Local behavior of the stochastic integral at random times).
Let X and Y be two semimartingales and let H G L(X) and J G L(Y). Let
U and V be two positive random variables with U < V. (U and V are not
assumed to be stopping times.) Define
A = {w : Ht(w) = JtH and Xt(u) - Xu{oj) = Yt(w) - Yu(w),
for all te [U(w),V(w))}.
Let Wt = H ¦ Xt and Zt = J ¦ Yt. Then a.s. on Л, Wt{u)) - Wu{uj) = Zt(w) -
Zv{uj) for all t G [U(oj),V(oj)).
Proof. We know by the Corollary to Theorem 26 of Chap. IV that the con-
conclusion of the lemma is true when U and V are stopping times. Let (u, v) be
rationale with 0 < и < v, and let
Auv - {u> : U(uj) < и < v < V(ui)} П A.
Then also Wt - Wu — Zt — Zu, all t G [u, v), a.s. on Auv, since u, v are fixed
times and hence a fortiori stopping times. Next let
A = iu> : P| {U(w) <u<v< V(u>), and Wt(u>) - Wu{w) = Zt{w) - Zu{w)
for all t e [u,v)} >.
The intersection in the definition of A is countable, so null sets do not accu-
accumulate. Finally let и decrease to U{uj) and then v increase to V(oj), which
gives the result. G

376 VI Expansion of Filtrations
Proof of Theorem 18. We first observe that without loss of generality we can
assume Xo = 0. Let H be a bounded F predictable process. We define stochas-
stochastic integrals at the random time L by
/ H3dX3 = (# • X)L, / Hs
Jo Jl
where H¦ X = (H • X4L>o is the F stochastic integral process. That is, we use
the usual definition of the stochastic integral, sampling it at the random time
L. When H is a simple predictable process, these are reasonable definitions.
Moreover we know from the lemma that if H and J are both bounded F pre-
predictable processes and if also H — J on (L, oo], then JL HsdXs — J^° JsdXs
a.s., so the definition is well-defined.
Since X is an F martingale in L2 with Xo = 0, we have E{fo°° HsdXs} = 0.
Applying the equalities (*) on page 372 to H ¦ X we have
E{ HsdXs} = E{ Hsd(X,ML)s} = E{ -^-d{X,ML)s} (f)
JO Jo Jo ^s-
where the second equality uses (**) on page 373. Recalling E{ JQ°° HsdXs} = 0
and combining this with the above gives (where we have changed the name
of the process H to К for clarity slightly later in the proof)
/>OO />OO
E{ KsdXs} = -E{ Ksd(X,ML)s}
Jl Jo
We next replace К with К\^г<ц to obtain
/>OO />OO
E{ KsdXs} = -E{ Ksl{Zs_<1}d(X,ML)s}
Jl Jo
К
because the predictable projection of j^|—1(l,oo)(s) is l{zs_<i}- Finally we
combine equalities (f) and (|), and define U = Hl^0 ц + Kl^oo] to get
E{ / HsdXs + / KsdXs}
Jo Jl
= E{ Usd{ d(X,ML)s-l{t>L} / z —d(X,ML)s}
Jo Jo As- JL L ~ ^s-
Up to this point we have not used the hypothesis that L is an honest time.
Now we do. Let us take U to be a simple G predictable process, and using
Theorem 17 we have

4 Time Reversal 377
where H and К are F predictable processes (not necessarily simple pre-
predictable, however). If we further take U bounded by 1 and take the supre-
mum over all such simple predictable processes of the elementary G stochas-
stochastic integrals /0 UsdXs, we get that the variation of X, Var(X) as defined in
Chap. Ill, is less than or equal the expected total variation of
=d{ d(X,ML)s + l{t>L} -
JO ^s- JL l
p / 1
dat=d{ d(X,ML)s + l{t>L} - —d(X,ML)s).
Thus the process XisaG quasimartingale, and hence also a semimartingale.
We also note that by the preceding we have that Е{Хт — ат} — 0 for all G
stopping times T, and thus it is a martingale for the G filtration. D
The next theorem is really a corollary of Theorem 18.
Theorem 19. Let X be a semimartingale for the F filtration. Then X is a
semimartingale for the G filtration if L is an honest time.
Proof. Let X = M + A be a decomposition of X in the F filtration, where M is
a local martingale and Л is a finite variation process. Since A remains a finite
variation process in the larger G filtration, we need only concern ourselves
with M. By the Fundamental Theorem of Local Martingales (Theorem 25
on page 125), we know that we can write M = N + V where N is a local
martingale with bounded jumps, and V is a martingale with paths of finite
variation on compacts. Again, V remains a semimartingale in the G filtration
since it is of finite variation, and further N can be locally stopped with F
stopping times Tn tending to oo a.s. to be a square integrable martingale for
each Tn. Then by Theorem 18 each XTn is also a G semimartingale. Since X
is locally a G semimartingale, it is an actual G semimartingale. D
4 Time Reversal
In this section we apply the results on initial expansions of filtration to some
elementary issues regrading time reversal of semimartingales, stochastic inte-
integrals, and stochastic differential equations.
Definition. Let Y = (Yt)o<t<i be a cadlag process. The time reversal of
Y on [0,1] is defined to be
Y =
Note that time as a superscript denotes reversal. Let F denote the forward
filtration, and correspondingly let F = (Jrt)o<t<i denote a backward filtration.
As an example, if В is a standard Brownian motion, J-^ = cr{Bs; s < t}, and

378 VI Expansion of Filtrations
Ft = rf V Af, where N are the P-null sets of T\, while F* = 1% VЫ is the
analogous backward filtration for B. Our first two theorems are obvious, and
we omit the proofs.
Theorem 20. Let В be a standard Brownian motion on [0,1]. Then the time
reversal В of В is also a Brownian motion with respect to its natural filtration.
Theorem 21. Let Z be a Levy process on [0,1]. Then Z is also a Levy process,
with the same law as —Z, with respect to its natural filtration.
In what follows let us assume that we are given a forward filtration F =
№)o<t<i and a backward filtration G = (?7*)o<t<i- We further assume both
filtrations satisfy the usual hypotheses.
Definition. A cadlag process Y is called an (F, G) reversible semimartin-
gale if Y is an F semimartingale on [0,1] and Y is a <G semimartingale on
[0,1).
Note the small lack of symmetry: for the time reversed process we do not
include the final time 1. This is due to the occurrence of singularities at the
terminal time 1, and including it as a requirement would exclude interesting
examples.
Let rt = (to, ...,tk) denote a partition of [0, t] with t0 = 0, tk = t, 0 < t <
1, and let
STt(H,Y) = H0Y0 + Y,(Hti+1 - Hti)(Yu+1 - Yti).
Definition. Let H, Y be two cadlag processes. The quadratic covariation
of H and Y, denoted [H, Y], is denned to be
when this limit exists in probability as mesh(rt) —> 0, and is such that [H, Y]
is cadlag, adapted, and of finite variation a.s.
Theorem 22. Let Y be an (F,<G) reversible semimartingale, and let H be
a cadlag process such that Ht ? Tt, and Ht ? Ql~l,Q < t < 1. Suppose the
quadratic variation [H, Y] exists. Then the two processes [H,Y] and Xt =
/0* Hs-dYs are (F, <&) reversible semimartingales. Moreover
~ r*
Jo
Proof. Since Ht e Я1~г,0 < t < 1, we have that #i_s e c^-d-a) - g^ an(j
thus it is adapted and left continuous, so the stochastic integral JQ Hi_sdYs
is well-defined. We choose and fix t, 0 < t < 1. Let r be a partition of
[1 — t, 1], r = (si,...,sn)) chosen such that ДЯ8( = 0 and AYSi = 0 a.s.,

4 Time Reversal 379
i = 2,..., n — 1. Note that one can always choose the partition points {s^} in
this manner because
{
Jo
= 0,
since for each u>, s н-» Ys(u>) has only countably many jumps, and hence
/o ^-{\AYs\>o}ds = 0 a.s. But then JQ Р(]ДУ8| > 0)ds = 0, which in turn
implies that Р(|ДУ8| > 0) = 0 for almost all s in [0,1]. We now define three
new processes:
n-2
n-2
Let тп be a sequence of partitions of [0,1] with limn^oo mesh(rn) = 0. Then
lim CT" =
n—>oo
Since Y is (F, G) reversible by hypothesis, we know that CT is <& adapted.
Hence [H, У] is G adapted and moreover since it has paths of finite variation
by hypothesis, it is a semimartingale.
Since H is cadlag we can approximate the stochastic integral with partial
sums, and thus
lim ATn =
n—>oo
Since У8,+1_ - У8,_ = -(У1- - У1-^), we have
ft
lim BTn = I .
n—^oo I n
J U
Combining ATn, 5Tti , and CTn, we get

380 VI Expansion of Filtrations
n-2
* + Х>в1+1(У,1+1-У,4)-
г=1
n-1
n-2
г=1
Since we chose our partitions rn with the property that AYSi — 0 a.s. for each
partition point Sj, the above simplifies to
t + ДЯ1(У,„_1 - У1_),
which tends to 0 since si decreases tol — t, and sn_i increases to 1. Therefore
0 = lim A7"" + lim B7"" + lim C
n—»oo n—>oo n—»oo
7"" = -X* + / H^s
Уд
This establishes the desired formula, and since we have already seen that
[H, Y]t is a semimartingale, we have that X is also a reversible semimartingale
as a consequence of the formula. D
We remark that in the case where У is a G semimartingale on [0,1] (and
not only on [0,1)), the same proof shows that X is a G semimartingale on
[0,1].
Example (reversibility for Levy process integrals). Let Z be a Levy
process on [0,1], and let H = (iit)o<t<i be the filtration generated by Z and
expanded by the addition of Z1 = Zo - Z\-. Then Z is an H semimartingale
on [0,1] by Theorem 21. Let Ht = f(Zt) with / such that the quadratic
covariation [H,Z] exists. Clearly Ht = f(Zt) G Tt- Since
W1-* D ct{ZA_u)_ - Z{1_v)_; 0 < u,v < 1} V a{Zx},
we have that Zt = Zi_A_t) € W1"*, since Zt = Zt_ a.s. Therefore Я4 G W1"*.
Using Theorem 22 we conclude that Xt = J* f(Zs_)dZs is an (F, H) reversible
semimartingale.
Example (reversibility for Brownian integrals). Let В be a standard
Brownian motion on [0,1]. Let Щ denote Brownian local time at the level
x and let ц be a signed measure on R. Let F be the minimal nitration of В
satisfying the usual hypotheses, and let (HI*)o<t<i be the minimal filtration
generated by B, expanded by adding B1 = —Bi, and satisfying the usual
hypotheses. Finally let / be a cadlag function on R of finite variation on
compacts with primitive F (that is, F+(x) = f(x)), and define

4 Time Reversal 381
Ut= f f(Bs)dBs + f L^(d
Jo Jr
We will show U is an (F, Й) reversible semimartingale.
The hypotheses on / imply that its primitive F is the difference of two
convex functions, and therefore Mt — F(Bt) - F(B0) is an F semimartingale.
Moreover M* — F(B\-t) —F(B{), and since we already know that Bi-t is an
H semimartingale by Theorem 3, by the convexity of F we conclude that M is
also an H semimartingale. Therefore /0 f(Bs)dBs is an (F, H) reversible semi-
semimartingale as soon as | JR Lfrj(da) is one, where tj is the signed measure 'sec-
'second derivative' of F. Finally, what we want to show is that At = \ /R L^fj,(da)
is an (F,H) reversible semimartingale, for any signed measure /x. This will
imply that U is an (F, Й) reversible semimartingale.
Since A is continuous with paths of finite variation on [0,1], all we really
need to show is that A* e Ti*, each t. But Л* = Ai-t - A\ = /R(?i_t -
L^)fj,(da), and
- L\ = lim(-i / l[e>e+e)(B,)de)
1 f1
= lim(— / l[a_Bl)a_Bl+?)(Bs - Bjds)
l[a-Bl,a-B1+e)(Bu)du)
[
where Af is the local time at level x of the standard Brownian motion B.
Therefore Л* = - jR(A^~Bl)fj,(da). Since A% is H adapted, and since B\ e
7i°, we conclude A is H adapted, and therefore U is an (F, Й) reversible
semimartingale.
We next consider the time reversal of stochastic differential equations,
which is perhaps more interesting than the previous two examples. Let В be
a standard Brownian motion and let X be the unique solution of the stochastic
differential equation
i-t i-t
Xt = X0+ / <i(Xs)dBs + / b(Xs)ds
Jo Jo
where a and b are Lipschitz, and moreover, a and b are chosen so that for h
Borel and bounded,
J
= / /i(z)ttA ~t,Xt,x)dx
where тгA — t,u, x) is a deterministic function. As in the example expansion
via the end of a stochastic differential equation treated earlier on page 367,

382 VI Expansion of Filtrations
we know (as a consequence of Theorem 10) that if F is the natural, completed
Brownian nitration, we can expand F with Xi to get H : Tit = f]u>t -^и V
a{Xi}, and then all F semimartingales remain H semimartingales on [0,1).
We fix (t,w) and define ф : R -> R by ф(х) = X(t,w,x) where Xt =
X(t,u),x) is the unique solution of
Xt = x-
for iel. Recall from Chap. V that ф is called the flow of the solution of
the stochastic differential equation. Again we have seen in Chap. V that
under our hypotheses on a and b the flow ф is injective. For 0 < s < t < 1
define the function ф$^ to be the flow of the equation
X.,t = x
f a{Xs<u)dBu + f b(XStU)du.
J s Js
It now follows from the uniqueness of solutions that Xt = фа,г(Ха), and
in particular X\ = фг,г(Х^, and therefore ф^\(Х\) = Xt, where ф^\ is of
course the inverse function of фа$. Since the solution Xa>t of equation (*) is
Q — cr{Bv — Bu; s < u,v < t} measurable, we have ф±г\ G ^rl~*, where
P = a{Bs;0 < s < t}V Я where Я are the null sets of T. Let H* =
C\u>t-^U V a{Xi\ and we have В is an И = Gi*)o<t<i semimartingale, and
that Xt = ф^\(Х{) is ~Н1~г measurable. Therefore Xt G Tt and also at the
same time Xt G Til~i. Finally note that since X is a semimartingale and a
is C1, the quadratic covariation [a(X),B] exists and is of finite variation, and
we are in a position to apply Theorem 22.
Theorem 23. Let В be a standard Brownian motion and let X be the unique
solution of the stochastic differential equation
Xt = Xo + [ <j{Xs)dBs + f b(Xs)ds
Jo Jo
for 0 < t < 1, where a and b are Lipschitz, and moreover, a and b are chosen
so that for h Borel and bounded,
= fh(xOT(l-t,Xt,x)dx
where тгA — t,u,x) is a deterministic function. Let H be given by Tit —
[\u>tTu Va{X{\. Then В is an (F,H) reversible semimartingale, and Xf =
Xi-t satisfies the following backward stochastic differential equation
Yt = Zi+ f a{Ys)dBs+ f a'(Ys)a{Ys)ds+ f b{Ys)ds,
Jo Jo Jo
for 0 < t < 1. In particular, X is an (F, Й) reversible semimartingale.

Bibliographic Notes 383
Proof. We first note that В is an (F, Й) reversible semimartingale as we saw
in the example on page 367. We have that [<r(X),B] = fQ cr'(Xs)cr(Xs)ds,
which is clearly of finite variation, since a' is continuous, and thus a'(X) has
paths bounded by a random constant on [0,1]. In the discussion preceding this
theorem we further established that cr(Xt) G 7Y1-t, and of course cr(Xt) ? Tf
Therefore by Theorem 22 we have
b(Xs)ds.
* = Xi-t -Xi= f a(Xi-a)dB
J
i-t
Observe that [a{X),B\t = J1_ta'(Xs)a(Xs)ds. Use the change of variable
и = 1 — s in the preceding integral and also in the term /lt b(Xs)ds to get
f
Jo
b(Xi-a)ds,
Jo
and the proof is complete. D
Bibliographic Notes
The theory of expansion of filtrations began with the work of K. Ito [100] for
initial expansions, and with the works of M. Barlow [7] and M. Yor [239] for
progressive expansions. Our treatment has benefited from some private notes
P. A. Meyer shared with the author, as well as the pedagogic treatment found
in [44]. A comprehensive treatment, including most of the early important
results, is in the book of Th. Jeulin [114]. Excellent later summaries of main
results, including many not covered here, can be found in the lecture notes
volume edited by Th. Jeulin and M. Yor [117] and also in the little book by
M. Yor [247].
Theorem 1 is due to J. Jacod, but we first learned of it through a paper of
P. A. Meyer [175], while Theorem 3 (Ito's Theorem for Levy processes) was
established in [107], with the help of T. Kurtz. Theorems 6 and 7 are of course
due to Th. Jeulin (see page 44 of [114]); see also M. Yor [245]. Theorem 10,
Jacod's criterion, and Theorem 11 are taken from [104]; Jacod's criterion is
the best general result on initial expansions of filtrations. The small result on
filtration shrinkage is new and is due to J. Jacod and P. Protter.
Two seminal papers on progressive expansions after M. Barlow's original
paper are those of Th. Jeulin and M. Yor [115] and [116]. The idea of using a
slightly larger filtration than the minimal one when one progressively expands,
dates to M. Yor [240], and the idea to use the process Zt —° l{L>t} originates
with J. Azema [3]. The lemma on local behavior of stochastic integrals at
random times is due to C. Dellacherie and P. A. Meyer [46], and the idea of
the proof of Theorem 18 is ascribed to C. Dellacherie in [44]. The results on
time reversal are inspired by the work of J. Jacod and P. Protter [107] and

384 Exercises for Chapter VI
also E. Pardoux [190]. Related results can be found in J. Picard [193]. More
recently time reversal has been used to extend Ito's formula for Brownian
motion to functions which are more general than just being convex, although
we did not present these results in this book. The interested reader can consult
for example H. Follmer, P. Protter and A.N. Shiryaev [77] and also H. Follmer
and P. Protter [76] for the multidimensional case. The case for diffusions is
treated by X. Bardina and M. Jolis [6].
Exercises for Chapter VI
Exercise 1. Let (?l,F,W,B,P) be a standard Brownian motion. Expand F
by the initial addition of a{B\}. Let M be the local martingale in the formula
Bt=B0+Mt + "\ "sds.
Jo
Bj-B,
1-s
Show that the Brownian motion M = (Mf)o<t<i ls independent of cr{Bi}.
Exercise 2. Show that the processes Jl denned in the proof of Ito's Theorem
for Levy processes (Theorem 3) are compound Poisson processes. Let N%
denote the Poisson process comprised of the arrival times of Jl, and let Gl
be the natural completed filtration of Nl. Further, show the following three
formulae (where we suppress the i superscripts) hold.
(a) {}^
(b) E{Jt\Jl,g1}=tJl.
(c) ??{Ji|J1}=tJi.
Exercise 3. In Exercise 1 assume Bo = x and condition on the event {Bi —
y}. Show that В is then a Brownian bridge beginning at x and ending
at y. (See Exercise 9 below for more results concerning filtration expansions
and Brownian bridges. Brownian bridges are also discussed in Exercise 24 of
Chap. II, and on page 299.)
Exercise 4. Let (ft, J7, F, Z, P) be a standard Poisson process. Expand F by
the initial addition on a{Z\\. Let M be the local martingale in the formula
= Mt+
i;
Show that M = (MtH<t<i is a time changed compensated Poisson process.
Exercise 5. Let В be standard Brownian motion. Show there exists an T\
measurable random variable L such that if G is the filtration F expanded
initially with L, then Qt — T\ for 0 < t < 1. Deduce that В is not a semi-
martingale for the expanded filtration G. Show further that L can be taken
to be of the form L = jQ f(s)dBs where / is non-random.

Exercises for Chapter VI 385
Exercises 6 through 9 are linked, with the climax being Exercise 9.
Exercise 6. Let В denote standard Brownian motion on its canonical path
space on continuous functions with domain R+ and range R, with Bt(u>) =
u)(t). F is the usual minimal filtration completed (and thus rendered right
continuous a fortiori). Define У/7 = P{Ba ? и\Тг}. Show that
where
.-{u-xf
= g(u,t,Bt) and g(u,t,x) =
for t < a. Show also that Y^ = l^B^eU} f°r t > a.
Exercise 7. (Continuation of Exercise 6.) Show that Уои_ = О except on the
null set {Ba = u} and infer that (У4")о<«о is a positive martingale which is
not uniformly integrable on [0, a].
Exercise 8. (Continuation of Exercises 6 and 7.) Show that
where
дд и-х -(и-a;J
-^-(u,s,x) = . = exp{~ f-j
dxy ' ^2n(a -1) X 2(a-t) S
for t < a.
Exercise 9. (Continuation of Exercises 6, 7, and 8.) Show that one can define
a probability Pu on C([0, a],M) such that Pu(Ba = u) = 1 with Pu absolutely
continuous with respect to P on Tt for t < a, and singular with respect to
P on Ta. (Hint: Sketch of procedure. Let Па = C([0, a),R), be the space of
continuous functions mapping [0, a] into R, and define M" = Ytu/Yt° on Tt for
(of course) t < a. Let Pu denote the (unique) probability on па = C([0, a), R)
given on Tt by dPu = M^dP, and confirm that
is a Brownian motion under Pu. Next show that
AZ=[-Ld[B,M»]. verifies M? ЩЩ "'*«
Finally show that

386 Exercises for Chapter VI
Bt - —Bo --u = (a-t) Г —d&
a a 'Jo a-s
under Pu, and thus (a -1) JQ (a - s)~ldf3s, which is defined only on [0, a), has
the same distribution under Pu as the Brownian bridge, whence
/"* 1
lim(a-t) / dCs =0
t^a Jo a- S
a.s., and deduce the desired result.)
Exercise 10 (expansion by a natural exponential random variable).
Let F be a filtration satisfying the usual hypotheses, and let Г be a totally
inaccessible stopping time, with P{T < oo) = 1. Let A = (^4t)t>o be the
compensator of 1{*>т} and let M be the martingale Mt = l{t>x} — A*. Note
that At = Aqo- Let G be the filtration obtained by initial expansion of F
with <r{Aoo}- For a bounded, non-random, Borel measurable function / let
М{ = /0* f(As)dMs which becomes in G,
{ =
ft
)l{t>T} - /
Jo
f
M{ = f(AT)l{t>T} - / f(As)dAs =
Jo
where F(t) = Jo f(s)ds, since A is continuous. Show that E{f(At)} =
E{F(At)}, and since / is arbitrary deduce that the distribution of the random
variable At is exponential of parameter 1.
Exercise 11. Let F, Г, A, and G be as in Exercise 10. Show that if X is an
F martingale with Хт- = Хт a.s., then X is also a G martingale.
Exercise 12. Show that every stopping time for a filtration satisfying the
usual hypotheses is the end of a predictable set, and is thus an honest time.
Exercise 13. Let В = (Bt)t>o be a standard three dimensional Brownian
motion. Let L = sup{t : \\Bt\\ < 1}, the last exit time from the unit ball.
Show that L is an honest time.
Exercise 14. Let ML be the fundamental L martingale of a progressive ex-
expansion of the underlying filtration F using the non-negative random variable
L.
(a) Show that for any F square integrable martingale X we have EXl =
EX^M^.
(b) Show that ML is the only F^ square integrable martingale with this
property for all F square integrable martingales.
Exercise 15. Give an example of a filtration F and a non-negative random
variable L and an F semimartingale X such that the post L process Yt =
Xt — XtAi = XtvL — Xt is not a semimartingale. [Hint: Use the standard
Brownian nitrations and let

Exercises for Chapter VI 387
/ f(s)dBs = \ I f(s)dBs}
o z Jo
where / is the non-random function of Exercise 5.)
Exercise 16. Let (п,Т,?,Р) be a filtered probability space satisfying the
usual hypotheses, let L be a non-negative random variable, let Goo — Foo V
cr{L}, and let G be given by
Qt = {Ae дж\ 3At eTt,An{t<L} = Atn{t< L}}.
Let X be a F semimartingale. Give a simple proof (due to M. Yor [239])
that Xtl{t<L,} (and hence X^l) 1S a semimartingale. (Hint: Show that it
is enough to consider X a uniformly integrable martingale, hence enough
to consider X a quasimartingale, and finally enough to consider X a positive
supermartingale. Let Zt = °1{l>*}, the F optional projection of 1{l>*}> which
is a supermartingale. Recall from Theorem 13 that P(Zl- > 0) = 1. Show
that
vz Xt
xt =
is a G positive supermartingale. Next show that XZ is an F special semi-
semimartingale, and let XZ — M + A be its canonical decomposition. Observe
that
Xtl{t<L} = —^—l{t<L} = ^ l{t<L) = Щ +At,
and argue that Mz is a G semimartingale as before, and that Az is the
product of the two G semimartingales A and (Zt)~1l[t<Ly, and hence it is
also a G semimartingale.)
Exercise 17. Let Z be a Levy process on [0, l] and let
Xt =
I a(Xs-)dZs + f b[Xs-)ds.
Jo Jo
Assume that a and b are smooth, and also assume (which is not always true!)
that the flows of X are injective. Find a stochastic differential equation solved
by Xi-t.
Exercise 18. Let (?l,Jr, F, B,P) be a standard Brownian space with В, а
standard Brownian motion. Let Z denote the random zero set of B, which is
a closed, perfect, nowhere dense set. Let {(Ln, Rn)}n>i denote the random
intervals contiguous to Z. Without loss of generality assume that the graphs
of Ln, denoted [Ln], are disjoint random sets. P. A. Meyer's conjecture was
that one could expand F to a filtration G in such a way that В would still
be a semimartingale and each Ln would be a G stopping time. Show this
is false (argument due to M. Barlow [8]) by taking Yt = \Bt\ and showing
(computed in the G filtration) that /0 l{Ys>0}dYs = Yt, which implies that
/0 l{ys=o}dYs — 0, which is a contradiction, since as a consequence of Tanaka's
formula /0 l^Ya=o}dYs — L°, the Brownian local time at 0.

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Symbol Index
AL 371 Vn 220,301
|| Л ||p 245 dx(H,J) 155
J™\dAs\ 101,245 ?(X) 85
AX,AX 180 SR{Z) 320
Л 154 SH{Z) 322
Л 118 ?f-s(?) 280
\A\, (\A\t)t>o 40,101 (|? •/)(*, w,x) 279
|Л|оо 101 №L>о 355
Е{\А\х}<оо 111 F-Л 40
?{|Л|т„}<оо 111 F 3,356
bL 56,155 FL 370
ЪО 102 1РМ 36
hV 102,154 TT 5
BMO 195 -7? 16
A 252 Tt- 105
D 56,244 ^ 293,309
Dn 250,301 F 377
Ducp 57 F-S acceptable 279

404 Symbol Index
F-S integral 82 1(M) 181,185
FV 39,101 Jx 58
9t 228 jp(N,A) 245
(H2,X)integrable 163 L(X) 163
HA 40 La{X)t 212,230
HX 58,60,156 Ц 212
HQX 60,165,170 L 56,101,102,153,155,244
HG-X 61 L(G) 61
He-X 175 UcP 57
№У] 378 Lo 52M6
ЖР 244 (M,N) 178
/Zf.dX, 58 Мь 372
J0°°H3dX3 58,60 мт- 172
J*Hs°dXs 82,271 M2 178,186
j*HsdXs 60,156 Л^2(Л) 182
/О'+Я3ЙХ3 60,78 \\М\\Вмо 195
H(L) 362 iV 154
H 357 С Ю2
H2 154,244,193 OL 372
W2(P) 169 -P" 36
W 193 V 102,154,206
H°° 314 P 293,309
(HI) 302,328 <2~P 131
(H2) 302,328 Qt{u,dx) 362
(H3) 329 Q 4,214
/x 52,144 Rn 19

Symbol Index 405
M+ 19 \\X\\sp 244
S 51 \\X\\H2 244
S(H) 53 (X,X) 70,122
Su 52,56 (Xе, Xе) 70
Sucp 57 {X- < a} 214
S(a) 248 °Xn 369
5(Л) 179 xVy 211
Tx 265 xAy 211
Гл 104 ж+ 211
[T] 104 ж" 211
иер 57 У 64,267
Var(X) 116 Ya+ 267
Varr(X) 116 Y- 64,159
Х(ст„) 269 У' 377
X* 11,244 [Z,Z] 84
Х+* 373 Zta 221
Х+* 373 (Z : Я)* 319
Xе 70,221 J*(dZs)Hs 319
Хт 10,178 П 293,309
Хт" 54,172 a-slice 248
Хт" 369 4 305
Xf 293,301 ф 301,382
[Х,Х] 66 <р 311
[Х,Х]С 70,70,216,271 тгп 17
[Х,У] 66 <t{L\L°°) 150
[Xе, Xе] 70 an 64
AXt 25,60,158 С,{х,ш) 303
\\X\\hp 245 < 131

Subject Index
cc-sliceable 248
BMO see bounded mean oscillation
F optional projection 369
F special, G semimartingale 369
F-S see Fisk-Stratonovich
FV see finite variation process
U2 norm 154
W norm 193
HF
(pre)locally in 247
converge (pre)locally in 260
norm 245
gp
(pre) locally in 247
converge (pre) locally in 260
norm 244
ucp see uniform convergence on
compacts in probability
absolutely continuous 131
Absolutely Continuous Compensators
Theorem 192
absolutely continuous space 191
accessible stopping time 103
adapted process
decomposable 55
definition of 3
space L of with caglad paths 155
with bounded caglad paths 56
with cadlag paths 56
with caglad paths 56
angle bracket 123
announcing sequence for stopping time
103
approximation of the compensator by
Laplacians 150
arcsine law 228
associated jump process 27
asymptotic martingale (AMART) 218
autonomous 250,293
Azema's martingale 204
and last exit from zero 228
definition of 228
local time of 230
Backwards Convergence Theorem 9
Banach-Steinhaus Theorem 43
Barlow's Theorem 240
Bichteler-Dellacherie Theorem 144
Bouleau-Yor formula 226, 227
bounded functional Lipschitz 257
bounded jumps 25
bounded mean oscillation
BMO norm 195
BMO space of martingales 195
Duality Theorem 199
bracket process 66
Brownian bridge 97, 299,384
Brownian motion
absolute value of 217
arcsine law for last exit from zero
228
Azema's martingale 228
Brownian bridge see Brownian
bridge
completed natural filtration right
continuous 19

408 Subject Index
conditional quadratic variation of
123
continuous modification 17
continuous paths 221
definition of 16
Fisk-Stratonovich exponential of
280
Fisk-Stratonovich integral for 288
geometric 86
Girsanov-Meyer Theorem and 141
Ito integral for 78
Levy's characterization of 88
Levy's Theorem (martingale
characterization) 86,87
last exit from zero 228
last exit time 46
local time of 217, 225
martingale characterization (Levy's
Theorem) 86,87
martingale representation for 187
martingales as time change of 88,
187
maximum process 23
on the sphere 281
Ornstein-Uhlenbeck 298
quadratic variation of 17, 67, 123
reflection principle 23, 228
reversibility for integrals 380
semimartingale 55
standard 17
starting at ж 17
stochastic area formula 89
stochastic exponential for see
geometric Brownian motion
stochastic integral exists 173
strong Markov property for 23
Tanaka's formula 217
tied down or pinned see Brownian
bridge
time change of 88
unbounded variation 19
white noise 141, 243
Burkholder's inequality 222
Burkholder-Davis-Gundy inequalities
193
cadlag 4
caglad 4
canonical decomposition
129,154
censored data 121
centering function 32
change of time 88,190
exercises in Chap. II 98
Lebesgue's formula 190
change of variables see Ito's formula,
see Meyer-Ito formula
Change of Variables Theorem
for continuous FV processes 41
for right continuous FV processes
78
classical semimartingale 102,127,144
Class D 106
closed martingale 8
Comparison Theorem 324
compensated Poisson process 31,42,
65
compensator 118
Absolutely Continuous Compensators
Theorem 192
Knight's compensator calculation
method 151
compensator of l{t>z,} for F 371
compound Poisson process 33
conditional quadratic variation 70,
122
Brownian motion 123
polarization identity 123
continuous local martingale part of
semimartingale 70,221
continuous martingale part 191
converge (pre) locally in H_p, S_p 260
convex functions
and Meyer-Ito formula 214
of semimartingale 210
countably-valued random variables
corollary 365
counting process 12
explosion time 13
without explosions 13
crude hazard rate 122
decomposable adapted process 55,101
diffeomorphism of Rn 319
diffusion 243,291,297
examples 298
rotation invariant 282
diffusion coefficient 298
discrete Laplacian approximations 150

Subject Index 409
Doleans-Dade exponential see
stochastic exponential
Dominated Convergence Theorem for
stochastic integrals
in Яг 267
in ucp 174
Doob class see Class D
Doob decomposition 105
Doob's maximal quadratic inequality
11
Doob's Optional Sampling Theorem 9
Doob-Meyer Decomposition Theorem
case without Class D 115
general case 111
totally inaccessible jumps 106
drift coefficient 143, 298
removal of 137
Duality Theorem 199
Dynkin's expectation formula 350
Dynkin's formula 56,350
Einstein convention 305
Emery's example of stochastic integral
behaving badly 176
Emery's inequality 246
Emery's structure equation see
structure equation
Emery-Perkins Theorem 240
enveloping sequence for stopping time
103
equivalent probability measures 131
Euler method of approximation 353
evanescent 158
example
Emery's example of stochastic
integral behaving badly 176
expansion via end of SDE 367
Gaussian expansions 366
hazard rates and censored data 121
Ito's example 366
reversibility for Brownian integrals
380
reversibility for Levy process integrals
380
example of local martingale that is not
martingale 37,74
example of process that is not
semimartingale 217
example of stochastic integral that is
not local martingale 176
examples of diffusions 298
Existence of Solutions of Structure
Equation Theorem 201
expanded filtration 370
expansion by a natural exponential r.v.
386
expansion via end of SDE 367
explosion time 13, 254,303
extended Gronwall's inequality 352
extremal point 183
Fefferman's inequality 195
strengthened 195
Feller process 35
filtration
countably-valued random variables
corollary 365
definition of 3
expansion by a natural exponential
r.v. 386
filtration shrinkage 367
Filtration Shrinkage Theorem 369
Gaussian expansions 366
independence corollary 365
Ito's example 366
natural 16
progressive expansion 355, 369
quasi left continuous 148,189
right continuous 3
filtration shrinkage 367
Filtration Shrinkage Theorem 369
finite quadratic variation 271
finite variation process
definition of 39,101
integrable variation 111
Fisk-Stratonovich
acceptable 278
approximation as limit of sums 284
integral 82,271
integral for Brownian motion 288
Integration by Parts Theorem 278
Ito's circle 82
Ito's formula 277
stochastic exponential 280
flow
of SDE 301
of solution of SDE 382

410 Subject Index
strongly injective 311
weakly injective 311
Fubini's Theorem for stochastic
integrals 207,208
function space 301
functional Lipschitz 250
fundamental L martingale 372
fundamental sequence 37
Fundamental Theorem of Local
Martingales 125
Gamma process 33
Gaussian expansions 366
generalized Ito's formula 271
generator of stable subspace 179
geometric Brownian motion 86
Girsanov-Meyer Theorem 132
predictable version 133
graph of stopping time 104
Gronwall's inequality 342, 352
extended 352
Hadamard's Theorem 330
Hahn-Banach Theorem 199
hazard rates 121
hitting time 4
Holder continuous 99
honest random variable 373
Hunt process 36
Hypothesis A 221
increasing process 39
independence corollary 365
independent increments
of Levy process 20
of Poisson process 13
index of stable law 34
indistinguishable 3,60
infinitesimal generator 349
injective flow see strongly injective
flow
integrable variation process 111
Integration by Parts Theorem
for Fisk-Stratonovich integrals 278
for semimartingales 68, 83
intrinsic Levy process 20
Ito integral see stochastic integral
Ito's circle 82
Ito's formula
for an n-tuple of semimartingales 81
for complex semimartingales 83, 84
for continuous semimartingales 81
for Fisk-Stratonovich integrals 277
for semimartingales 78
generalized 271
Ito-Meyer formula see Meyer-Ito
formula
Ito's example 366
Ito's Theorem for Levy processes 356
Jacod's Countable Expansion 366
Jacod's Countable Expansion Theorem
53, 356
Jacod's criterion 363
Jacod-Yor Theorem on martingale
representation 199
Jensen's Inequality 11
Jeulin's Lemma 360
Kazamaki's criterion 139
Knight's compensator calculation
method 151
Kolmogorov's continuity criterion 220
Kolmogorov's Lemma 218
Kronecker's delta 305
Kunita-Watanabe inequality 69
Levy Decomposition Theorem 31
Levy measure 26
Levy process
associated jump process 27
bounded jumps 25
cadlag version 25
centering function 32
definition of 20
has finite moments of all orders 25
intrinsic 20
is a semimartingale 55
Ito's Theorem for 356
Levy measure 26
reflection principle 49
reversibility for integrals 380
strong Markov property for 23
symmetric 49
Levy's arcsine law 228
Levy's stochastic area formula 89
Levy's stochastic area process 89
Levy's Theorem characterizing
Brownian motion 86,87

Subject Index 411
Levy-Khintchine formula 31
last exit from zero 228
Le Jan's Theorem 124
Lebesgue's change of time formula 190
Lenglart's Inclusion Theorem 177
Lenglart-Girsanov Theorem 135
linkage operators 329
Lipschitz
bounded functional 257
definition of 250, 293
functional 250
locally 251,254,303
process 250,311
random 250
local behavior of stochastic integral
62,165,170
local behavior of the stochastic integral
at random times 375
local convergence in H_p, S_p 260
local martingale
BMO 195
cadlag martingale is 37
compensator of 118
condition to be a martingale 38, 73
continuous part of semimartingale
221
decomposable 126
definition of 37
example that is not a martingale
37,74
fundamental sequence for 37
Fundamental Theorem 125
intervals of constancy 71, 75
not locally square integrable 126
not preserved under shrinkage of
filtration 126
pre-stopped 172
preserved by stochastic integration
see stochastic integral, preserves
reducing stopping time for 37
time change of Brownian motion 88
local property 38,162
local time
and Bouleau-Yor formula 227
and change of variables formula
see Meyer-Ito formula, see
Meyer-Tanaka formula
and delta functions 217
continuous in t 213
definition of L$ 212
discontinuous, example of 225
occupation time density 216, 225
of Azema's martingale 230
of Brownian motion 217
of semimartingale 212
regularity in space 224
support of 214
locally bounded process 164
locally in Kp,gp 247
locally integrable process 359
locally integrable variation process
111
locally Lipschitz 251, 254,303
locally special semimartingale 149
Memin's criterion for exponential
martingales 352
Metivier-Pellaumail inequality 352
Metivier-Pellaumail method 352
Markov process
definition of 34
diffusion 243, 291, 297
Dynkin's expectation formula 350
Dynkin's formula 56
equivalence of definitions 34
infinitesimal generator 349
simple 291
strong 292
time homogeneous 35, 292
transition function 35, 292
transition semigroup 292
martingale
L2 projection 9
W norm 193
BMO 195
asymptotic (AMART) 218
Azema's 204,228
Backwards Convergence Theorem 9
Burkholder's inequality 222
cadlag modification 8
closed 8
continuous martingale part 191
definition of 7
Doob's inequalities 11
fundamental L martingale 372
in L2 178
Jacod-Yor Theorem 199
local 37

412 Subject Index
Martingale Convergence Theorem 8
martingale representation 203
maximal quadratic inequality 11
measures M2(A) 182
not locally square integrable 126
Optional Sampling Theorem 9
orthogonal 179
predictable representation property
182
purely discontinuous part 191
quasimartingale 116
sigma martingale 233
space M2 of 178
square integrable 73, 74
strongly orthogonal 179
submartingale 7
supermartingale 7
uniformly integrable 8
with exactly one jump 124
Martingale Convergence Theorem 8
martingale representation 203
mathematical finance theory 137
maximal quadratic inequality 11
Meyer's Theorem 104
Meyer-Ito formula 214
Meyer-Tanaka formula 216
modification 3
Monotone Class Theorem 7
monotone vector space 7
closed under uniform convergence 7
multiplicative collection of functions 7
natural filtration 16
natural process 111
net hazard rate 122
Novikov's criterion 140
oblique bracket 123
occupation time density 216, 225
optional <7-algebra 102,372
optional projection 367, 369
Optional Sampling Theorem 9
Ornstein-Uhlenbeck process 298
orthogonal martingales 179
Ouknine's formula 240
partition 116
path space 142
path-by-path continuous part
70
Perkins-Emery Theorem 240
Picard iteration 255
pinned Brownian motion 299
Pitman's Theorem 240
Poisson process
arrival rate 15
compensated 31,42, 65,118
compound 33
definition of 13
independent increments 13
intensity 15
stationary increments 13
polarization identity 66,123, 271
potential 105,150
pre-stopped local martingale 172
predictable cr-algebra 102, 154
predictable compensator of l{t>z,} for F
371
predictable process
predictable <7-algebra 102
simple 51
predictable projection 368
predictable representation property
182
predictable stopping time 103
predictably measurable process 102,
105
prelocal convergence in i?p, S_p 260
prelocally in i?p, §? 247
preserved by stochastic integration see
stochastic integral, preserves
process Lipschitz 250,311
process stopped at T 10, 37
progressive expansion 355, 369
projections
optional projection 367
predictable projection 368
property holds locally 38,162, 247
property holds prelocally 162, 247
purely discontinuous part of martingale
191
quadratic covariation 66, 378
polarization identity 66,271
quadratic covariation process 270
quadratic pure jump 71
quadratic variation
conditional 122
continuous part of 70, 271

Subject Index 413
covariation 66
finite 271
of a semimartingale 66
polarization identity 66, 271
quasi left continuous filtration 148,
189
quasimartingale 116
random Lipschitz 250
random partition
definition of 64
tending to the identity 64
Rao's Theorem 118
Ray-Knight Theorem 225
reduced by stopping time 37
reflection principle
for Brownian motion 23, 228
for Levy processes 49
for stochastic area process 92
regular conditional distribution 362
regular supermartingale 150
representation property 182
reversibility for Brownian integrals
380
reversibility for Levy process integrals
380
reversible semimartingale 378
Riemann-Stieltjes integral 41
right continuous filtration 3
right stochastic exponential 319, 320
right stochastic integral 319
rotation invariant diffusion 282
sample paths of stochastic process 4
scaling property 238
semimartingale
(F, G) reversible 378
(H2,X) integrable 163
H2 norm 154
absolute value of see Meyer-Tanaka
formula
bracket process 66
canonical decomposition 129
classical 102,127,144
continuous local martingale part 70,
221
convex function of 210
definition of 52
equivalent norm 244
example of process that is not 217
Integration by Parts Theorem 68,
83
local martingale as 127
local time of 212
locally special 149
preserved by stochastic integration
see stochastic integral, preserves
quadratic pure jump 71
quadratic variation of 66
quasimartingale as 127
space H2 of 154
special 129,154
stochastic exponential of 85
submartingale as 127
supermartingale as 127
topology 264
total 52
semimartingale topology 264
separable measurable space 189
sharp bracket 123
sigma martingale 233
preserved by stochastic integration
see stochastic integral, preserves
sign function 212
signal detection 140
simple Markov process 291
simple predictable process 51
Skorohod topology 220
Skorohod's Lemma 239
smallest filtration making r.v. a
stopping time 119,370
special semimartingale 129,154
canonical decomposition 129
square integrable martingale 178
preserved by stochastic integration
see stochastic integral, preserves
stable law 34
index 34
stable process
defintion of 33
scaling properties 34
symmetric 34
stable subspace
generated 179
of M2 178
stable under stopping 178
standard Borel space 362
stationary Gaussian process 299

414 Subject Index
stationary increments
of Levy process 20
of Poisson process 13
statistical communication theory 140
Stieltjes integral see Riemann-Stieltjes
integral
stochastic area formula 89
stochastic area process
definition of 89
density 91
Levy's formula 89
properties 92
reflection principle 92
stochastic differential equation 255,
321
flow of 301
flow of solution 382
weak solution 201, 240
weak uniqueness 201
stochastic exponential
appproximation of 269
as a diffusion 298
definition of 85
Fisk-Stratonovich 280
for Brownian motion see geometric
Brownian motion
for semimartingale 85
invertibility of 335
right 319,320
uniqueness of 255
stochastic integral
X integrable, L(X) 163
Associativity Theorem 62, 159, 165
behavior of jumps 60,158,165
does not preserve FV processes in
general 241
Dominated Convergence Theorem in
Ш 267
Dominated Convergence Theorem in
ucp VTA
example that is not local martingale
176
for L 59
for bV 156
for S 58
for V 163
for Brownian motion 78
Fubini's Theorem 207, 208
local behavior at random times 375
local behavior of 62,165,170
preserves
FV processes, integrands in L 63
continuous local martingales 173
local martingales 128,171,174
locally square integrable local
martingales 63, 171
semimartingales 63
sigma martingales 234
square integrable martingales 159,
171
right 319
said to exist 163
stochastic process
adapted 3
decomposable 55,101
definition of 3
finite variation 39,101
increasing 39
indistinguishable 3,60
locally bounded 164
locally integrable 359
modification of 3
natural 111
potential 105
sample paths of 4
stopped 10,37
strong Markov 36
stopped process 10, 37
stopping time
cr-algebra 5,105
accessible 103
announcing sequence 103
change of time 98
definition of 3
enveloping sequence 103
explosion time 13, 254, 303
fundamental sequence of 37
graph 104
hitting time 4
hitting time of Borel set is 5
predictable 103
reducing 37
totally inaccessible 103
Stratonovich integral see Fisk-
Stratonovich, integral
Stratonovich Integration by Parts
Theorem 278
strengthened Fefferman inequality 195

Subject Index 415
Strieker's Theorem 53
strong Markov process 36, 292
strong Markov property 36, 292
for Brownian motion 23
for Levy process 23
strongly injective flow 311
strongly orthogonal martingales 179
structure equation 200, 238
Existence of Solutions Theorem 201
submartingale 7
supermartingale 7
regular 150
symmetric Levy process 49
symmetric stable process 34
Tanaka's formula 217
tied down Brownian motion 299
time change 190
time change of Brownian motion 88
time homogeneous Markov process
292
time reversal 377
total semimartingale 52
total variation process 40
totally inaccessible stopping time 103
transition function for Markov process
292
transition semigroup 292
truncation operators 265
uniform convergence on compacts in
probability 57
uniformly integrable martingale 8
usual hypotheses 3, 291
variation 116
variation along т
116
weak solution of SDE 201, 240
weak uniqueness of SDE 201
weakly injective flow 311
white noise 140, 243
Wiener measure 141
Wiener process 137,140

Applications of Mathematics (continued from page и>
51 Asmussen, Applied Probability and Queues Bnd ed. 2003)
52 Robert, Stochastic Networks and Queues B003)
53 Glasserman, Monte Carlo Methods in Financial Engineering B003)