Stochastic
Differential Equations
and Applications
Volume 1
Avner Friedman
Department of Mathematics
Northwestern University
Evanston, Illinois
ACADEMIC PRESS New York San Francisco London 1975
Л Subsidiary of Harcourt Brae* Jocanovich, Publisher*

Copyright © 1975, by Academic Press, Inc.
all rights reserved.
no part of this publication may be reproduced or
transmitted in any form or by any means, electronic
or mechanical, including photocopy, recording, or any
information storage and retrieval system, without
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ACADEMIC PRESS, INC.
Ill Fifth Avenue, New York, New York 10003
United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1
Library of Congress Cataloging in Publication Data
Friedman, Avner.
Stochastic differential equations and applica-
applications.
(Probability and mathematical statistics series)
Bibliography: p.
Includes index.
1. Stochastic differential equations. I. Ti-
Title.
QA274.23.F74 519.2 74-30808
ISBN 0-12-268201-7
AMS(MOS) 1970 Subject Classifications: 60H05, 60H10,
35J25, 35K15, 93E05, 93E15, 93E20.
PRINTED IN THE UNITED STATES OF AMERICA

Contents
Preface ix
General Notation xi
Contents of Volume 2 xiii
1 Stochastic Processes
1. The Kolmogorov construction of a stochastic process 1
2. Separable and continuous processes 6
3. Martingales and stopping times 9
Problems 15
2 Markov Processes
1. Construction of Markov processes 18
2. The Feller and the strong Markov properties 23
3. Time-homogeneous Markov processes 30
Problems 31
3 Brownian Motion
1. Existence of continuous Brownian motion 36
2. Nondifferentiability of Brownian motion 39
3. limit theorems 40
4. Brownian motion after a stopping time 44
5. Martingales and Brownian motion 46
6. Brownian motion in n dimensions 50
Problems 53

vi CONTENTS
4 The Stochastic Integral
1. Approximation of functions by step functions 55
2. Definition of the stochastic integral 59
3. The indefinite integral 67
4. Stochastic integrals with stopping time 72
5. Ito's formula 78
6. Applications of Ito's formula 85
7. Stochastic integrals and differentials in n dimensions 89
Problems 93
5 Stochastic Differential Equations
1.
2.
3.
4.
5.
6.
Existence and uniqueness
Stronger uniqueness and existence theorems
The solution of a stochastic differential system
as a Markov process
Diffusion processes
Equations depending on a parameter
The Kolmogorov equation
Problems
98
102
108
114
117
123
125
Elliptic and Parabolic Partial Differential
Equations and Their Relations to
Stochastic Differential Equations
1. Square root of a nonnegative definite matrix 128
2. The maximum principle for elliptic equations 132
3. The maximum principle for parabolic equations 134
4. The Cauchy problem and fundamental solutions
for parabolic equations 139
5. Stochastic representation of solutions of partial
differential equations 144
Problems 150
7 The Cameron-Martln-Glrsanov Theorem
1. A class of absolutely continuous probabilities 152
2. Transformation of Brownian motion 156
3. Girsanov's formula 164
Problems 169

CONTENTS vll
8 Asymptotic Estimates for Solutions
1. Unboundedness of solutions 172
2. Auxiliary estimates 174
3. Asymptotic estimates 180
4. Applications of the asymptotic estimates 185
5. The one-dimensional case 188
6. Counterexample 191
Problems 193
9 Recurrent and Transient Solutions
1.
2.
3.
4.
5.
6.
7.
Transient solutions
Recurrent solutions
Rate of wandering out to infinity
Obstacles
Transient solutions for degenerate diffusion
Recurrent solutions for degenerate diffusion
The one-dimensional case
Problems
Bibliographical Remarks
References
Index
196
200
203
207
213
217
219
222
224
226
i

Preface
The object of this book is to develop the theory of systems of stochastic
differential equations and then give applications in probability, partial
differential equations and stochastic control problems.
In Volume 1 we develop the basic theory of stochastic differential equa-
equations and give a few selected topics. Volume 2 will be devoted entirely to
applications.
Chapters 1-5 form the basic theory of stochastic differential equations.
The material can be found in one form or another also in other texts.
Chapter 6 gives connections between solutions of partial differential equa-
equations and stochastic differential equations. The material in partial differential
equations is essentially self-contained; for some of the proofs the reader is
referred to an appropriate text.
In Chapter 7 Girsanov's formula is established. This formula is becoming
increasingly useful in the theory of stochastic control.
In Chapters 8 and 9 we study the behavior of sample paths of the solution
of a stochastic differential system, as time increases to infinity.
The book is written in a textlike style, namely, the material is essentially
self-contained and problems are given at the end of each chapter. The only
prerequisite is elementary probability; more specifically, the reader is as-
assumed to be familiar with concepts such as conditional expectation, inde-
independence, and with elementary facts such as the Borel—Cantelli lemma. This
prerequisite material can be found in any probability text, such as Breiman
[1; Chapters 3,4].
I would like to thank my colleague, Mark Pinsky, for some helpful
conversations.
Ix

General Notation
All functions are real valued, unless otherwise explicity stated.
In Chapter n, Section m the formulas and theorems are indexed by (m.k)
and m.fc respectively. When in Chapter I, we refer to such a formula (m.fc)
(or Theorem m.fc), we designate it by (n.m.k) (or Theorem n.m.k) if 1фп,
and by (m.fc) (or Theorem m.fc) if I— n.
Similarly, when referring to Section m in the same chapter, we designate
the section by m; when referring to Section m of another chapter, say n, we
designate the section by n.m.
Finally, when we refer to conditions (A), (Aj), (B) etc., these conditions are
usually stated earlier in the same chapter.
xl

Contents of Volume 2
10. Auxiliary Results in Partial Differential Equations
11. Nonattainability
12. Stability and Spiraling of Solutions
13. The Dirichlet Problem for Degenerate Elliptic Equations
14. Small Random Perturbations of Dynamical Systems
15. Fundamental Solutions for Degenerate Parabolic Equations
16. Stopping Time Problems and Stochastic Games
17. Stochastic Differential Games
xlll

1
Stochastic Processes
1. The Kolmogorov construction of a stochastic process
We write a.s. for almost surely, a.e. for almost everywhere, and a.a. for
almost all.
We shall use the following notation:
Я" is the real euclidean n-dimensional space;
®n is the Borel ст-field of Я", i.e., the smallest ст-field generated by the
open sets of Й";
Я °° is the space consisting of all infinite sequences (xx, x2, . . . , xn, . . .)
of real numbers;
5 x is the smallest o-field of subsets of Я °° containing all fc-dimensional
rectangles, i.e., all sets of the form
{(*!, x2, . . . ); x^e/i, . . . , xkG.Ik), к > 0,
where Ilt . . . , Ik are intervals.
Clearly <¦$ ж is also the smallest ст-field containing all sets
{(xi.% ...);(*i, ...,хк)&в], ве«(.
An n-dimensional distribution function Fn(xlt . . . , xn) is a real-valued
function defined on Я" and satisfying:
(i) for any intervals Ik = [ak, bk), 1 < к < n,
V • • \Fn(x)>0 A.1)
where
\f(X) " /(*1> *k-l» felt' X* + l' *n)
~ f{xi> • • • • xk-i> °k> xk+v • • • > ^n);
(ii) if xfk) t x, as к | oo A < / < n), then
Frt(*<*>, . . . , **„*>) T Fn(xlt...,xn) as fc|°o;

2 1 STOCHASTIC PROCESSES
(Hi) if x, I — oo for some /', then Fn(x1, . . . , xn) j, 0, and if x^ | oo for
all /, 1 < / < n, then Fn(xv . . . , xn) | 1.
Unless the contrary is explicitly stated, random variables are always
assumed to be real valued, i.e., they do not assume the values + oo or — oo.
If X1? . . . , Xn are random variables, then X = (Xlf . . . , XJ is called an
n-dimensional random variable, or a random variable with values in fi", The
function
Fn(Xl, ...,xn) = P(X, <xl,...,Xn<xn) A.2)
is an n-dimensional distribution. Conversely, for any n-dimensional distribu-
distribution Fn there is a probability space and a random variable X with values in
Я" such that its distribution function is Fn, i.e., A.2) holds; see, for instance,
Breiman [1].
A sequence {Fn(xv . . . , xn)} of distribution functions is said to be
consistent if
ton Fn(x1, . . . , xn) = Fn_1(x1, . . . , x,^), n > 1. A.3)
Let (B, *&, P) be a probability space, and let (Xn) be a sequence of
random variables. We call such a sequence a discrete time stochastic process,
or a countable stochastic process, or a stochastic sequence. The distribution
functions of (Xj, . . . , Xn), n > 1, form a consistent sequence. The converse
is also true, namely:
Theorem 1.1. Let {Fn} be a sequence of n-dimensional distribution func-
functions satisfying the consistency condition A.3). Then there exists a probabil-
probability space (В, 5Г, P) and a countable stochastic process {Xn} such that
, <Xl,...,Xn<xn) = F(xlt . . . , xJ. n > 1.
In fact, one can take В - Я °°, ?F = %K,
P{(x1, x2,...)-x1<y1,...,xn< yn) = Fn( ylt . . . , yn)
and more generally
P{(xlf X2,...);a1<x1<b1,...,an<xn<bn}
is given by the left-hand side of A.1), and
Хп{(хъ х2, ...)) = xn.
The consistency condition guarantees that P is well defined on all rectangles.
The main nontrivial point here is the fact that P can be extended into a
probability in (B, §), and this follows from the Kolmogorov extension
theorem:

1. THE KOLMOGOROV CONSTRUCTION
If P is defined on all finite-dimensional rectangles, and:
(i) P > 0, P(R°°) = 1;
(ii) if Sl5 . . . , Sm are disjoint n-rectangles and S = Sx U • • • LJSm
7^
(iii) if (S^j is a nondecreasing sequence of n-dimensional rectangles and
S, t S as / | oo, then lim^Q0P(S/) = P(S);
then there is a unique extension of P into a probability on (Я °°, SJ.
For proof, see Breiman [1].
A stochastic process is a family of random variables {X(t)} defined on a
probability space (fi, <5, P), where t varies in a real interval I (I is open,
closed, or half-closed). We denote it also by {X(t), t&I), {X(t)} (t&I) or
X(t), t&I.
A stochastic process defines a set of distribution functions
) < «i, • • • , X(tJ < xj A.4)
where tY < • • • < ?n, t €E Z, n = 1, 2, . . . ; they are called the finite-dimen-
finite-dimensional joint distribution functions of the process.
A set of distribution functions
{Fti...K(Xl, ...,xn)} A.5)
defined for all ^ < • • • < tn, Ц, GI, 1 < / < n, n = 1, 2, . . . , is said to be
consistent if
4m Fh ¦ ¦ ¦ tSxi> ¦ ¦ ¦ ' *«) = F*i ¦ ¦ • <*-i<*+i ¦ ¦ ¦ Jxi' • • ' ' x*-i> xk+i> • ¦ ¦ > xn)-
The set given by A.4) is consistent. We shall now prove the converse.
Theorem 1.2. Given a consistent set of distribution functions as in A.5),
there exists a stochastic process {X(t), tG.1] defined on a probability space
(fl, S7, P) such that the given distribution functions are the joint distribution
functions of the process, i.e., such that A.4) holds.
Proof. Denote by Я1 the set of all real valued functions *(•) on I. A set of
the form
B = {xf);D),x(f2),...ND}, ?>?$« A.6)
is called a cylinder set with countable base {tf}. Denote by 5z the class of
all cylinder sets with countable base. This class is clearly a ст-field, and it
coincides with the ст-field generated by all the finite-dimensional rectangles
in Я',
{x{-);x(t1)el1,...,x(tn)GIn},
where /j, . . . , In are intervals.

1 STOCHASTIC PROCESSES
We take В = Я1, ?F = ®x. Given a sequence T
r the class of all sets
in /, denote by
By Theorem 1.1 and the remark following it, there exists a unique probabil-
probability measure PT on (B, 5 T) such that
FT{x(-); x(*i) < «i, • • • , *(O < *„} = F,,,....^*! *„)> n > 1.
A.7)
Set P(B) = Pr(B) if В G <Sr. For this definition to make sense we must
show that if В G <Sr where 7" = {*•} then
P^B) = Fr(B). A.8)
Setting S-ГиГ we have TcS, <Src<Ss. Since PT = Ps on all
finite-dimensional rectangles with base in T, PT = Ps on any set of 5 r. Thus
Pr(B) = PS(B). Similarly, Pr(B) = PS(B), and A.8) follows.
Having defined the probability space (В, |3Г, Р), we now take
A.4) then immediately follows.
We refer to the stochastic process constructed in this proof as the process
obtained by the Kolmogorov construction.
Definition. A v-dimensional countable stochastic process (or a v-dimen-
sional stochastic sequence) is a sequence (Xn) of ^-dimensional random
variables. A family {X(t), t&I) of ^-dimensional random variables is called a
v-dimensional stochastic process.
A j'-dimensional stochastic sequence (Xn) gives rise to functions
Fn(xlf . . . , xn) = P(XX <x1,...,Xn<xn) A.9)
where xt G Я" and a < b means that af < Ц for 1 < / < v, where a =
(av . . . , a,), b = (Ь» . . ., by). Setting
xn_v x*)
fc < v)
A.10)
where
П -
X*
(X(n-i)r
it is clear that the sequence
distribution functions.
k, 00, .... 00),
. . , x,)} is a consistent sequence of

1. THE KOLMOGOROV CONSTRUCTION 5
Definition. A function FnCcj, . . . , xn) defined for all -с(ей' is called an
n-dimensional distribution function with respect to R" if:
(i) condition A.1) holds with ak, bk replaced by ak = (a^, . . . , оь),
^k = (bkv • • • > fyb,), where ak < bk, and x;- replaced by ? in the definition of
4 '(ii) Fn(x[k), 5Ц4) T *»(*i. ¦ ¦ • > *J « *W t ^ for 1 < fc < n (i.e.,
each component of x/*) increases to the corresponding component of x,);
(iii) if some fcth component of some x{ decreases to — oo, then
Fn(x1> . . . , xn) I 0; and if all components of all the Щ increase to oo, then
ад *«) т i.
Set
Fn(*l> • • • > *„-!> **) = ,lim Fn(*l> • • • . *„-!> «„*) A-11)
к—>oo
where the first к components of x*, x* agree, whereas the last v — к
components of x* and x* are equal, respectively, to oo and k. Then,
obviously, the set of distribution functions G^n_1-jv + 1, . . . , Gny defined by
A.10) is consistent.
A sequence of distributions (Fn) with respect to Я" is said to be
consistent if
lim Fn{xlt ... ,xn) = Fn_1(x1, . . . , x^),
where xn | oo means that each component of xn increases to oo. This is
clearly the case if and only if the corresponding sequence of distributions
G{(xlt . . . , x{), defined by A.10), A.11), is consistent.
If we apply Theorem 1.1 to the latter sequence, we conclude that for any
consistent sequence of distribution functions, with respect to Я", there exists
a countable ^-dimensional stochastic process {Xn} whose distribution func-
functions are the Fn, i.e., A.9) holds. The probability space (B, 5", F) can be
taken such that В = Я"' °° = the space of all infinite sequences (xv x2, . . . ),
х„еЯ", and ?F = ®„ „, the smallest ст-field containing all finite-dimensional
rectangles
{(х1г X2, . . . );X1€.I1,X2€.i2, . . .}
where Z? is an interval in Я".
A set of distribution functions {F/i...<>(x1, . . . , 3cn)} where tY <
¦ ¦ ¦ < tn, t{ ? I, n = 1, 2, . . . , is said to be consistent if
i F«. ¦ • ¦ J*l *.)-*"«,¦•. 4_Л+1 ¦ • • J*i. • • • . *k-i. )
Theorem 1.3. Given a consistent set of distribution functions with respect
to R>, { Fh .. . ^ (xu . . . , xj; *!<•••< tn, t{ G I }, there exists

1 STOCHASTIC PROCESSES
probability space (fi, ?F, P) and a v-dimensional stochastic process X(t)
(t G I) defined on it such that
< *„] = F,,...^, . . . , xj. A.12)
The functions Fti... ^ are called the finite-dimensional joint distribution
functions (with respect to Я') of the process X(t).
The proof is similar to the proof of Theorem 1.2; it is based on the
extension of Theorem 1.1 to distribution functions with respect to Я",
mentioned above. Here fi is the space RrI of all functions *(•) from I into
Я* and <5 = <&„! consists of all sets A.6) with D G %r< x. The construction
of the process X(t) in this proof is called the Kolmogorov construction.
2. Separable and continuous processes
Definition. A ^-dimensional stochastic process {X(t), t€.I) is called separ-
separable if there exists a countable sequence T = {t} that is a dense subset of I
and a subset N of fi with PBV) = 0 such that, if w ? 2V,
{X(t, u)?Ffor all *G/} = {X(^, to) GF for all ^G/}
for any open subset ] oil and for any closed subset F of Я". Г is called a set
of separability, or briefly, a separant.
Two ^-dimensional stochastic processes (X(t)} and (X'(f)} (teZ) defined
on the same probability space are said to be stochastically equivalent if
P[X{t) ф X'(t)] = 0 for all
We then say that [X'(t)} is a uersion of
Theorem 2.1. Any v-dimensional process is stochastically equivalent to a
separable stochastic process.
The random variables of the separable process may actually take the
values ±oo, even though the random variables of the original process are
real valued.
For proof the reader is referred to Doob [1]. Theorem 2.1 will not be used
in the text of this book.
From Theorem 2.1 it follows that when one constructs a stochastic
process from a given set of distribution functions (as in the Kolmogorov
construction), one may always take this process to be separable.
Let X(t) be a r-dimensional stochastic process for tGl. Thus, for each t,
X(t) is a random variable X(t, to). The functions t-*X(t,u) (to fixed) are
called the sample paths of the process. If for a.a. to the sample paths are

2. SEPARABLE AND CONTINUOUS PROCESSES 7
continuous functions for all t€.I, then we say that the stochastic process is
continuous. If for a.a. w the sample paths are right (left) continuous, then we
say that the stochastic process is right (left) continuous.
It is easy to see that a right (or left) continuous stochastic process is
separable and any dense sequence in I is a set of separability.
A stochastic process [X(t), tG.1) is said to be continuous in probability if
for any sGI and € > 0,
P[\X(t) - X(s)\ > c]->0 if t?I, t^s.
The following theorem is due to Kolmogorov.
Theorem 2.2. Let X(t) (tG.1) be a v-dimensional stochastic process such
that
E\X{t) - X(s)\fi < C\t- s\1+a B.1)
for some positive constants C, a, /?. Then there is a version of X(t) that is a
continuous process.
Proof. Take for simplicity I - [0, 1]. Let 0 < у < a/ /? and let 8 be a
positive number such that
A - S)(a + 1 - py) > 1 + 8. B.2)
Then, by B.1),
;<2n
for some 0 < i < / < 2", / - i < 2"*}
where ц > 0 by B.2), and Cl5 C2 are positive constants. Since 22~nAl < oo,
the Borel-Cantelli lemma implies that for a.a. w
|X(f2-") - X(i2"")| < h((j - iJ"") if 0 < i < j < 2",
/ - i < 2"", n > m(u)
B.3)
where Л(<) ™ fY and m(w) is some sufficiently large positive integer.
Let tlt t2 be any rational numbers in the interval @, 1) such that tr < t2,

8 1 STOCHASTIC PROCESSES
t = h ~ h < 2~mA~e) where m = m(u>) is as in B.3). Choose n > m such
that
We can expand tv t2 in the form
ty = i2~n - 2-P' - • • ¦ - 2-л
t2 = /2~B + 2"'1 + • • • + 2~«
where n < p1 < ¦ ¦ ¦ < pk, n < qx < ¦ • ¦ < qr. Then tx < i2~n <
/2~" < t2 and / - i < Й" < 2"*.
By B.3),
|X(t2-" - 2~P' - • • • - 2~»)
Hence
it
|Х(*!> - X(i2"")| < 2 bB-ft) < C3h{2'n) (C3 constant).
r=l
Similarly,
|X(g-X(/2"")|< C3hB-).
Finally, by B.3),
Hence
|X(t2) - X(fx)| < Ch(t2 - tx) (C constant). B.4)
Let T = {^} be the sequence of all rational numbers in the interval @, 1).
From B.4) it follows that
|ХЦ) - X((j)| < Ch(l» " 0 if 0 < ^ - ^ < 2-mA-S), m = m(«).
B.5)
Thus X(t, w), when restricted to te Г is uniformly continuous. Let X(t, w)
be its unique continuous extension to 0 < t < 1. Then the process X(t) is
continuous. We shall complete the proof of Theorem 2.2 by showing that
X(t) is a version of X(t).
Let t*G[0, 1] and let tf&T, tf-*t* if /'->oo. Since X(f) is a continuous
process,
X(tf,w)-+X(t*,w) a.s. as /'-»oo.
From B.1) it also follows that
fy, u)-*X(t*, u) in probability, as /'-»oo.

3. MARTINGALES AND STOPPING TIMES
Since X(tf, w) = X(tf, w) a.s., we get X(t*, w) = X(t*, w) a.s.
From B.5) we see that
- X{s)\ < Ch{t - s) = C{t - s)y
if t-s<2~ml-1-s\ m = m(w). Thus:
Corollary 2.3. Let X(t) (t&I) be a v-dimensional process satisfying B.1).
Then for any bounded subinterval J c.1 and for any e > 0, a.a. sample paths
of the continuous version of X(t) satisfy a Holder condition with exponent
{a/p)-ionj.
3. Martingales and stopping times
Definition. A stochastic sequence {Xn} is called a martingale if E\Xn\ < oo
for all n, and
E(Xn+1|X1, . . . , Xj - Х„ a.s. (n - 1, 2, . . . ).
It is a submartingale if E\Xn\ < oo for all n, and
1|X1,...,XB)> Х„ a.s. (n=l,2, ...).
Theorem 3.1. If {Xn} is a submartingale, then
P( max Xk > X) < ^ EXn+ for any X > 0, n > 1. C.1)
1 < Jl< П Л
This inequality is called the martingale inequality.
Proof. Let
\ = П (x,. < \)n (Xj > x) (i< fc < n),
A = ( max X,- > \).
Then A is a disjoint union U nk^Ak. Therefore
XP(A) = 2 AP(AJ < 2 E(X)tXAj C.2)
Jt—l fc-i
where Хв is the indicator function of a set B. Since Afc belongs to the a-field

10 1 STOCHASTIC PROCESSES
Of Xv . . . , Xk,
ex: > i
n n
> 2 ExAkE(Xn\X1, . . ., Xk) > 2 J
ib-l fc-l
Comparing this with C.2), the assertion C.1) follows.
Corollary 3.2. If {Xn} is a martingale and E\Xn\a < oo for some a > 1
and all n > 1, then
P{ i max jXk\ >\)<±E \Xn\a for any X >0, n> 1.
This follows from Theorem 3.1 and the fact that {|Xja} is a submartin-
submartingale (see Problem 8).
Definition. A stochastic process {X(t), t&I} is called a martingale if
JS|X(t)| < oo for all* El, and
E{X(f)|X(T),T<s,T6l} = X(s) for all t <= I, s <= I, s < t.
It is called a submartingale if E \X (t)\ < oo and
E{X(t)\X(r),r < s,rEl] > X(s) for all F 1, s <= I, s < t.
Corollary 3.3. (i) If {X(t)} (t&I) is a separable submartingale, then
P{ max X(s) > X) < ^ EX+ {t) for all X > 0, t<EI.
1 s<t,sei 'A J
(ii) If {X(t)} (t&I) is a separable martingale and if E\X(t)\a < oo for
some a > 1 and all t&I, then
P{ max X(s) >X} < -^ E\X{t)\a for all X > 0, t&I.
This follows by applying Theorem 3.1 and Corollary 3.2 with Xf = X(sj) if
1 < / < n — 1, X(. = X(t) if; > n, where s1 < ¦ • • < sn_j < t, taking the
set (sj, . . . , sn_1) to increase to the set of all t with tf < t, where {^} is a
set of separability (cf. Problem l(b)).
We denote by ^(X(X), Xsl) the a-field generated by the random
variables X(X),Xe I.

3. MARTINGALES AND STOPPING TIMES 11
Definition. Let X(t), a < t < /J, be a p-dimensional stochastic process. (If
/J = oo, we take a < t < oo.) A (finite-valued) random variable т is called a
stopping time with respect to the process X(f) if a < т < /J and if, for any
a < t < /J, (t < t} belongs to <3(X(\), a < X < t). If J3 = oo and т may
also assume the value + oo, then we call т an extended stopping time.
Notice that the sets (т > t}, {s < т < t) (where s < i) also belong to
), a<\< t).
Lemma 3.4. If т is a stopping time with respect to a v- dimensional process
X(t), t > 0, then there exists a sequence of stopping times т„ such that
(i) т„ has a discrete range,
(ii) т„ > т everywhere,
(iii) т„ 4 т everywhere as n 1 oo.
Proo/. Define тп = 0 if т = 0 and
i + 1 .r i. . i + 1 1, 1O ч
T« = ^- lf n<T<— (l = 1'2"--)-
Then clearly (i)-(iii) hold. Finally, т„ is a stopping time since
{'„<*>- U
= u
If X(t) is right continuous, then
X (t А т„)-* X (t a t) a.s. as n -» oo.
Since X(t а т„) is obviously a random variable, the same is true of X (t л т).
We shall prove:
Theorem 3.5. If т is a stopping time with respect to a right continuous
martingale X(t), t > 0, then the process Y(t) = X(t А т), t > 0, is also a
right continuous martingale.
Before giving the proof, we need some preliminaries.
Definition. A sequence of random variables Xn is said to be uniformly
integrable if ?|Xn| < oo for any n > 1, and
Urn f |XJ dP-*0 as \-»oo. C.3)

12 1 STOCHASTIC PROCESSES
Lemma 3.6. If {Xn} is uniformly integrable, then lim ?|Xn| < oo. If
further Х„-*Х a.s. or in probability, then E\X - Xn|->0 and EXn-*EX.
Proof. For any X > 0,
E\Xn\<\ + [ \Xn\dP.
From this and C.3) it follows that ?|Xn| < C, where С is a constant
independent of n. To prove the last assertion, notice, by Fatou's lemma, that
if Xn->X a.s., then
j \X\dP <hm j |Х„|<?Р< oo.
Next, by Egoroffs theorem, Xn—*X almost uniformly, i.e., for any с > О
there is an event A with P(A) < e such that Х„-»Х uniformly on fi\A.
Hence
Tim f |X_ - XI dP = liin f |X_ - XI dP < lim f |XJ dP + [ \X\ dP.
n-*oa J ' ^ ' n-»oo Уд n-»oo Ja •'A
Now, by C.3),
\XjdP<[ \Xn\dP+( \Xn\dP < r,(\) + Ac,
/
JA
where tj(\)-»O if \->oo, and т/(\) does not depend on n. By Fatou's lemma,
the same inequality holds for X. Taking e \. 0, we get
liin f IX. - XI dP < 2tj(\)-»0 if
If Xn—*X in probability, then any subsequence Xn, of Xn has a subsub-
sequence Xn» which converges a.s. to X. By what we have proved,
Е\Хп„ - X|-»0 if n"^oo. But this implies that E\Xn - X|-»0 as n-^oo.
Pwo/ o/ Theorem 3.5. Let 0 < tx < t2 < • • • < fm_! < tm = s < t,
m
A = П [-Х («i Л т) e B,], Bf Borel sets in R \ C.4)
and let т„ be the stopping times constructed in the proof of Lemma 3.4.
Assume first that there is an integer / such that s — (j + l)/n. On the set
т < (/ + l)/n, т„ < (/ + l)/n so that
Consequently
( X(tArn)dP. C.5)
(^ + 1)/]

3. MARTINGALES AND STOPPING TIMES
13
Next,
м
/ X(tArn)dP= 2 /
A П [t > (/ + l)/n] /-/ + 1 JA П [l/n< т < (/+ l)/n]
¦i
An[r>(M
X(t)dP
]
C.6)
where (M + l)/n < t < (M + 2)/n.
Notice that
А П т >
M + 1
n
П
П
M +
Since each set on the right belongs to ^(X(X), X < (M + l)/n), we can use
the martingale property of X(t) to deduce that
f X(t)dP=(
JAn[r>(M + l)/n] •/А
Hence, from C.6),
Ап[т>(М
Г
•/Ап[т>(
Г
JA
An[l/n<r<(l
dP
•/А
JAn[r>M/n]
Proceeding in this manner step by step, we arrive at
f X(t лт„) dP= f
JAn[r>(j + i)/n] Ja
since s = (/ + l)/n,
X(s Arn) dP,
l
А П т >
n
n
П т >
is in ^(X(X), X < (/ + l)/n), and j л т„ = s on this set.
Recalling C.5) we conclude that
f X(sArn)dP= f X(tArn)dP.
Ja Ja
C.7)
So far we have assumed that the number s is in the range of т„. If this is
not the case, i.e., if ;'/n <*<(/ + l)/n for some /, then we modify the

14 1 STOCHASTIC PROCESSES
definition of rn, by taking
s if - < т < s,
n
/ + 1 r / + 1
if s < т < .
n n
The assertion C.7) then follows as before, and the т„ satisfy the properties
asserted in Lemma 3.4.
We shall now complete the proof of the theorem in case т is bounded, i.e.,
т < N < oo (N constant).
The range {sin} of т„ lies in [0, N + 1).
Notice now that \X(t)\ is a submartingale. If we apply the proof of C.7) to
\X(t)\, instead of X(t), we obtain
f \X(sArn)\dP<f \X(tArn)\dP
JA JA
whenever t > s. Taking, in particular,
we get
/ \X(sArn)\dP<( \X(rn)\dP.
But since |X (t)\ is a submartingale,
Г \X(rn)\dP<[ \X(N + l)\dP.
Summing over i,
\X(SATn)\dP<f \X(N+l)\dP. C.8)
Now, since X (t) is right continuous and т is bounded, the random variable
sup sup|X(s л т„)|
n s>0
is bounded in probability. This implies that the right-hand side of C.8)
converges to 0 if X -» oo, uniformly with respect to n. Thus the sequence
{X(s л т„)} is uniformly integrable. Similarly, one shows that the sequence
{X(t /\ тп)} is uniformly integrable. By Lemma 3.6 it follows that
Е|Х(*Лт)| < oo.
If we now take n f oo in C.7) and apply Lemma 3.6, we get
X(s at) dP= f X(t at) dP. C.9)
Since this is true for any set Л of the form C.4), {X(t л т)} is a martingale.

PROBLEMS 15
We have thus completed the proof in case т is bounded. If т is un-
unbounded, then т Л N is a bounded stopping time. Hence E\X(t Л т)| < oo if
N > t. Further, C.9) holds with т replaced by т Л N, and taking N > t the
relation C.9) follows. This completes the proof.
Remark. Let {X(t), 0 < t < T) be a right continuous process and let ^t
@ < t < T) be an increasing family of cr-fields such that ^(X(X),
0 < X < t) is a subset of ?,. If E\X(t)\ < oo and
E(X(t)\$s) = X(s) forall 0 < s < t < T,
then we say that X (t) is a martingale with respect to ^t @ < t < Г). This
clearly implies that [X(t), 0 < ? < Г} is a martingale. The proof of
Theorem 3.5 shows that if X (t) is a martingale with respect to ^t
@ < t < Г), then the process X(t л т) is also a martingale with respect to
^t; here т is any random variable such that т > 0 and {т < ?} S ?F( for all
0 < t < T.
PROBLEMS
1. Let X(t), t > 0 be a stochastic process and let {*•} be a set of
separability. Prove that for a.a. w:
(a) lim inf X(t, w) < X(t, w) < lim sup X(fc, w)
7 "-00 \t,-t\<i/n v'' ' n-»» |((|^/ x'
(t > 0).
(b) sup X(t, w) = sup X(^, w), inf X(f, w) = inf ХЦ, w).
a<t<b a<tf<b a<t<b a<tf<b
(Notice that, as a consequence, supa<(<fcX(?) and infa<(<bX(f) are finite- or
infinite-valued random variables.)
(c) supa<(<fc|X(?)|, lim sup(_^X(?), and lim vait__>sX(t) are finite- or in-
infinite-valued random variables.
(d) For any positive numbers e, 8,
sup |X(f4, «) - X(tf, «)| < ?
|t,-/>T<e
if and only if
sup |X(t, u) - X{s,w)\ < e.
\t-s\<S
2. Let X(t), X'(t) (t > 0) be j'-dimensional stochastic processes in
E2, ?+, F) and (Я', If', P') respectively, having the same finite-dimensional
distribution functions. Then for any sequence {?„}, tn > 0, and for any set
*,,..
'(t,), X'(ta), . . . )?D}.

16 1 STOCHASTIC PROCESSES
3. Suppose two ^-dimensional stochastic processes X(t) and X'(t) (t > 0)
defined in probability spaces (Я, §, P) and (fi', §', P'), respectively, are
separable and have the same finite-dimensional distribution functions. If X(t)
is continuous, then X'(t) is continuous.
[Hint: Let v = 1. It suffices to prove continuity for 0 < t < f\ t* < oo.
Let T = {tj} be a set of separability for both processes for 0 < t < t*. The
continuity of X(t) implies that
{00 00 \
П U Sup |X@ ~ X(t,)| < ^ = 1.
„-1 m=l k-^l<l/m « J
By Problem 2, with
о=П U П Щ, 01
n m \t,-t,\<l/m
00 00
П U sup |x'@ - хщ < I
Now use Problem l(d).]
4. Let X(t) and X'(t) (t > 0) be j'-dimensional separable stochastic pro-
processes defined in probability spaces (Я, $", P) and (Я', S7', P') respectively,
and having the same finite-dimensional distribution functions. Then:
(a) P{|X(t)|-»oo as t-*oo} = P'{|X'(*)|-»<» as t->oo};
(b) P{|X(t)|/*"->M as *-*oo} - P'{|X'(f)|/fa-^M as f^oo},
for any a > О, М > 0.
[Hint: {|X@H« as t-^oo} = { П „ Uwinf,>jX(t,)| > n}.]
5. If two stochastic processes are stochastically equivalent, then they have
the same finite-dimensional distribution functions.
6. Let Xn, n > 1 be independent random variables with ?|Xn| < oo,
EXn = 0. Then the sequence of partial sums X1 + • • • + Xn is a martingale.
7. Let X be a random variable in (Я, f, P) with E|X| < oo, and let ф(х)
be a convex function defined on the real line such that E |ф(Х)| < oo. Prove:
(a) ф(ЕХ) < Еф(Х) (Jensen's inequality);
(b) ф(Е(Х\%)) < Е(ф(Х)\%) a.e., where % is any cr-subfield of <5.
8. If {Xn} is a martingale and ф(х) is a convex function on the real line,
and if ?|<f>(XJ| < oo for all n, then {Ф(Х„)} is a submartingale.
9. If. {Xn} is a submartingale and ф(х) is a convex function and monotone
nondecreasing on the range of the Xn, and if Е|ф(Х„)| < oo for all n, then
{Ф(ХП)} is a submartingale.
10. If т is a bounded stopping time with respect to right continuous
martingale X(t), then ЕХ(т) - EX@).

PROBLEMS 17
11. Prove that a continuous process is continuous in probability. Give an
example showing that the converse is not true in general.
12. Let X (t), X (t) (t E I) be two stochastically equivalent and separable
processes. Show that if one of them is continuous, then the other is also
continuous, and
P{X(t) = X(t)foralltEl} = 1.
13. A stochastic process {X(t), tGl) is said to be measurable if the
function (t, ui)-*X(t, w) (from the product measure space into R1) is measur-
measurable. Show that a continuous stochastic process is measurable.

2
Markov Processes
1. Construction of Markov processes
If x(t) is a stochastic process, we denote by ^(x(X),XE]) the smallest
ст-field generated by the random variables x(t), t E].
Definition. Let p(s, x, t, A) be a nonnegative function defined for
0 < s < t < oo, xER", A E ®p and satisfying:
(i) p(s, x, t, A) is Borel measurable in x, for fixed s, t, A;
(ii) p(s, x, t, A) is a probability measure in A, for fixed s, x, t;
(iii) p satisfies the Chapman-Kohnogorov equation
p(s, x, t,A)=f p(s, x, X, dy)p(X, y, t, A) for any s < X < t.
A.1)
Then we call p a Markov transition function, a transition probability func-
function, or a transition probability.
Theorem 1.1. Let p be a transition probability function. Then, for any
s > 0 and for any probability distribution ir(dx) on (R", <$„), there exists a
v-dimensional stochastic process {x(t), s < t < oo} such that:
Р[*(*)еА]-*г(А), . A.2)
P{x{t)EA\<3{x{X), s < X < s)} = p{s, x(s), t, A) a.s. (s < s < t).
A.3)
Proof. Let M[[0, oo); Я"] be the space of all functions и from [0, oo) into
Я". Let ^ be the smallest ст-field with respect to which all functions u(u)
are measurable, s < и < t, i.e., 5J is the ст-field generated by all the sets
{«; ы(и)Е.А} where s < и < t,Ae%f.
18

1. CONSTRUCTION OF MARKOV PROCESSES 19
Let ^ be the smallest ст-field with respect to which all functions «(a)
are measurable, s < и < oo. We take Я = M[[0, oo); Я*], Ф = f^,.
We next define a family of finite-dimensional distribution functions with
respect to Я": if * = t0 < tx < ¦ • • < tn and х,ей' (О < i < n), then
F
Г^1 Г*1 r*°
¦ ¦ • <,(% «I,---. *J = I • • • / I
•' — oo •' —oo —
toh ¦ ¦ • <,(% «I,---. *J = I • • • / I p(*.-i, yn-i, '„, d«/J
•' — oo •' —oo —oo
• • • p(t1, t/л, t2, dy2)p{t0, i/0, tls di/1)w(di/o)-
It is easy to verify that the Fn are in fact n-dimensional distribution
functions with respect to Я". Using the Chapman-Kolmogorov equation it
follows that
if Xk t 00;
thus the family of distribution functions in A.4) is consistent.
By the Kolmogorov construction, there exists a probability Ps in (Я, 5^,)
and a v-dimensional stochastic process x(t), s < t < oo, such that the
finite-dimensional distribution functions of this process coincide with the
given distributions functions Ftg... ^.
Notice that x(t, u) — o)(t).
Now, if s = t0 < tx < ¦ ¦ ¦ < tn,
Ps{x{t0)EB1,...,x{tn)EBn}
= [¦¦( f Pta-i» У--1. t» rfyJ • • • P(*6. Уо. *i. dyjiridyo)
Jen jb1 jb0
A.5)
provided each Bf is a v-dimensional interval (- oo, х(). It is easily seen that
both sides of A.5) coincide also when the Bt are any v-dimensional intervals
(z(, x(), or disjoint unions of such intervals. Since two probabilities that agree
on a field & agree also on the a-field generated by B, it follows that A.5)
holds for all B{ in <&„.
Taking n = 0 in A.5) we get the assertion A.2). Thus it remains to prove
A.3). Since the random variable on the right-hand side of A.3) is measurable
with respect to Sr(x(X), s < X < s), it remains to prove that
f XW.NA] dP. - f P(*. *(*). *, A) dPs if D S ^(«(X), s < X < a).
A-6)
If we prove A.6) for any set D of the form
D- («((Jeeo,*(t,)EB1 ^(OgbJ A.7)

20 2 MARKOV PROCESSES
where s — t0 < tt < ¦ • ¦ < tn — s, then, by the countable additivity of
both sides of A.6), the assertion A.6) will follow for any D G
&(x(\), s < X < s).
Set fn+i " *, Bn+1 - A. When D is given by A.7), the left-hand side of
A.6) becomes
PS[D П(x(t)EA)] = P,{x(OEBo, . . . , x(OeBn, x(tn+l)eBn + 1)
•4+J hn jb1 -4
*/o> *i> dyjiridyo)
f p(tn, yn, tn+1, A)p{tn_v yn_v tn, dyn)
Bn b1 Jb0
¦ " - P(*o> «/с *i» dyjiridyo).
Therefore it suffices to show that
f
f - ' " f f /(«/Jp(*n-i. У«-1. *n. dyJ • • • p(*o> Уо. *i. dyj
•'д •'в! •'Bo
= / •••/ /WOX A-8)
where / is a Borel measurable function. It clearly suffices to prove A.8) when
/ is any indicator function Xf- Setting B'n = Bnr\ F, A.8) becomes
f f ¦ ¦ ¦ ( f P(*n-i> У--1' *»• d^n) ¦ ¦ • P(*o> Уо. *i. dy^idyo)
- f f • • • Г ^- A-9)
Since the right-hand side is equal to
A.9) is a consequence of A.5). This completes the proof of A.3).
For any xGRn let чтх be defined by
.0 if x ? A.
Denote by Px s the probability Ps constructed on the measure space (Я, <S*a0)
when it = itx, and denote the expectation by Ex s. Set ?F = ^. Then we
have the following situation:
(a) (fl, f) is a measure space, % are ст-subfields such that % С ^ if
s' < s, t < f, and ?F is the smallest ст-field containing all the ^.
(b) There is a function x(t, a) from [0, oo) X Q into Я" such that x(t, a)
is ^\ measurable for each t > s.
(c) For each xGR',s > 0 there is a probability Px , defined on (fl, fi^)

1. CONSTRUCTION OF MARKOV PROCESSES 21
such that
Px, ,{*(*, со) = *} = 1, A.10)
p,,,{*(*+ К w)SA|^} = p(t,x(t, to), t + h, A) a.s. (t > s, h > 0).
A.11)
Definition. A collection {fl, f, f*, x(t), Px s} satisfying (a), (b), (c), where
p is a transition probability function, is called a (v-dimensional) Markov
process corresponding to p.
The property A.11) is called the Markov property.
Notice that in the model constructed in Theorem 1.1 4i*t is the smallest
a-field generated by x(u), s < и < t. Notice also that x(t, со) — u(t).
The model constructed in Theorem 1.1 will be called the Kolmogorov
model.
Taking the expectation Ex s of both sides of A.11), we get (when t = s
and t + h is denoted by t),
{ p(s,x,t,dy)f(y) A.12)
H f — Xa- By approximation, this relation holds also for any bounded Borel
measurable function /.
We can therefore write A.11) in the form
Ex,s{M* + *O)|SJ} - f P(*. *(')> t + h, dy)f( y)
JR'
= Ex(tXtf(x(t+h)) a.s. A.13)
if / = Xa- By approximation, this relation holds also for any bounded Borel
measurable function f.
We have the following generalization of A.5) (in case it = <ir^)\
' ' P^' *' *13 ^ ^L14^
where D is any set in %vn (®,;„ is the а-field of Borel sets in
H* X Я' X • • • X Я" (n times)). Indeed, if we denote the left-hand and
ri^ht-hand sides of A.14) by P(D) and Q(D) respectively, then P(D) and
^)(D) define probability measures on $,_„. Since, by A.5), these measures
agree on all rectangular sets, they must agree on ®, „.
Theorem 1.2. If {B, I3r, §J, x(t), Ps x) is a v-dimensional Markov process,
then
Px..(B\x(t)) a.s. forany В 6 ЭДЛ), X > t). A.15)

22 2 MARKOV PROCESSES
Proof. Let 0 < hx < h2 and let fx, f2 be bounded Borel measurable func-
functions in R". Using A.13), we have
l(*(* + K)) f P(t + hv x(t + hj, t + h2, ]
= f p(t, x(t), t + hv Л/ОЛЫ / P(* + К Vv * + К dy2)f2(y2).
Similarly one can prove by induction on n that, if 0 < ht < • ¦ • < hn,
-J- •
• • • p(t + K-v «Л.-1. * + Л», <ty»)/»( У-)- AЛ6)
For any set D ? §F|11 its indicator function f(xt xn) —
Xd(xv • • • > xn) can be uniformly approximated by finite linear combina-
combinations of bounded measurable functions of the form /^a^) • • • /„(*„) (each x{
varies in R"). Employing A.16) we deduce (by the Lebesgue bounded
convergence theorem) that
¦¦¦p(t+ hn_lt «/„_!, t + hn, dyn) a.s. A.17)
Setting
B={(x(A),.,.,x(O)?D), A.18)
we conclude that PX_,(B|S*) is #(*(*)) measurable and A.15) holds for this
B; here ^(x(f)) denotes the smallest ст-subfield with respect to which x(t) is
measurable.
Since the class & of sets В for which A.15) holds is a monotone class
containing all the sets of the form A.18), it must contain the a-field
generated by these sets, i.e., & = l3r(x(X), X > t).
Theorem 1.3. Let x(t) and y(t) be v-dimensional processes defined for
t > 0 and having the same set of joint distribution functions. Suppose x(t)
satisfies the assertions in Theorem 1.1. Then y(t) also satisfies the same
assertions.
An outline of the proof is given in Problem 4.

2. THE FELLER AND THE STRONG MARKOV PROPERTIES 23
2. The Feller and the strong Markov properties
Denote by M(R') the space of all bounded Borel measurable functions /
from R" into R1 with the norm ||/|| = ess supxSH.|/(x)|. Consider the
mappings Ts t from M(R") into M(Rr):
(Ts ,/)(*) - f p(a, x, t, dy)f( y) @ < s < t < oo).
It is easily seen that \\TS_J| = 1, and
Г,.Л«=Г..« if 0<s<t<u.
We define Tt t = identity. The family {Ts ,} is called the semigroup
associated with the transition probability function p (or with the correspond-
corresponding Markov process).
Definition. If for any bounded continuous function f(x), the function
(t, z)MTt t+J)(z) = f p{t,z,t + K dy)f( y)
is continuous, for any X > 0, then we say that the transition probability p (or
the corresponding Markov process) satisfies the Feller property.
We denote by ?pj+ the intersection of the ст-fields %, t' > t.
Similarly we denote by Щ- the smallest a-field containing 5J,, t' < t.
Theorem 2.1. Let {B, I3r, 4Pt, x(t), Px s) be a v-dimensional Markov
process, right continuous and satisfying the Feller property. Then (fl, I3r,
^\*, x(t), Px s) is also a Markov process with the same transition probability
function.
Proof. Let / be a bounded continuous function, and let с > 0, h > 0. By
A.13),
E*,AMt + h + e))|fJ+€} ~fp(t + €, x(t +e),t+h + t,dy)f(y).
Taking the conditional expectation with respect to 3^+, we get
{t + t,x(t + t),t+h+e,dy)f{y)\%}. B.1)
Let e J, 0. Since x(t + e)->x(t) a.s., the Feller property implies that
fp(t + e, x(t + €),t + h + e,dy)f(y)^Jp(t, x{t), t + h, dy)f(y) a.s.

24 2 MARKOV PROCESSES
By the Lebesgue bounded convergence theorem for conditional expecta-
expectation, we then get
*,,{ / p(* + «. *(* + «). * + ^ + «.
p(t, *(*), t + h, dy)f(y)\% } =fp(t, x(t), t + h, dy)f(y),
since x(t) is $J+ measurable.
On the other hand
/(*(* + h + ?))->/(*(*)) as €|0,
so that
Thus, upon taking e | 0 in B.1), we get
EX,S{M* + h))\<5*+ } =fp(t, x(t), t + h, dy)f(y).
This completes the proof.
From Theorem 2.1 it follows that whenever one has constructed, from a
given transition probability function, a Markov process that is right con-
continuous and satisfies the Feller property, one may assume (without loss of
generality) that 3J+ = 5J.
Remark. If a Markov process is left continuous, and if 9^ = ^{^(X);
s <X < t}, then %- = %. This follows by noting that, for any a G R", the
set {a • x(t) < X} coincides with the set
П П U [«
m=l n=\ k=n
which belongs to %-.
Definition. An extended real-valued random variable т defined on Й is
called an s-Markov time or an s-stopping time with respect to a given
v-dimensional Markov process {fl, I3r, Щ, x(t), Px ,} if т > s and the sets
{т < t} belong to ^ for every t > s.
Any constant function т — t with t > s is an s-stopping time.
Since {t < t} is the complement of {т > t), т is an «-stopping time if
and only if т > s and the sets {т > t) belong to 9^, t > s. Notice that the
sets
{X< т < t} - {t < t}\{r < X}
also belong to %.

2. THE FELLER AND THE STRONG MARKOV PROPERTIES 25
Consider the events Ac^ such that
{т<()е^ for all t > s.
They form a ст-field ^.
Definition. A «--dimensional Markov process (fl, f, Щ, x(t), Pxs) with
transition probability p is said to have the strong Markov property if for every
x E Я' and for every real-valued s-Markov time т,
Рж.,{*(* + т)еА|^т} = р(т, х(т), t + т, А) а.е. B.2)
For т — h (h constant), this coincides with the Markov property A.11).
Using A.12) it is clear that the strong Markov property is equivalent to
E,,s{f(t+ t)|S»t} -/р(т,*(т), t + r,dy)f(y) ~ Ex(TlTf(x(t + t)) B.3)
for any/ = Xa; hence for any bounded Borel measurable function/.
We shall give some examples of «-stopping times.
Theorem 2.2. Let (fl, ?F, S^, x(t), Px s) be a v-dimensional continuous
Markov process and let F be a nonempty closed subset of R". Let
t(w) = first time t > s such that x(t, w)SF. Then т is an s-stopping time,
for any s > 0.
If s — 0, т is called the (first) hitting time of the set F or the exit time
from the open set Я"\F; if s > 0, т is called hitting time, after s, of F (or the
exit time, after s, from fl"\F).
Proof. Let {tj} be a dense sequence in the interval [s, oo). Denote by
р(ж, F) the distance from x to F. Let {^} be a dense sequence in F. Since
{\x{t) - ty < a} - {<o; x(t, <o)?D} (a > 0)
where D = (xefi*, \x - f,| < a), it follows that the set {\x(t) - f,| < a)
is in ^"t,\{ t > s. Consequently, also the set
П U {рЗД.^)<|} = П U U
п = 1 tj<t n = l tf<t к
is in 5J. But as is easily seen, this set coincides with the set {т < t).
Let G be a nonempty open set in Я". Let
t(w) = inf {t; t > s,x{t,u)eG).
Then т is called the (first) hitting time, after s, of G. Notice that, for a
continuous process,
t(w) - inf [t; t > s, x(t, a)eG}
- inf {U t > s, meas [s < t' < t; x(t\ to) G G] > 0} a.s.

26 2 MARKOV PROCESSES
For any set G, the extended random variable defined by the last expression is
called the (first) penetration time, after s, of G.
Theorem 2.3. Let {Я, f, Щ, x(t), Pxs] be a v-dimensional continuous
Markov process with $J+ = 5J, and let т be the penetration time, after s, of
an open set G. Then т is an s-stopping time, for any s > 0.
Proof. Let {t} be a dense sequence in [s, oo), and let
a»- U U [р(*0»),я'\с)>!|.
(<t+l/n fc=l J
It is clear that if t(w) < t, then иЕД,; and if й?А„, then t(w) < t + l/n.
Hence (t < t) - 1ипА„. Since А„е^+1/„, it follows that the set (t < t)
belongs to ?Fj}+.
Remark. Theorems 2.2, 2.3 are clearly valid not only for a continuous
Markov process, but also for any continuous process x(t); the definition of
(extended) stopping time is given in Chapter 1, Section 3.
Theorem 2.4. Let {Й, Sr, <S\, x(t), Pxs} be a v-dimensional Markov pro-
process, right continuous and satisfying the Feller property, and let 5J+ = SJ.
Then it satisfies the strong Markov property.
We begin with a special case.
Lemma 2.5. The strong Markov property B.2) holds for any Markov process
provided т is an s-stopping time with a countable range.
Proof. Denote by {^} the range of т. Since (т < ^} and
{r < t,} - U (т < Ь)
belong to ?P , it follows that
с^{т = {(}е§;.
Note next that G, belongs to ^, since
{t= tt}n{r < t} e^ if t > tt,
= фе1^ if t < tt.
Hence, for any A?§,,
(x(r)GA}- U {r-t,} isin?PT.

2. THE FELLER AND THE STRONG MARKOV PROPERTIES 27
This shows that х(т) is ^ measurable.
Now let E<=<$ST. Then
E n Gj = E n (т < ^)пС(е^.'
The Markov property gives
P*,s{*(t + t,)eA\^} = p{tf, x{t,), t, + t, A) a.s.,
and since ?пС(ЕЙ,
K.{ W* + '/) e A]n E П G,} = | P(tj, xitf), t, + t, A) dPxs.
Noting that т = fc on Gj and summing over /, we get, upon recalling that
U q = a,
[(м(т),т+ t,A)dPxs. B.4)
To complete the proof of B.2) we have to show that the integrand on the
right-hand side of B.2) is ^ measurable. Since т and x(r) are S^ measur-
measurable, the Feller property implies that
fp{r, х(т), т + t,dy)f(y)
is 5J measurable if /(«/) is a bounded continuous function. Taking a
sequence / = fm of uniformly bounded and continuous functions that con-
converge a.e. to Xa> we conclude that
dy)/m( У) = Р(т, *(т), т + t, A)
is 5J measurable.
Froo/ o/ Theorem 2.4. Choose т„ as in Lemma 1.3.4. We claim that
fj С ^. Indeed, since т„ > т,
Ап(т„ < t) =[Ап(т < *)]п(т„ < t)
for any set A and for any t > s. If A S %, then А П (т < t)?5}. Since also
(т„< t)eSJ,
Ап(т„ < (JeS), i.e., A?fTi.
If / = Хв' ^еп ^е strong Markov property for т„ gives
*U/W' + tJ)|^J =/р(т„, х(тп), t + rn, dy)/(y) a.s. B.5)
But then this relation holds also for any bounded continuous function /.
Let E G #J. Then (as proved above) EE^ and therefore, by B.5),
( V «(О. t + т„, <%)/( y)] dPx>s. B.6)

28 2 MARKOV PROCESSES
The function J* p(X, x, t + X, dy)f( y) is continuous in X, x. Also, if n f oo,
т„ | т and х(тп)-*х(т) (by the right continuity of the x(t)). Hence, if we let
n f oo in B.6) and employ the Lebesgue bounded convergence theorem, we
get
f(x(t + t)) dPXtS =fE [ fp(r, x(t), t + т, dy)f( y)] dPx>.. B.7)
This relation holds also if / = Xa> -A ^ <$>р.
If we show that х(т) is ^ measurable, then the proof of the theorem
follows from B.7) and the argument following B.4).
Let «Ей'. Then, by the right continuity of x(t),
00 00 00
[а-*(т)<Х] = П П U
m*=Nk = l n = k
for any positive integer N. We also have
00 00 00
[t<«]- n n и k<« + r
and, more generally,
Q =[a • x(t) < X] П[т < t]
00 00 00
= n n и f«• *w
Since ж(тя) Е ^,
[ а ¦ х(т„) <A+
It follows <?ef^+1/N for any N. Hence Qe^+. Since f^+ = <3*t, we
conclude that [a • х(т) < А]6?Т. But since a and X are arbitrary, x(r) is 'З7*
measurable.
Corollary 2.6. Let (Я, ?F, ^\, x(t), Px s) be a v-dimensional continuous
Markov process satisfying the Feller property. Then it satisfies the strong
Markov property.
Notice that we do not assume here that 5^+ = Щ.
Proof. The proof of Theorem 2.4 applies here except for the last step,
namely, the proof that x(r) is % measurable. To prove this, suppose for

2. THE FELLER AND THE STRONG MARKOV PROPERTIES 29
simplicity that s = 0. Let {t,} be a dense sequence in [0, oo). For any a6fi'
and Л real,
[a • x(t) < \]n [t < t]
oo oo n— 1
- n n и* и
n = l m=l < = 0 (i,
[i
П| ~t
[
where * indicates that for i = 0 one takes 0 < т < t/n. Since each set
[ • • • ] on the right-hand side is in <?t°, it follows that [a • х(т) < \] E. <3?.
But since a and Л are arbitrary, х(т) is If? measurable.
It can be shown (see Stroock and Varadhan [1]) that, for a continuous
Markov process, ^ is actually generated by x(t Л т), t > s.
The next result is the Blumenthal zero-one law.
Theorem 2.7. Let {Я, Sr, 9t, x(t), Pxs] be a v-dimensional Markov pro-
process, right continuous and satisfying the Feller property. If, for some s > 0,
x(s) = x a.s. (x a point in Rn), then
Px,s(B) = Oorl for any BE^+. B.8)
Proof. By Theorem 2.1, formula A.17) remains valid when Щ is replaced
by <f*+. Taking t = s, we get
jp, x,s+ hv dyt) • ¦ ¦ p(s + hn_b yn_ly s + hn, dyn) a.s.
By A.14), the right-hand side is equal to
UW» + ki)
Hence,
x,) = Px,s(B) as. B.9)
for any set В of the form A.18) with t{ > s. But then B.9) follows also for
any set В in ЧИ'Ж. Taking, in particular, B6f|t and noting that
a.s.
Since the right-hand side is constant, \в = const a.s. Hence either
a.s. or Хв ш 1 a.s. This gives B.8).

30 2 MARKOV PROCESSES
3. Time-homogeneous Markov processes
Definition. A Markov process is said to have a stationary transition prob-
probability function p if
p(s, x, t, A) = p@, x, t - s, A).
We then also say that the Markov process is time-homogeneous, or tempo-
temporally homogeneous.
It is clear that, when p is stationary, Px s = Px 0, Ex s = Ex 0 for all s > 0.
Therefore, from the probabilistic point of view it suffices to work just with
px,s> Ex,*' % when « = 0. We set
P* = P*, o> К - К0' P(*. x' A) " P(°. x- f' A)> % - ^°-
A 0-stopping time т will be called, simply, a stopping time or a Markov time.
(Notice that т is actually an extended stopping time according to the
definition of Chapter 1, Section 3.) The Markov property now takes the form
Ex{f{x(t + h))\%} = E^x(h)) a.s.; C.1)
A.12) for s = 0 becomes
Ej(x(t))=fp(t,x,dy)f(y), C.2)
and the strong Markov property takes the form
Ex{f(x(t + т)Ш = W(*W) C.3)
where т is a stopping time and <€r = f^0.
We shall denote a time-homogeneous Markov process by (Я, fr, %t,
For a time-homogeneous Markov process, the Feller property states that
the function
z—>
f p(t,z,dy)f(y)
is continuous for any bounded continuous function f and for any t > 0.
The semigroup associated with a stationary transition probability function
is given by
(Tj)(x)=fp(t,x,dy)f(y). C.4)
Denote by Mo the space of all functions from M(R") into R1 for which
1|т/-/|М> if t-»o,
where ||f|| = ess supxSH.|/(x)|.
Definition. The infinitesimal generator & of the time-homogeneous Mar-

PROBLEMS 31
kov process is defined by
Tf - f
&f = lim — - . C.5)
Its domain D& consists of all the functions in Mo for which the limit in C.5)
exists for a.a. x.
Theorem 3.1. Let & he the infinitesimal generator of a continuous time-
homogeneous Markov process satisfying the Feller property. If f E D& and
&f is continuous at a point y, then
Шу) н M^M (M)
where U is an open neighborhood of у and tv is the exit time from U.
This is called Dynhn's formula. Since we shall not use this formula in the
sequel, we relegate the proof to the problems.
For nonstationary transition probability function, one defines the infinite-
infinitesimal generators by
&f iim
>J no t
The Dynkin formula extends to this case (with obvious changes in the proof).
PROBLEMS
1. Let (fi, <?, Щ,х(г),РХг,} be a continuous Markov process. Let т be
the first time that t > s and \t, x(t)) hits a closed set F in R"+1. Prove that т
is an s-stopping time, for any s > 0.
2. Let (fi, 'S', Wt> x(t), Px s} be a ^-dimensional Markov process with
transition probability p. Let X denote a variable point (x0, x) in Rr+1 and let
X(t, w) = (t, x(t, w)). Prove that (fi, <5, %, X(t), Px s} is a Markov process
with transition probability p,
p{s, X,t,I X A) = Xr(*b)p(«. x, t, A)
where / is any Borel set in [0, oo).
3. If (fi, *$, Wt, x(t), Px} is a time-homogeneous Markov process and if
for some a > 0, « > 0, С > 0, then x(t) has a continuous version.
[Hint: Use Theorem 1.2.2.]
4. Prove Theorem 1.3.

32 2 MARKOV PROCESSES
[Hint: Since p(s, y(s), t, A) is <3( y(X), s < X < s) measurable, it suffices
to show that if tx < • • • < tn = s < tn+1 = t,
„, y{tn+1)<EA]
Denote by Qx the probability distribution of (ж(^), . . . , x(O> x(tn+1)) and
by Q2 the probability distribution of (y(t^), . . . , у(tn), y(tn+1))- Make use of
the relations
/ Xa,(*i) • •
f Ф(ж1, . . . , xn+1) dgx = /Ф(*1; . . • , *„+1) ^g2,
where Ф is bounded and measurable.]
5. A ^-dimensional stochastic process x(t) (t > 0) is said to satisfy the
Markov property if
P[x(t) e А\Цх(Х)), X < s)] - P[*(*) 6 A|*(s)] a.s. (•)
for any 0 < s < t and for any Borel set A. It is well known (see Breiman [1])
that there exists a regular conditional probability p(s,x, t, A) of (*), i.e.,
p(s, x, t, A) is a probability in A for fixed (s, x, t) and Borel measurable in x
for (s, t, A) fixed. Prove that p satisfies
I p(«, ж, Л, dy)p(X, y, t, A) = p(s, x, t, A) a.s. (s < Л < t).
6. Let x(t) be a process satisfying the Markov property. Then, for any
0 = tQ < tY < ¦ ¦ ¦ < tn and for any Borel sets Bo, . . . , Bn,
P{x{tn) 6Д, xfo) ЕВХ, *(*o) eB0)
= /•••[ I P(*n-i. ^-1. ^ dxn) ¦ ¦ ¦ p{t0, x0, tlt dx1)ir(dx0)
where 17 (dx) is the probability distribution of x@).
7. Let (Я, 1$, §st, x(t), Px s} be a p-dimensional continuous Markov pro-
process satisfying the Feller property. Let т be a real-valued s-stopping time.
Denote by W^ the o-field generated by x(t + т), t > 0. Prove that
Кш(ВI%) - К,(BI^(T)) for апУ В^оо. C-8)
[Hint: Prove that for any bounded measurable function f(xv . . . , xn),
*U/W'i + т), . . . , x(tn + t))|5»t] - EMJ(x(tl + т), . . . , x(fn + t))
C.9)

PROBLEMS 33
where s < tY < • • • < tn; cf. the proof of Theorem 1.2.]
In Problems 8-16, (Я, *$, ^t, x(t), Px} is a continuous time-homogeneous
p-dimesional Markov process satisfying the Feller property. By Corollary 2.6,
the process also satisfies the strong Markov property.
8. Prove that for every open (closed) set G, p(t, x, G) is lower (upper)
semicontinuous.
9. Let A be an open set if R" with boundary ЗА, and let Г be a closed
subset of ЗА. Assume that there is an increasing sequence of closed subsets
Г^ of ЗА such that Г^ f Г =ЭА\Г as то f oo. Assume that Px{x(t) hits
ЗА} = 1, and let E = {x(t) hits Г before it hits Г'}. Then there exists a
sequence/„(a^, . . . , xn) of bounded measurable functions such that
fn{x{h), ..., x{tn))->XE a.s.,
and the functions fn(xh . . . , xn) do not depend on the particular process
x(t).
[Hint: Let {\} be a dense sequence in [0, oo). Check that
e= и nln и
nun
where ;', то, p, i, r, I vary over 1, 2, ... . Show that P(E) = lim P(Ea) where
Ea is defined as in E but with ;, то, р, i, r, I varying only over a finite set.
Deduce that Xe = lim Xe,, Хе„ = /n(*(*i). • • • » *(*„)) for some n = n(a),
and/n(x1, . . . , xn) does not depend on the particular process x(t).]
10. Let A and В be open sets with boundaries ЗА and ЭВ respectively, and
assume that A U ЗА is contained in B, and that ЗА consists of a finite number
of manifolds. Denote by tA and tB the exit times from A and В respectively.
Suppose that Px(tB < oo) = 1 for all xE. A U ЗА. Prove that for any bounded
measurable function f,
EJM*b)) = ExEx{tj(x(tB)) (*EA). C.10)
[Hint: For time-homogeneous processes, C.9) becomes
Ex[Mh + т), . . . , x(tn + t))\%] = ExD{x(tx), ..., *(O). C.11)
Suppose / = Хг> Г closed, and introduce the events
E = (x(f) hits Гп ЭВ before it hits ЭВ\Г},
E' = {x(t + tA) hits Гп ЭВ before it hits ЭВ\Г}.

34 2 MARKOV PROCESSES
By Problem 9,
Xe - lim fn{x{h), ..., *@), Xe' = lim /„(*(*i + tA), . . . , xn{tn + tA)),
Apply C.11) to /„ to obtain
EJ(*(<b)) - ЕгХе' = Я*Е*л)Хе = ExEx(tj(x(tB)).]
11. Let т be the exit time from an open set G. Suppose Рх(т > a) < /? for
all x e G. Prove that
(i) ?x(t > not) < j8»;
(ii) Ехт < a/(l - j8);
(iii) ЕхеХт < e*"/(l - е^Д) if j8 < 1, Л < (- a log 0).
[Hint: Write
X[x(t)eGlf t<al = l™/m(*(*l)> • • • > X(O) U < «)»
X[x(t)eGif «<а + /П = lim/m(X(*l + ^). • • • . ^(^ + ^))X[x(t)eGif t<a]
for any /? > 0. Use C.11) and induction to verify that
12. Suppose v = 1 and т = inf (s, x(e) ¦= y) is finite valued, where у is a
real number. Prove, for any closed set A,
P[x(t) e А, т < t] = Е1Х
= Е,Хг<Лт)[*(* " т) 6 A] - f Px(t 6 ds)Pv[x(t - s) e A].
[ffinf: To prove the fourth equality, choose a sequence of functions as in
Problem 10 with E = {x(t - т) е A}.]
13. A point у is called absorbing if p(t, y, { y}) = 1 for all t > 0. It follows
that (T/)( y) = /(y). Hence, if /6D4, then ((?/)( y) = 0. Show that if у is
nonabsorbing and U is a sufficiently small neighborhood of у, т^ = exit time
from 17, then suplS[7 Extv < oo.
[Hint: p(t0, y, N) < 1 for some tQ > 0 and a closed neighborhood N of y.
Hence p(t0, x, R"\N) > S > 0 if \x - y| is sufficiently small, and then
14. Prove that Mo is a closed subspace of M(R'), that T(MocM0 if t > 0,
and that TJ is a continuous function in t e[0, oo) for any/EM0.
15. From the theory of semigroups (see, for instance, Dynkin [2]) one
knows that:
(i) De is dense in №„, &f e Af0 if / e De;

PROBLEMS 35
(ii) TtD&cD&;
(iii) if /EDj, dTtf/A - Tt6B/ = 6BTJfor t > 0;
(iv) the operator RJ = ^e~u(TJ) dt (/ e Mo) is defined for all X > 0
and satisfies:
(a) RJ6DS if/eM0;
(b) над < Х-Ч1/Ц;
(с) (XI- ffi)RJ = /if/eM0, RX(XZ- 6B)/-/if/6De.
Now let у be nonabsorbing and take a neighborhood U of у with
suPi6t; ?cTu < °°- Let W"" M ~ ®f- Check that
f(x) = ЯхЬл(х)
= E, fTu e-*\(x(t)) dt+ Ex\e~^ Г е^ЫФ + т„)) dt\
Use the strong Markov property to deduce
16. Use Problems 13 and 15 to complete the proof of Dynkin's formula.

3
Brownian Motion
1. Existence of continuous Brownian motion
Definition. A Brownian motion or a Wiener process is a stochastic process
x(t), t > 0 satisfying;
(i) *@)-0;
(ii) for any 0 < t0 < tx < • • • < tn, the random variables
x(tk) — x(tk_1) A < к < п) are independent;
(iii) if 0 < s < t, x(t) — x(s) is normally distributed with
E(x(t) - x(s)) - (t - s)^, ?(*(*) - *MJ= (t - s)a2
where ju, a are real constants, а Ф 0.
ju is called the drift and a2 is called the variance.
Property (ii) implies that x(t) — x(s) is independent of 9(x(X), X < «) or,
more generally, that ^(a^fi) — x(s), ju > s) is independent of <3(x(X),
\ < s).
If x(t) is a Brownian motion with drift ju and variance a2, and if
0 < t0 < *! < • • • < tn> then
Г(*,, tk) s Е(я(^) - M*/)(*U) - fd») - <^2 min (t,, tk). (l.l)
Indeed, if t > tk,
Щ, tk) = E[x(t,) ~ x(tk) - p(t, - tk) + x(tk) - ixtk][x{tk) -
If ju, = 0, a2 = 1, then we speak of normalized Brownian motion. Notice
that for any Brownian motion with drift ju and variance a2, (x(t) — fit)/a is a
normalized Brownian motion.
From now on we consider only normalized Brownian motion and refer to
it briefly as Brownian motion.
36

1. EXISTENCE OF CONTINUOUS BROWNIAN MOTION 37
Let Xv . . . , Xn be random variables with joint normal distribution
N@, Г) where Г = (Г(/), Г^ = min (t,, tj), 0 < *x < • • • < *„. It is easily
checked that det Г^О. Let (Г(у*) be the inverse matrix to Г. Let Xo be the
random variable Xo = 0; its distribution function is
0 if xo < 0,
1 ii xo> 0.
Then (see Problem 1) the joint distribution function F^...^ of Xo,
Xx, . . . , Xn (where t0 — 0) is given by
), x\, ¦ ¦ ¦ , Xn)
X(*o)
Г ¦¦/"'
-'-00 •'-00
exp
-2 S
A.2)
It follows that
if xk t oo.
Thus, the functions F^ ... ^ given by A.2) form a consistent family of
distribution functions. By the Kolmogorov construction there exists a prob-
probability space (fi, 4$, P) and a process x(t) such that the joint distribution
functions of x(t) are given by A.2). But then, by Problem 2, x(t) is a
Brownian motion.
We have thus proved that the Kolmogorov construction produces a
Brownian motion.
It is easily seen that a Brownian motion is a martingale. Thus the
martingale inequality can be applied to Brownian motion. Another important
inequality is given in the following theorem.
Theorem 1.1. Let x(t) be a Brownian motion and let 0 = t0 < tx <
• • • < tn. Then, for any X > 0,
P\ max x{t) >X}< 2P[x{tn) > X], A.3)
I- 0< / < n J
Proof. Let ;* = first;' such that x(t.) > X. Since x(tn) - *(«:.) @ < ; < n)
has normal distribution, its probability distribution is symmetric about the
origin, i.e., P[x(tn) - x(tj) > 0] = P[x(tn) - x(tj) < 0]. Using also the facts
that x(tn) - x(tt) is independent of ?f(x(t0), . . . , x(tt)) and that {/* = ;}

38 3 BROWNIAN MOTION
belongs to this a-field, we can write
P[ max x(t,) > X, "
L 0<;<n '
< ? P[/* = fc, x@ - x(tk) < 0] = ? ?(Г - k)P[x(tn) - x(tk) < 0]
k-1 k-1
- ? p(/* - *)p[*(o - х(^) > о] = ? p[r = *. *@ - *(«*) > o]
< ? p[/* = k, x(o > a] < pko > 4
k-1
On the other hand,
P[ omaxnx(<i) > К *(O > A] - P[*(A.) > Ы
Adding these inequalities, A.3) follows. Noting that —x(t) is also a
Brownian motion and employing A.3), we get
P[ max |x(t)| > Л] < Pf max x(t) > xl + p[ max (-x(fc)) > xl
< 2F[x(O > X] + 2P[(-x(tn)) > X] = 2P[|x(g| > X].
If X is a random variable with normal distribution N@, oz), then
?X2" = ^a2", EX2n+1 = 0 (n = 0,1,2,...). A.5)
In particular, for any Brownian motion x(t),
E\x{t) - x{s)\2n = Cn\t - s\n (С„ constant) A.6)
for n = 1, 2, .... Taking n = 2 and applying Theorem 1.2.2, we get:
Theorem 1.2. There is a continuous version of a Brownian motion.
Corollary 1.3. Every separable Brownian motion is continuous.
This follows from Theorem 1.2 and Problem 3, Chapter 1.
From now on, when we speak of a Brownian motion, it is always tacitly
assumed that we speak of a continuous version.
From A.6) we see that Corollary 1.2.3 can be applied to a Brownian
motion with /? = 2n, a = n — 1, for any n > 1. Consequently:
Corollary 1.4. For any a < ?, almost all the sample paths of a Brownian
motion are Holder continuous, with exponent a, on every bounded t-interval.

2. NONDIFFERENTIABILITY OF BROWNIAN MOTION 39
2. Nondlfferentlablllty of Brownian motion
We have proved the existence of a Holder continuous Brownian motion x(t),
with any exponent a < J.
Levy [1] has proved that the sample paths of x(t) satisfy:
\x(t) — x(s)\
lim т-^г = 1 a.s., for any T > 0;
— {2(t-s)\ogl/(t-s)}1/2
for proof see also McKean [1].
We shall prove here only a weaker result:
Theorem 2.1. For any a > \, almost all simple paths of a Brownian motion
are nowhere Holder continuous with exponent a.
Proof. Let T be any positive integer and let N be a positive integer such
that N(a - ?) > 1. If x(t, со) is Holder continuous at s (s E [0, Г]) with
exponent a, then
\x(t, со) - x(s, a)\ < P0\t - *|" if \t - s\ < N/n
for some /So > 0 and some positive integer n.
Let /? be a positive number and consider the event
An = {there exists an s E [0, f] such that \x(t) - x(s)\ < /3\t - s\"
if |t - s\ < N/n}.
Then А„ | A as n | oo- If we prove that, for any /?, T, P(A) = 0, then the
assertion of the theorem follows.
Let
4
max
Bn = {«; Zfc(«) < 2PNa/na for some fe}.
Note that if ы E An and |x(t, «) - x(s, w)| < y8|t - s|a when |t - s|
N/n, and if we take к to be the largest integer such that k/n < s, then
Consequently А„ с Bn, so that »
P(A)=nbmP(An)
<]imP(Bn). B.1)

40 3 BROWNIAN MOTION
Next,
Since the random variables
are independent and identically distributed,
2BNa 11 N
P(Bn) < пГ-1" x ! " P M
'-y/n«
Substituting nax = t/, we get
I \N
P(Bn) = nT i Г e-!/2/^-1) dy _»о if n ^ oo,
since 2V(a - 1/2) > 1. Using B.1) we then conclude that P{A) = 0. This
completes the proof.
Corollary 2.2. (i) Almost all sample paths of a Brownian motion are
nowhere differentiable.
(ii) Almost all sample paths of a Brownian motion have infinite varia-
variation on any finite interval.
The assertion (i) follows from the fact that if a function is differentiable at
a point, then it is Lipschitz continuous at that point. Since a function f(t)
with finite variation is almost everywhere differentiable, (ii) is a consequence
3. Limit theorems
Theorem 3.1. For a Brownian motion x(t),
lim V - 1 a.s., C.1)
П° V2tloglog(l/t)
x(t)
lim — 1 a.s. C.2)
'Teo V2f log log t

3. LIMIT THEOREMS 41
These formulas are called the laws of the iterated logarithm.
Proof. We first prove that
xlt)
lim — < 1. C.3)
n0 V2tloglog(l/t)
Let 8 > О, ф(*) = \2t log log(l/t) , and take a sequence tn J, 0. Consider
the event
An = {x(t) > A + S)ip(t) for at least one t E [*„ + 1, *„]}.
Since <#>(*) t if * t.
А„с{ sup
By Theorem 1.1, if x > 0
sup
=cV^ 1 < 2P[x(O > xVTn ] = 2P
J L J
>
e-«V*
it x Jx у it
Taking x = дс„ = A + 8)фЦп+1)/^ , we get
Now take tn = qn where 0 < q < 1 and \= q(l + 8J > 1. Then
xn = {2 log[a(n + 1)X]} , a = log(l/q).
It follows that
P{An) < -^ (C constant).
(n + 1) log(n + 1)
Hence 2 P(AJ < oo, so that, by the Borel-Cantelli lemma, P{An i.o.) = 0
(where "i.o." stands for infinitely often). But this means that
lmT '< 1 + 8.
n° Vatbgiog(i/t)
Since 8 is arbitrary, C.3) follows.
We shall next prove that
lim > 1. C.4)

42 3 BROWNIAN MOTION
We again start with a sequence tn J, 0. Let Zn = x(tn) — x(tn+1). For any
X > 0, € > 0,
By integration by parts,
so that
Thus
Taking *„ - qf», 0 < q < 1,
¦ V2 log[n log(l/qf)] - Vl3 log(an)
'1-q
where a = log(l/9), /3 = 2A — e)z/(l - q), and choosing q sufficiently
small so that /? < 2, we get
P[Zn > A - е)ф(О1 > ^ (c positive constant)
and, consequently,
2 p[zn >(i - с)ф(О1
n-l
Since the events [Zn > A — е)ф(*„)] are independent, the Borel-Cantelli
lemma implies that
P[Zn > A - ?)ф(*„) i.o.] - 1. C.5)
By the proof of C.1) applied to the Brownian motion — x(t),
P[x{tn+1) < -A + с)ф(*п+1) i.o.] = 0.
Putting it together with C.5), we deduce that a.s.
*(*„).= Zn + x(*n+1) > A -

3. LIMIT THEOREMS 43
For any given 8 > 0, if we choose e and q so small that
1 - e - A + t)Vq~ > 1 - S,
and note that Ф(*„+1)/ф(*„)-»"^ if n—»oo, we get that
P[x(tn) > A - 8)<t>(tn) i.o.] = 1.
Since S is arbitrary, this completes the proof of C.4).
In order to prove C.2), consider the process
if <>0'
0 if t = 0.
It is easy to check that x(t) satisfies the conditions (i)-(iii) in the definition
of a Brownian motion. Since it is clearly also a separable process, it is a
(continuous) Brownian motion. If we now apply C.1) to x(t), then we obtain
C.2).
Corollary 3.2. For a Brownian motion x(t),
lim — 1 a.s., C.6)
V
lim — ¦¦ -1 a.s. C.7)
'T°° Л/it log log t
This follows by applying Theorem 3.1 to the Brownian motion — x(t).
If f(t) is a continuously differentiable function in an interval [a, b], and if
nn={'n,i>---><»,-J C-8)
is a sequence of partitions of [a, b] with mesh |П„| = max(tn ;. - tn ^_x)
converging to 0, then
S [/(Q-/K.,-i)]"-»0 if n^oo.
/-1
Such a conclusion does not hold for a Brownian motion. Instead we have:
Theorem 3.3. Let x(t) be a Brownian motion, and Пп a sequence of
partitions C.8) of a finite dosed interval [a, b] with mesh |ПП|—»0 if n—>oo.
Let
si,- 2NU-*(a,,-i)]2
1-1
Then Sn—*b — a in the mean.

44 3 BROWNIAN MOTION
Proof. Write tf = tn ,, m — mn. Then
m о
S. - (b ~ a) = 2 [(*(<») ~ *(',-i)) ~ (t, ~ *,-г)]-
Since the summands are independent and of mean zero,
where
Since the Y. are equally distributed with normal distribution,
!2 - l)" • (b - а)|П„|-»0 if n^oo.
4. Brownian motion after a stopping time
Let т be a stopping time for a Brownian motion x(t). Denote by <^T the
а-field of all events A such that
Ап(т<«) is in ^(x(X), 0 < X < t).
From the considerations of Section 1.3 we know that х(т) is a random
variable.
Theorem 4.1. If т is a stopping time for a Brownian motion x(t), then the
process
y(t) = x(t + т) - x(t), t > 0
is a Brownian motion, and % (y(t), t > 0) is independent of ?fT.
Thus the assertion is that a Brownian motion starts afresh at any stopping
time.
Proof. Notice that if т = s (a constant), then the assertion is obvious.
Suppose now that the range of т is a countable set {s(}. Let BEff and

4. BROWNIAN MOTION AFTER A STOPPING TIME 45
x < t2 < • • • < tn. Then, for any Borel sets Ax, . . . , А„,
P[y(t1)SA1,...,y(tn)eAn,B]
= 2P[t/(*i)eA1;..., i/(OeAn, т = «t, в]
- 2 рЫк + %) - *(*ik
It
еА„,т = %В]. D.1)
Since
(t = sk)n В = [В n (т < sfc)]n (t = sfc) is in Щх(\), 0 < \ < sk)
and 3F(x(A + sj - *(st)), X > 0) is independent of 3F(x(A), 0 < A < sfc),
and since the assertion of the theorem is true if т = sk, the fcth term on
the right-hand side of D.1) is equal to
P[(*(*i + **) - x(sk)) 6А„ . . . , {x{tn + sk) - x{sk)) e AjP[r = *fc; B]
= Ptxfo) e Ах,..., x(O e aJp[t = et, B].
Summing over fe, we get
Piyit^A,, ..., t/(OeAn, B] - РШеА^, ..., x(OeAjP(B). D.2)
Taking В = Й, it follows that the joint distribution functions of the
process y(t) are the same as those for x(t). Since y(t) is clearly a continuous
process, it is a (continuous) Brownian motion.
From D.2) we further deduce that the a-field '$( y(t), t > 0) is indepen-
independent of %.
In order to prove Theorem 4.1 for a general stopping time, we approxi-
approximate т by a sequence of stopping times т„, defined in Lemma 1.3.4. We have
(see the first paragraph of the proof of Theorem 2.2.4)
Set yn(t) = x(t + т„) — х(т„). By what we have already proved, if В е ?FT,
then
h) <xv..., yn(tk) < xk, B] = Pfcfo) < xv . . . , x(tk) < xk]P(B)
for any 0 <*!<••• < tk. Notice that yn(t) -> y(t) a.s. for all t > 0, as
n -» oo. Hence, if (дс1( . . . , xk) is a point of continuity of the fe-dimensional
distribution function Fk(xb . . . , xk) of (х(^), . . . , x(tk)), then
) <xlt..., y(tk) < xk, В] = ?[дс(О <«!,..., x(tk) < xk]P{B).
D.3)

46 3 BROWNIAN MOTION
Since (see Problem 1) Fk(xv . . . , xk) is actually an integral
I • • • I P(zu . . . , %) dz1 ¦ ¦ ¦ dzk,
J-(X> •'—00
it is continuous everywhere. Thus D.3) holds for all xv . . . , xk. This implies
that
)ea1, ..., y(tk)eAk, В] = Р[х(ОеА1;..., x(tk)eAk]P(B)
for any Borel sets Al5 . . . , Ak, and the proof of the theorem readily follows.
5. Martingales and Brownian motion
If x(t) is a Brownian motion and <§t = <§(x(X), 0 < X < t), then
jE["(x(t) — зсE)J| ^T 1 = t — s E 2)
a.s. for any 0 < s < t. Note that E.1), E.2) hold if and only if x{t) and
x2(t) — t are martingales.
We shall now prove the converse.
Theorem 5.1. Let x(t), t > 0 be a continuous process and let ?t (t > 0) be
an increasing family of a-fields such that x(t) is S^ measurable and E.1),
E.2) hold a.s. for all 0 < s < t. Then x(t) is a Brownian motion.
Proof. For any e > 0 and a positive integer n, let т0 be the first value of t
such that
;_max |x(*') - x(*")| = e;
0 < s\ s" < t
if no such t exists, set т0 = oo. Let т = т0 л 1- Since, for 0 < s < 1,
00
{T>*}= U П
0<s',s" <s
т is a stopping time.
Let t/(t) = x(t л т). Since x(t) and %2(f) — t are continuous martingales,
Theorem 1.3.5 implies that y(t) and y2(t) — t are also continuous martin-
martingales. Hence,
f[y2(s)-sAr]dP=f[y2{t)-tAr}dP if AE#,, s<t

5. MARTINGALES AND BROWNIAN MOTION
47
where #, = f (y(X), 0< A < s). It foUows that
fA{E[ y«(«) - уЩ\§,} dP= jT [ y*(t) - y>(s)} dP
= f [t at ~ s At] dP< f (t- s)dP.
JA JA
Since the integrand on the left is Ws measurable, we conclude that
E {[ y2(t) - t/2(*)]|fs} < t- s a.s., s < t. E.3)
Our aim is to prove that if 0 < t0 < tY < • ¦ • < tk, then
Eexp < 2
к
-2
E.4)
for any real numbers Xf. In view of the result stated in Problem l(a), this will
show that x(t) is a Brownian motion.
We begin with the special case к = 1, t0 = 0, tx — 1. Thus, we want to
prove that
EeiX*m = e~x*/2. E.5)
Our approach will be to write
*(i) = 2
and make use of E.1), E.2) while at the same time, letting n increase to
infinity. However, a technical difficulty arises due to the fact that the
random variables x(j/n) — x((j — l)/n) are not uniformly bounded. It is
for this reason that we shall work, instead, with the process y(t) and with the
random variables
In view of the definitions of т and y(t),
Since y(t) is a martingale and since y2(t) satisfies E.3),
E-6)
^Si, . . . , S,-i) =0 B< / < n),
- ВД < ^ .
E.7)

48
3 BROWNIAN MOTION
a.s. Hence, if / > 1,
n) = E
-aff A + 0A))
where, in view of E.6), o(l) -» 0 if с -»0, uniformly with respect to X, n
provided X varies in a bounded interval.
It follows that
< E
< CE\[ - -
we have used here E.7). The symbol С denotes any one of various different
positive constants which do not depend on n, e, X, provided X is restricted to
bounded intervals, e < 1, n > 1.
The last estimate implies that
Summing over / and noting that
E.8)

5. MARTINGALES AND BROWNIAN MOTION 49
we get
Consequently,
e~x*/2]\ < C[\ - E(y2(l))] + o(l). E.9)
Since x(t) is a continuous process,
p(T = 1)^.1 and P[y(l) = x(l)]^l if n-»oo.
Hence, by the Lebesgue bounded convergence theorem,
EeA«a) ^ Ee**w if n^oo.
Therefore from E.9) it follows that, for any у > 0, if n > no(e, y), then
|E[e«M» _ e-x«/2]| < C[l - ?(;/2(l))] + o(l) + Y. E.10)
Note next that
1 - E( y*(l)) = E[x2(l) - y2(l)] = / И1) - у2(т)] dP
•/т<1
< f x2(l) d?^0
•/т<1
if Р(т = 1)-»1. Thus, if n > ni(?, y),
Using this in E.10) we get
Taking y-»0 (and n > no(e, y), n > nx{t, y)) and then e-*0, the assertion
E.5) follows.
Similarly one can prove that
E ехр[1А,.(хЦ) - х(г^у))] = exp[- ?A,2] E.11)
for any 0 < f;_! < t^ Xj real.
By reviewing the proof of E.11) one finds that the same proof with minor
changes actually gives a better result, namely,
E{exp[iA, (*(*,) - x(t,-1))]\?(x(\),0 < X < t,^} - exp[ - |Л,2] a.s.
E.12)
(Notice that if r^_x is defined analogously to т, with t = 0 replaced by
t — tj_v and if y(t) — x(t л Tj-i)< then
Ц y(X), 0 < X < ^_0 - ^(*(X). 0 < X <^)
since т^_! > ^_i.)
Now let 0 < t0 < • • • < tk_l < tk and let Xx Xk be real numbers.

50
3 BROWNIAN MOTION
Applying E.12) successively, we find that
[ ? 1
E exp i 2 \(x(*j) "" x(*/-i))
E I exp
X ?
= E I exp
), 0 < X
fc-i
- exp - 2
This completes the proof of E.4).
Remark. By the remark at the end of Section 1.3 we have that y(t) and
y2(t) — t (introduced in the last proof) are martingales with respect to 45t.
Hence instead of E.3) we actually have
<t-s a.s., s<t.
This implies
.} = exp[- |X2(f - s)}. E.13)
6. Brownian motion in n dimensions
Definition. An n-dimensional process w(t) = {w^t), . . . , wn(t)) is called
an n-dimensional Brownian motion (or Wiener process) if each process w{(t)
is a Brownian motion and if the a-fields 45{wt(t), t > 0), 1 < i < n, are
independent.
Theorem 6.1. Let w(t) be an n-dimensional Brownian motion. Then
P hm
°
V2tloglog(l/t)
PlinT
1 a.s.,
1 a.s.
F.1)
F.2)
log log

6. BROWNIAN MOTION IN n DIMENSIONS 51
Proof. It is enough to prove F.1). If F.1) is false, then there is a 5 > 0
such that
\w(t)\
lim > 1 + 5 with positive probability.
'4° V
Given any с > 0, cover the unit sphere by a finite number of spherical
regions K{ with opening e with respect to the origin. Then, for some i,
P\ lim
But if г is the center of K, and A+5) cos e > 1, then
[ z-w(t) }
P] lim >A + 5) cose > l[ > 0.
I'"*0 V2floglog(l/f) J
This is impossible since z • w(t) is a. Brownian motion (cf. Problem 7).
Theorem 4.1 and its proof extend immediately to n-dimensional Brownian
motion.
We shall now extend Theorem 5.1 to n-dimensions.
Theorem 6.2. Let x(t) = (x^t), . . . , xn(t)), t > 0 be a continuous, n-di-
n-dimensional process and let 'f, (t > 0) be an increasing family of a-fields
such that x(t) is ?Ff measurable and, for all 0 < s < t < oo,
E[x(t)-x(s)\%] = 0 a.s.,
Е[(ф) - ф))(ф) - ф))] = Sti(t - s) a.s.
where 8^ = 0 if i ф \, 8U = 1. Then x(t) is a Brownian motion.
Proof. For any уЕЙ", \у\ = l.the conditions of Theorem 5.1 are satisfied
for у • x(t). Hence у • x(t) is a Brownian motion. Thus, in particular, each
xt(t) is a Brownian motion. It remains to show that these motions are
mutually independent. For this it suffices to show that ^(х{(г), t > 0) is
independent of f (xj(t), t > 0) if i ф j. Take, for simplicity, i = 1, / = 2.
Since YjXj + Y;X2 is a Brownian motion if yf + yf = 1,
[УЛ@ W)] *•
Since also Exf(t) = t,
EXl(t)x2(t) = 0.
But then
Ex,(t + s)x2(t) - ?{^(«)Е[х,(* + s)\%]} - Ex2(t)xt(t) = 0,

52 3 BROWNIAN MOTION
i.e., x^t + s) is independent of x2(t). Similarly x2(t + s) is independent of
Xj(t). Since t and s are arbitrary nonnegative numbers, it follows that
f \xx(X), X > 0) is independent of 9{х2(Х), X > 0). This completes the proof.
One can easily check that an n-dimensional Brownian motion satisfies the
Markov property with the stationary transition probability function
By the method of Section 2.1 we can construct a time-homogeneous
Markov process corresponding to the transition probability F.3). The sample
space consists of all Д "-valued functions on [0, oo). But we can also
construct another model (fl0, CDTL, 91tt, ?(f), Px) where Bo is the space of all
continuous functions x(-) from [0, oo) into R", (D1tf is the smallest a-field
such that the process
€(«,*(¦))=*(*) F.4)
is measurable for all 0 < s < t, and
x{();?(l)v,i(k)k}
= P{w; x + ю{^)GAlt . . . , x + w(tk)<EAk). F.5)
Theorem 6.3. {Яо, 911, 91tt, i-(t), Px} is a time-homogeneous Markov
process with the transition probability function given by F.3).
Proof. For any set В in 91L, define
PX(B) = ?{w; x + w{-, w) is in B).
It is easily seen that Px is a measure on (Bo, СУИ), satisfying F.5). The rest
follows from Theorem 2.1.3.
To any continuous p-dimensional Markov process (Я*, 9r*, 9r**, x*(t),
P* s) we can correspond a Markov process {Я, 9И, 91LJ, |(t), Px>#} having
the same transition probability function p. Я is the space of all continuous
functions x(-) from 0 < t < oo into R", 911* is the smallest a-field such that
x(X) is measurable for any s < X < t, 91L = %?,, ?(t, %(¦)) = x(t) and,
finally, if s <*!<••• < tfc,
PIiS{x(-); {(fjEA,, . . . , €(tt)GAt}- ^.{«; ^(^EA,, .... *•(**) EAt}.
F.6)
In fact, for any set В in 91L, define
*UB) -!*.{«; **(-,«) SB}.
It is easily seen that Px , is a measure on (B, 91L), and it satisfies F.6). In

PROBLEMS
53
view of Theorem 2.1.3, {B, 91L, 91L*. ?(t), Px s} is then a Markov process
with the same transition probability function as for (B*, §"*, '&**,
o }
PROBLEMS
1. Recall that n random variables Xj, . . . , Xn are said to have joint
normal distribution (ft, Г), where ft = (ft1; . . . , fij, Г = (rjf)"^=1 sym-
symmetric, if the characteristic function
(«!, . . . , «„))
has the form
[n n
2 ft,«j - i 2 ¦
/-1 ,-,*-i
Prove that if this is the case then:
(a) The random variables Xv . . . , Xn are independent if and only if
Ги = 0 whenever i ф /.
(b) If Xt, Xj are independent for any pair (t, /), i ф j, then Xv . . . , Xn
are mutually independent.
(c) If Yj = 2"=1aj;X(- A < i < to) where atj are constants, then
Y1; . . . , Yn have joint normal distribution.
(d) ft, = EX,, Ttj = E(X, - ftf)(X; - ft,.).
(e) If det Г^О, then (Xx — ft1; . . . , Xn - fij has a distribution func-
function p( y) given by
"(У)
exP
«,/=i
where (ГГ1) is the inverse matrix to Г.
2. If x(f), t > 0 is a process with x@) = 0 such that for any
0 < t0 < tx < • • ¦ < ?n the random variables x(t0), . . . , x(tn) have joint
normal distribution N(ji, Г) where ft = (ft, . . . , ft), Г = (Г^), Т^к =
a2 min(^, ffc), then x(f) is a Brownian motion with drift ft and variance a2.
3. Verify A.5) for a random variable X with normal distribution N@, a2).
4. Give a direct proof of C.2). [Hint: Apply the method of proof of C.1)
with q > 1.]
5. If x(t) is a Brownian motion, [x(t)/t]—>0 a.s. as t-±oo.
6. Let x(t) be a Brownian motion and let / be an interval in [0, oo). Then
for any 5 > 0, P{x(n8)Gli.o.} = 1.
7. If w{(t) — 2"_iaj#u?;(f) where (aif) is an orthogonal matrix and

54 3 BROWNIAN MOTION
(w-^t), . . . , wn(t)) is an n-dimensional Brownian motion, then (wv . . . , wn)
is also an n-dimensional Brownian motion.
8. If w(t) is a Brownian motion then aw(t/a2) is also a Brownian motion,
for any positive constant a.
9. If w(t) is a Brownian motion and z(t) = exp[aw(t) — (A/2], a positive
constant, then
(i) Ez(t) = l.
(ii) z(t) is a martingale; in fact, if §s = ^(w(X), 0 < X < s), then
(iii) P{maxo<x<f[u;(s) - as/2] > /3 < e~afi if a > 0, /3 > 0.
10. Let un(t) be Brownian motion defined on the same probability space,
p
and suppose that un(t)-*u(t) for each t > 0, as n—»oo. Prove that u(t) is a
Brownian motion.
11. If w(t) is a Brownian motion and A is a positive constant, then, for any
T >0,
EeA\w(t)\ < C if 0 < « < T,
where С is a positive constant depending on А, Г.
12. Let f,,t >0 be an increasing family of a-fields and let x(t) be
an n-dimensional process such that x(t) is ^t measurable and
E[x(t + s)\&t] = x(t) for all t > 0, s > 0. Prove that if у ¦ x(t) is a
Brownian motion for any у ? R", \y\ = 1, then x(t) is an n-dimensional
Brownian motion. [Hint: Cf. the proof of Theorem 6.2.]

4
The Stochastic Integral
In this chapter we shall define the integral
I(T)=fJf(t)dw(t)
where w(t) is a Brownian motion and f(t) is a stochastic function, and study
its basic properties. One may define
= f(T)w(T) - fTf'(t)w(t) dt
if / is absolutely continuous for each w. However, if / is only continuous, or
just integrable, this definition does not make sense.
Since w(t) is nowhere differentiable with probability 1, the integral
Jo fit) dw(t) cannot be defined in the usual Lebesgue-Stieltjes sense.
1. Approximation of functions by step functions
We shall call a stochastic process also a stochastic function or, briefly, a
function.
Let w(t), t > 0 be a Brownian motion on a probability space (Я, 5", P).
Let ^t (t > 0) be an increasing family of a-fields, i.e., 9^ с %2 if tx < t2,
such that ff с f, f («;(s), 0 < s < t) is in <&t, and
€F(ty(X + t) - u>((), X > 0) is independent of ^
for all t > 0. One can take, for instance, <$t = ^(w(s), 0 < s < t).
Let 0<a<^<oo.A stochastic process/(f) defined for a < t < /? is
called a nonanticipative function with respect to ^ if:
(i) /(f) is a separable process;
55

56 4 THE STOCHASTIC INTEGRAL
(ii) f(t) is a measurable process, i.e., the function (t, u)—>f(t, w) from
[a, /3] X Я into R1 is measurable;
(iii) for each tE.[a, /}], f(t) is ?t measurable.
When (iii) holds we say that /(t) is adapted to f (. We denote by L? [ a, /3 ]
A < p < oo) the class of all nonanticipative functions f(t) satisfying:
P\ (Р \f(t)\P dt< oo I = 1 (p{esssup|/(t)| < oo) = lifp = ooY
[Ja j \ la<t</S J /
We denote by M^[a, /3] the subset of L^[a, /3] consisting of all functions
/with
E Г^ |/(*)lp Л< °° Desssup|/(*)|l < ooifp- oo).
Definition. A stochastic process f(t) defined on [a, /3] is called a step
function if there exists a partition a = t0 < tx < • • • < ir = /3 of [a, /3]
such that
/(t)-/@ if f<< *<W 0< i< r-1.
Lemma 1.1. Let/ 6 L*[a, /3]. 77ien:
(i) f/iere exists a sequence of continuous functions gn in L^[a, /3] such
that
lim Г"|/(*)-S.(*)I2* = O a.*.; A.1)
(ii) Лете exists a sequence of step functions fn in L^[a, /3] such that
Urn fP\f(t)-fn(t)\2dt = 0 a.s. A.2)
Proof. Let
([*2)] if
(
U 10 if \t\ > 1
where с is a positive constant determined by the condition /?0Op(t) df = 1.
Define f(t) = 0 if t < a and let
;е)Ф Bе<1). A.3)

1. APPROXIMATION OF FUNCTIONS BY STEP FUNCTIONS 57
Then JJ is clearly continuous, and
(/«/)(*) = 7 f p( '~ж~е к*) * = f p(*)/(* - * -«) ^-
A.4)
By Schwarz's inequality,
f (Jjf A < f{ Г P(Z) dz Г p(z)f(t ~Z-€)dz}dt
< f p(z){ fa[j*(t) dt} dz.
Hence
Й ft
[ (JJJ dt< f f(t) dt. A.5)
For fixed w for which /ff2(f, u) dt < oo, let «„be nonrandom continuous
functions such that un(t) = 0 if t < a and
f^|«n(f)-/(t,w)|2d^0 if n->oo. A.6)
Since ып is continuous, it is clear that
(/e«„)(*)-»«„(*) uniformly in te[a, /3], as e-»0.
Writing
and using A.5) with / replaced by / — un, we get, after taking e —»0,
Taking n—»oo and using A.6), we obtain
= 0 a.s.
Since the integrand on the right-side of A.4) is a separable process that is
l.'7( measurable, the integral is also <5t measurable. Consequently the asser-
assertion (i) holds with g,, = Jl/nf.

58 4 THE STOCHASTIC INTEGRAL
To prove (ii), let
Jt) - g-( — ) ii a + — < t< a + ^-±-i @ < к < ml p - a)).
V,m(<)-gn(<)N' = 0 a.s. A.7)
К
Then
Now, for any 5 > 0,
g(«)l2 * > f
f } < | for *ome » "o-
f } < f
From A.7) with n = n0 we get
Hence
Taking 8 = 1/k and denoting the corresponding hn<) m<> by \pk, it follows
^a, /}] and
But then there is a subsequence {/„} of {^} satisfying the assertion (ii).
Lemma 1.2. Let /EM*[a, j8]. Then:
(i) Лете exists a sequence of continuous functions kn in M*[a, ft] such
that
\f(t) ~ kn(t)\2 dt-*O if n~*oo; A.8)
(ii) there exists a sequence of bounded step functions ln in M*[a, /3] such
that
\f{t)-ln{t)\2dt-*0 if n-+oo. A.9)
Proof. Let g^ be as in Lemma 1.1. For any N > 0, let
t if \t\ < N,
Nt/\t\ if \t\ > N.

2. DEFINITION OF THE STOCHASTIC INTEGRAL 59
Notice that \<t>N(t)- <S>N(s)\ < \t - s\. Therefore
'" IM/O) - *»(Ш\2 dt <// I/O - g-OI" dt^° a.s.
¦'a
Since
the Lebesgue bounded convergence theorem gives
?(V(/(«))-M6,(f*^" if »^«- A-Ю)
Next,
? / I*n(/(')) ~ /C*)l" dt < 4 // |/(*)|2 dt dP-^0
{|/|>2V}
as N->ao, since /EM,2,[a, j8]. From this and A.10) it follows that for every
positive integer к there are N = N(k) and n = n(fc, N) such that
Taking hk - <t>N(gn) where N = N (к), п = n(N, k) and noting that hk(t) is
nonanticipative, the assertion (i) follows.
The proof of (ii) is similar. The ln are of the form флг(/„) where the /n are
as in Lemma 1.1. Notice that these are in fact step functions.
2. Definition of the stochastic Integral
Definition. Let f(t) be a step function in L^[a, /3], say f(t) = ft if
tt < ( < ti + 1, 0 < i < r - 1 where a = t0 < tx < • • • < tr = /3. The ran-
random variable
is denoted by
Г
•'a
and is called the stochastic integral of / with respect to the Brownian motion
w, it is also called the ltd integral.

60
4 THE STOCHASTIC INTEGRAL
Lemma 2.1. Let fx, /2 be two step functions in L^[a, /3] and let \t, X2 be
real numbers. Then X1f1 + X2f2 is in L^[a, ft] and
X2f2(t)]dw(t) = X, f № dw(t) + X2 f /a(t) dw(t).
a Ja Ja
B.1)
The proof is left to the reader.
Lemma 2.2. If f is a step function in M%[a, )8], then
E (P f(t) dw(t) = 0,
•'a
ff{t)dw{t)
B.2)
B.3)
Proof. Since
i = 0
is finite, by assumption, we deduce that Ef2(tt) < oo. In particular,
E\f(tt)\ < oo. Also, E\w(ti+1) — ю(^)\ < oo. But since f(t{) is f^ measur-
measurable whereas w(ti+1) - w(tt) is independent of tSti,
0.
^)J are independent and have finite
has finite expectation. By Schwarz's
If к > i, then и>(^ + 1) - w(tk) is independent of f(tk)f{ti)(w(ti + 1) -
In view of the last inequality and the finiteness of E\w(tk + 1) — w(tk)\, we
deduce that
Summing over i, B.2) follows.
Next, since f2(t{) and (w(ti + l) —
expectation, also/2(?j)(u>(ti + 1) — ю(
inequality it follows that

2. DEFINITION OF THE STOCHASTIC INTEGRAL
61
Hence
dw(t)
= 2*
i=O
i = O
г
by B.4), and B.3) is proved.
Lemma 2.3. For any step function f in L* [a, (i ] and for any e > 0, N > 0,
Proof. Let
dw(t)
if «fc
0 if tk<t<t
^. B.5)
k + 1
and
and
/=o
where /(t) = f(tf) if ^ < t < t/+1; f0 = a < fx <
фд, e I^[o, /8] and
N,
< tr = j8. Then
(\l(t)dt=± /2(^)
•'« / = о
where r is the largest integer such that
2
j-0
»¦ -
Hence
E I ф2(() л < N.
'a
Further, /(() - <j>N(t) = 0 for allt e [a, )8) if /ff2(«) dt < N. Therefore
fPf(t)dw(t)
> ? < P
+ P
f%N(t)dw(t)
>
(t)dt > N .

62
4 THE STOCHASTIC INTEGRAL
Since, by Chebyshev's inequality, the first integral on the right is bounded by
2
lip I ¦ / л\ J I t\ ^" *• '
(%N{t)dw(t)
the assertion B.5) follows.
We shall now proceed to define the stochastic integral for any function/
in L2w[a, p].
By Lemma 1.1 there is a sequence of step functions fn in L^[a, /?] such
that
f \fn(t) - f(t)\2 dt ^ 0 if n-+oo.
B.6)
Hence
By Lemma 2.3, for any € > 0, p > 0,
P
>
It follows that the sequence
[f
fn{t)Av(t)
is convergent in probability. We denote the limit by
ff(t)dw{i)
and call it the stochastic integral (or the Ho integral) of f(t) with respect to
the Brownian motion w(t).
The above definition is independent of the particular sequence { /„}. For
if { gn} is another sequence of step functions in L^[a, /3] converging to f in
the sense that
then the sequence {hn} where h2n = fn, h2n+1 = g^ is also convergent to /
in the same sense. But then, by what we have proved, the sequence

2. DEFINITION OF THE STOCHASTIC INTEGRAL
63
is convergent in probability. It follows that the limits (in probability) of
f?fn dw and of /fgn dw are equal a.s.
Lemmas 2.1—2.3 extend to any functions from ^
Theorem 2.4. Let fx, f2 be functions from L^[a, /3] and let \1г Х2 be real
numbers. Then Xx/X + X2 f2 is in L^[a, ft] and
[ iKfiit) + Kf2(t)]dw(t) = Xi Г fi(t) dw(t) + X2 f f2{t) dw{t).
•'a •'a •'a
B-7)
Theorem 2.5. If f is a function in M*[a, /?], then
E I f{t) dw{t) = 0,
•'a
rP
ff(t)dw(t) =E ff(t)dt.
B.8)
B.9)
Theorem 2.6. If f is a function from L^[a, /?], then, for any e > 0, N > 0,
dw(t)
> *} <
dt > л/) + ^ . B.10)
The proof of Theorem 2.4 is left to the reader.
Proof of Theorem 2.5. By Lemma 1.2 there exists a sequence of step
functions /n in M*[a, /3] such that
E fP\fn(t)-f(t)\2dt-^O if n-,00.
This implies that
By Lemma 2.2,
E ffi(t) A->E ff(t) dt.
E fPfn(t) dw(t) = 0,
Ja
fPfn(t)dw(t
J"fn(t) dw(t) -J"fm(t) dw(t)
B.11)
B.12)
B.13)
\fn(t) - fm(t)f A-0
if ntm—> oo.
B.14)

64 4 THE STOCHASTIC INTEGRAL
From the definition of the stochastic integral,
Г fn(t) dw(t) Л Г /(о dw(t).
Using B.14) we conclude that actually
in
Hence, in particular,
E f f{t) dw(t) = lira E ( fn{t) dw(t),
fPf(t)dw(t)
— lim E
and using B.12) and B.13), B.11), the assertions B.8), B.9) follow.
Proof of Theorem 2.6. By Lemma 1.1 there exists a sequence of step
functions fn in L^[a, ft] such that
B.15)
B.16)
By definition of the stochastic integral,
Applying Lemma 2.3 to /„ we have
> ?' < P
> N'\ +
N'
@'
Taking n -» oo and using B.15), B.16), we get
{ JT'/•(') A > N J + -2
@1
for any с > с', N < N'. Taking c' | «, N' 12V, B.10) follows.
Theorem 2.7. Let /, /n be in L^[a, j8] and suppose that
f l/n(*) ~ /(*)l2 dt -* 0 as n-»oo. B-17)
37ien
1 fn(t)dw(t) Z. f*f(t)dw(t) as n->oo. B.18)

2. DEFINITION OF THE STOCHASTIC INTEGRAL
Proof. By Theorem 2.6, for any 6 > 0, p > 0,
65
(/„(«) - f(t)) dwit)
> e} < ?
{ Jf" |/n(t) -
dt > €2p} + P.
Taking n—»oo and using B.17), the assertion B.18) follows.
The next theorem improves Theorem 2.5.
Theorem 2.8. Let feM*[a, fi]. Then
f{t) dw(t)
= 0,
B.19)
B.20)
We first need a simple lemma.
Lemma 2.9. If /EL2 [а, /J] and fis a bounded and fFa measurable
function, then f/ is in L^[a, y3] and
[P Sf(t) dw(t) = ? f fit) dwit). B.21)
Proof. It is clear that f/ is in L2[a, y3]. If / is a step function, then B.21)
follows from the definition of the stochastic integral. For general / in
L^[u, /J], let /„ be step functions in L^[a, /J] satisfying B.17). Applying
B.21) to each fn and taking n—*oo, the assertion B.21) follows.
Proof of Theorem 2.8. Let f be a bounded and <5a measurable function.
Then ?f belongs to M*[a, /3] and, by Theorem 2.6,
f Sf{t)
0.
Hence, by B.21),
f(t) dw(t)} =0,
i.e.,
0.
This implies B.19). The proof of B.20) is similar.

66 4 THE STOCHASTIC INTEGRAL
Theorem 2.10. If /?I^[a, ft] and f is continuous, then, for any sequence
Пп of partitions a = tn 0 < tn x < • • • < tn ^ = ft of [a, ft] with mesh
|П„|->0,
1 f(tn k)[w(tn k+1) - w(tntk)] Л f{t) dw(t) as n-*oo. B.22)
Proof. Introduce the step functions g^.
&.(*) " /(<**) if *п,к < * < tn,k+v 0<k<mn-l.
For a.a. u, gn(t) -*f(t) uniformly in t E [a, ft) as n -» oo. Hence
By Theorem 2.7 we then have
(Pgn(t)dw(t) Л fPf(t)dw(t).
Ja •'a
Since
the assertion B.22) follows.
Lemma 2.11. Let /, g belong to L^[a, ft] and assume that f(t) = g(t) for
alia < t < ft, wGfi0. Then
f f{t)dw{t)=f g{t) dw{t) fora.a.ueu0. B.23)
•'a ¦'a
Proof. Let \pk be the step function in L%\a, ft] constructed in the proof of
Lemma 1.1, satisfying
Similarly let фк be step functions in L^,[a, ft] satisfying
From the construction in Lemma 1.1 we deduce that we can choose the
sequences <j>k, $к so that, if wGfl0, <t>k(t, u>) = \f/k(t, to) for a < t < ft. Hence,

3. THE INDEFINITE INTEGRAL 67
by the definition ff the integral of a step function,
I $k(t) dw(t) = I <j>h(t) dw(t) if w?fi0.
•'a •'a
Taking fc~»oo, the assertion B.23) follows.
3. The indefinite integral
Let/EL?[0, Г] and consider the integral
f{s) dw{s), Q< t < T C.1)
[
where, by definition, fof(s) dw(s) — 0. We refer to I(t) as the indefinite
integral of/. Notice that I(t) is <St measurable.
If / is a step function, then clearly
Г f{s) dw{s) + С f(s) dw{s) = С f(s) dw(s) if 0<a<P<y<T.
Ja Jp Ja
C.2)
By approximation we find that C.2) holds for any / in L?[0, T].
Theorem 3.1. I//EM^[0, Г], then the indefinite integral Щ @ < t < Г)
is a martingale.
Proof. Let 0 < t' < t < T. By C.2) and Theorem 2.8,
f(s) ds\%) = I(t').
Theorem 3.2. I//EL?[0, Г], then the indefinite integral I(t) @ < t < Г)
has a continuous version.
Proof. Suppose first that / e M,f[0, Г]. Let {/„} be a sequence of step
functions in M^ [0, Г] such that
E С\f(t) - /n(t)J dt^O if n-*oo. C.3)
Notice that the indefinite integrals
Ш - f' /.(*) dw(s) @<t<T)

68
4 THE STOCHASTIC INTEGRAL
are continuous functions. By Theorem 2.8, In(t) - Im(t) is a martingale, for
each n, m. Hence, by the martingale inequality (Corollary 1.3.3)
P{ sup \In(t) - Im(t)\ >
fT(fn(s)-fm(s))dw(s)
Jn
if n,
Taking e = 1/2* it follows that for some nk sufficiently large,
sup Jljt) - Im(t)\ > ^ } < ^ if m > nk.
Q p
We can choose the nk in such a way that nk f if к f. Hence,
Since 2fc~2 < oo, the Borel-Cantelli lemma inplies that
i.e., for a.a. ы
l1^*) - ^+1('I < i forall 0 < f < T, if fc > *„(«).
But then, with probability one, {^(i)} is uniformly convergent in
tG[O, T]. The limit J(t) is therefore a continuous function in tG[O, T] for
a.a. со. Since C.3) implies that
it follows that
С fn(s) dw(S)^ С f(s) dw(s) in L2(Q),
}(t) = С f(s) dw(s) a.s.
•'o
Thus, the indefinite integral has a continuous version.
Consider now the general case where /EL^[0, T]. For any N > 0, let
1 if z < N,
0 if z > N,
C.4)
and introduce the function
It is easily checked that fN belongs to M^[0, T]. Hence, by what was

3. THE INDEFINITE INTEGRAL
69
already proved, a version of
Ш = f Ш dw(s) @< t < T)
is a continuous process.
Let
f(t)dt<N .
If coG^, then /^(f) = /M(t) for 0 < t < T, M > N. By Lemma 2.11 it
follows that for a.a. coEfi*,
Therefore
Ш = Ш if 0 < t < T.
/(*)= Jim
is continuous in t ? [0, T] for a.a. ш (= ?LN.
Since QN f, P(fijv) 1 1 if N 1 <x>, J(t) @ < t < T) is a continuous process.
But since for each tG@, T],
¦\
= P
as M—»oo, we have, by Theorem 2.7,
Consequently, I(t) has the continuous version J(t).
Remark. From now on, when we speak of the indefinite integral C.1) of a
function/eL^[0, T] we always mean a continuous version of it.
Theorem 3.3. Let /GL?[0, Г]. T/ien, /or ani/ e > 0, N > 0,
P sup
I o<t<r
f2
Proof. With the notation of C.4), C.5) we have
sup
0< t<T
fj(s)dw(s)
> e
< P{ sup
0< t<T
ftf(s)dw(s)-ftfN(s)dw(s}
>0
+ P \ sup
,0<t<T
> €
^ A + B.

70
4 THE STOCHASTIC INTEGRAL
By Theorem 3.1, /oijv(s) dw(s) is a martingale. Hence, by the martingale
inequality,
Next, on the set
Q» - {fTf2
N
we have/(t) = fN(t) for 0 < t < T. Hence, by Lemma 2.11, for each fixed t
in [0, T]
ft ft
I f(s)dw{s)=l fN(s) dw(s) fora.a.
•'0 J0
Since both integrals are continuous processes, the last relation holds for all
te[0, T] and for all u,ett'NCUN where P(toN\u'N) - 0, i.e.,
A < P(Q\QW) = P\ j f(s) ds > N\.
Combining the estimates on A, B, C.6) follows.
Theorem 3.4. Let fn, f belong to L*[0, T] and assume that /J|/n - /|2
dt Д 0 t/n-»oo. Then
ft ft p
I fn(s) dw(s) - I f(s)dw(s) -» 0 if n-»oo.
•'о •'о
This is a consequence of Theorem 3.3.
Theorem 3.5. Let f G Л0О, T]. T/ien /or any X > 0
P [ sup I ff(S) dw(s) > \) < ^ E (Tf\s) ds. C.7)
sup
о <«< г
Proof. By Theorems 3.1, 3.2, the indefinite integral of / is a continuous
martingale. The inequality C.7) then follows from the martingale inequality.
Another estimate is given in the following theorem.
Theorem 3.6. Let f G Л0О, T]. Then
sup
0< t< T
Cf{s)du>(s) < 4? (Tf(t)dw(t) =4? (Tf(t)dt.
Jo J Л) Л)
C.8)
Like Theorem 3.5, the present estimate is also a consequence of the fact

3. THE INDEFINITE INTEGRAL 71
that /o/(*) dw(s) is a continuous martingale. The general result for martin-
martingales is given in the following theorem.
Theorem 3.7. If X(t) @ < t < T) is a separable martingale, then, for any
a > 1,
E{ sup |x(t)|«} <(-5Ц-)в?|Х(Г)|«. C.9)
The proof follows immediately from the following lemma.
Lemma 3.8. If {Xn} is a submartingale and Xn > 0 for all n, then, for any
a > 1,
Proof. Let Y = тах1<(<пХ(. For any X > 0, set x*(X) = 1 if X, < X for
1 < i < fc - 1, Xfc > X, and x*(ty = 0 otherwise. Then 2? = 1x*(X) = 1 if
X < Y and2t=1x*(X) = 0 if X > Y. It follows that, for any /? > 0,
fj X. C-11)
Since Хх*(Х) < Хкхъ(\),
X"-1 2 Л(Х) < 2 ^-%X,W. C-12)
Observing that X/t(^)is measurable with respect to ^(Xj, . . . , Xk) and using
the submartingale assumption, we get
Taking the expectation in C.12) and using the last inequality, we find that
± EX-2 2
Integrating with respect to X and using C.11), we get
-EYa < ——EYa-%.
a a - 1 "
By Holder's inequality,
?y« < _^ (?Ya)(")/a(EXaI/a
a — I "
and C.10) follows.
Remark. It is clear that all the results derived so far in this section

72 4 THE STOCHASTIC INTEGRAL
regarding the indefinite integral C.1) extend to indefinite integrals
С f(s) dw(s).
Theorem 3.9. Let f G M*[a, /8]. Then
sup
ff(s)dw(s)
a<t< i
The proof is left to the reader.
r« • C-13)
Definition. Let т0 be a random variable, 0 < т0 < Т. If /?L^[0, T], we
define
f°f(s) dw(s
to be the random variable I(t0), where I(t) is given by C.1).
From Theorems 3.1, 3.2, and 1.3.5 we deduce:
Theorem 3.10. If f G M*[0, т] and т is a stopping time with respect to ?Ft,
0 < т < T, i.e., (t < t) G. % for all 0 < t < Г, then ihe process
fTAt f{s) dw{s), 0 < t < T
is a martingale and
E fTAtf(s)dw(s) = 0. C.14)
4. Stochastic integrals with stopping time
If feLl[a, T] for all Г > 0, then we say that / belongs to L%[a, oo).
Similarly we define/GM^[a, oo).
Let/G L? [0, Г] and let fl5 f2 be random variables, 0 < f: < f2 < T. We
define
/(t) dw(t) — I /(t) dw(t).
0 •'O
Lemma 4.1. Define x{(t) = 1 if t < f(, %(*) = 0 i/ * > J, (i = 1, 2). 27ien
are ^t measurable and
f fV(*) <Mt) - (TX2(t)f(t) dw(t) - fTXl(t)f(t) dw(t). D.1)

4. STOCHASTIC INTEGRALS WITH STOPPING TIME 73
Proof. It is enough to prove the lemma in case ?\ = 0. It is clear that Хг(*)
is 4Ft measurable; in fact, it is a nonanticipative function. In case / is a step
function and f2 is a simple function, D.1) follows directly from the definition
of the integral. In the general case, let fn be step functions such that
ГТ|/п(*)-Л*I2А ^0 if n-*oo,
•'o
and let f2n be simple stopping times such that f2n I f2 everywhere if n | oo
(cf. Lemma 1.3.4). We have
(iinfn(t) dw(t) =fTx2n(t)fn(t) dw(t) D.2)
•'o •'o
where Xtni*) = l if f < L. X2n@ = ° if f > ?2n- It: is clear
^ f2(w)> as n—>oo. Hence
I IX2n@ ~ Хг(*I /(*) dt-*O a.s. as n —» oo,
by the Lebesgue bounded convergence theorem. We also have
I 1Хгп@|2|/@ ~ /nW|2 dt -^ 0 a.s. as n—»oo.
Putting these together, we find that
I \X2n(t)fn(t) ~ X2w/@l dt —* 0 as n—»oo.
•'o
Hence, by Theorem 2.7,
/• T p /»T
•'0 •'0
By Theorem 3.4,
fn(s) dw(s) — I f(s)dw(s) -> 0 if n->oo.
It follows that
' fn (s) dw(s~j — I " /(*) du)(s) -> 0 if n-> oo.
Since from the continuity of the indefinite integral we also have
I 2" /(s) dw(s) -» I * f(s) dw(s) a.s. as n -» oo,
we find that
f (s) dwls) —» I /(*) ^u^(*) if я —> oo.
« " •'о
Combining this with D.2), D.3), the formula D.1) (in case f, = 0) follows.

Theorem 4.2. Let f G M *[0, T] and let f1; f2 be stopping times, 0 <
f! < f2 < T. 77ien
E f * f(t) dw(t) - 0, D.4)
= E Г 2f{t) dt. D.5)
Proof. Let Xj(*) = 1 if t < ?„ x<(*) = 1 if * > ?,. By Lemma 4.1,
[''fit) dw(t) - ГT(X2W - XiW)/U) <M*)-
•'fi •'o
Applying Theorem 2.5 to the right-hand side, the assertions D.4), D.5)
readily follow.
Note that D.4) follows also from Theorem 3.10.
Theorem 4.2 is a generalization of Theorem 2.5. The next theorem is a
generalization of Theorem 2.8.
Theorem 4.3. Let f G Л0О, T] and let f1; f2 be stopping times {with
respect to ff), 0 < ft < f2 < T. T7ien
0, D.6)
D-7)
fl f*f{t)dw{t)
v I ^i
Here ?Fj denotes the a-field of all events A such that
An(?<s) is in <5г, for all s > 0.
Denote by 5^* the a-field generated by all sets of the form
An(f > t), AE%, t > 0.
We shall need the following lemma.
Lemma 4.4. For any stopping time f, Щ = ^.
Proof. Let В = АП (f > t),AEf,. For any s > 0, let
C= Bn(f < *) - An(f > t)n(? < s).
If s < t, then С = 0 G §,. If s > i, then
C=[Acu(f < ttfntf < *)
where Dc is the complement of D. Since Ac G f f and (f < f) e ?F,, it follows
that [ACU(? < t)]cG?fr Hence C??F,. We have thus proved that
В П (? < *) is in <?F, for all s > 0, i.e., В e V7f. It follows that V7f* с '.Tf.

4. STOCHASTIC INTEGRALS WITH STOPPING TIME 75
Conversely let Bef{. Then
Bs = В П (f < s) is in §,, for all s > 0.
Hence Bsc = BCU (? > s) is in <»,, and since
it follows that Bsc is one of the sets that generate ff*. Therefore Bsc G ff*.
Since Sj* is a a-field, Bs belongs to ^*, and also limsTooBs G ff*. But this
limit is the set B. Hence Bef{*.
Proof of Theorem 4.3. Let С = А П (ft >j),Aef,. Its indicator func-
function Xc = XaXi(*) is 5, measurable. Consider the function
Xc(x2(*) - Xi(O) = XaXi(s)(x2(*) ~ Xi(*))-
If s < t, each factor on the left is in cSt, and so is then the product. If s > t,
then Xi(t) — 1. so that Хг(*) ~ Xi(*) = 0. Thus the product is again in I3rf.
We have thus proved that
Хс(хг(*) — Xi(*)) is ^t measurable for any set C. D.8)
Let В 6 f{. From the proof of Lemma 4.4 we have that B/ has the form
of the set С for which D.8) holds. Hence also
XB,(X2(*) - Xi(*)) is ^ measurable.
Taking s t oo we conclude that
Хв(Хг(*) ~Xi(*)) is % measurable.
We can now proceed as in the proof of Theorem 4.2 to prove that
2 f(t) dw(t) = 0,
2
f[t) dw(t) — E I Xb/2(*) dt,
and the assertions D.6), D.7) follow.
As an application of Theorem 4.3 we shall prove:
Theorem 4.5. Let /GL?[0, oo) and assume that f™f2(t) dt = oo with
probability 1. Let
r(t) = infl s; I /2(X) d\ = t
then the process
«@-/T<0/(*
•'o
is a Brownian motion.

76 4 THE STOCHASTIC INTEGRAL
Let
t*(t) = f f2(s)ds.
Then т is the left-continuous inverse of t*, i.e., r(t) = min(s, t*{s) = t). t* is
called the intrinsic time (or intrinsic clock) for l(t) = fof(s) dw(s). Theorem
4.5 asserts that there is a Brownian motion u(t) such that u(t*(t)) = I(t).
Proof. It is easily verified that r(t) is a stopping time. Notice next that
fT(([)C %{h) if t1 < t2. Indeed, if A G fT((i), then An [т(^) < s] is in <$, for
all s > 0. But since r(t2) > т(?х),
An[r(t2) < s] = (An [t(^) < s]} П [т(У < s] is in %.
Thus A G fT((s). We shall now assume that / G M*[0, oo) and /(*) = 1 if
t > X, for some X > 0. Then r(tz) < t2 + X and, by Theorem 4.3,
=0,
dw(s)
= E
If we prove that u(t) is continuous, then Theorem 3.5.1 implies that u(t)
is a Brownian motion. From Theorem 3.6 we deduce that
^suptju(t)-u(s)\*\ <4?{ / \ 7«E)Л}-4с.
f' < S < f + E
Since f is arbitrary,
sup |«(t)-u(*)|>e) <
f > 0, s > 0
Taking a=J,e = l/fc4 and using the Borel-Cantelli lemma, we find that
P{ sup \u(t) — m(s)| > 1/fc i.o. I = 0, i.e. и is continuous.
l|t-s|<l/*4 I
t>0,s>0
For a general / G Mj[0, oo), apply the preceding result to /X where fx(t)
if t < X, fx(t) = lift> X. Since (by Theorem 4.7 below)
(fx() M) r(°/(s) * a-e- on the set X > T(*)>
•'0 •'O
where тх is the т function of/2, Problem 10, Chapter 3 implies that u(t) is a
Brownian motion.
So far we have proved the theorem only in case / E M*[0, oo). Now let

4. STOCHASTIC INTEGRALS WITH STOPPING TIME 77
/ G L^[0, oo) and define Xn, In as in C-4)> C-5)' Then fN e Mwi°> °°)> so that
is a Brownian motion. But rN(t) — r(t), fN(s) = f(s) if 0 < s < r(t), t < N.
Hence (by Theorem 4.7 below)
fNtfN(s)dw(s)= fTt f(s)dw(s)
ii N > t. It follows that the function on the right is Brownian.
Corollary 4.6. Let /GL^[0, oo), |/| < К where К is a constant. Then, for
any a > ?,
4r f /(*) dw{s)->0 if t-»oo. D.9)
Proof. Let M > К and define
**(*) = Г (/(*)+ мJ*>
0
a{t) = Г (/(s) + MJ du>(s) (t = inverse of t*).
0
Since |/+ M| > M — К >0, the function t*(t) is strictly monotone in-
increasing to oo as 11 oo. Hence a (t) is defined for all t > 0 and, by Theorem
4.5, it is a Brownian motion. But then, by the law of the iterated logarithm,
a(t)
ta
or
a(t*(t))
a.s. if t-*oo,
, . ..a ->0 a.s. if t-*oo.
Noting that *¦(*) < (K + MJ*, we get
\ (' (/(*) + M) dw{s)->0 a.s. if
t Jo
Using the fact that (u>(*)/*")-»0 if t-*ao, the assertion D.9) follows.
Theorem 4.7. Let f G L?[0, oo), g G L^ [0, oo), and set
h(t) - С f(s) dw(s), k(t) - Г' g(s) dto(e).
•'o •'o
Let r be a random variable, т > 0, and assume that f(s) — g(s) if s < т.
Then h(t) -'fc@ a.s. i/f < т.

78 4 THE STOCHASTIC INTEGRAL
Proof. Construct, by Lemma 1.1, step functions fn, gn which belong to
Z4[0, T] and which satisfy
C\fn(t)-f(tfdt Л 0, jT\gn(t) - g(t)\2dt Л 0
->o Jo
as n -» oo. That construction is such that
/»(*.«)-g»(*.«) if s<r(U)AT. D.10)
Let
From the definition of the integral of a step function and from D.10) it
easily follows that
К(*, «) = K(*. «) if * < т(«) Л Г. D.11)
By Theorem 3.4,
sup \hn(t) - h(t)\ Л 0, sup |^(t) - fc(t)| - 0
0< К Г 0< t<T
if n-»oo. Hence, for some subsequence (n'};
sup \hn,(t) - /i(t)|-»0, sup |^.(t) - fc(t)|->0 a.s.
0<(< Г 0< t<T
if n'-»oo. Recalling D.11), we find that for a.a. w,
h(t,a) = k(t,a) if * < t(w) д Г.
The proof is now completed by taking a sequence T = Гп | oo.
5. ltd's formula
Definition. Let |(f) @ < f < Г) be a process such that for any
0 < *! < f2 < Г
€(*2) - €(*i) = ГЧ a(t) dt + fh b(t) dw{t)
where a?L^[0, Г], bGL^[0, Г]. Then we say that ?(*) has stochastic
differential di, on [О, Г], given by
a(t) dt + b(t) dw(t).
Observe that ?(t) is a nonanticipative function. It is also a continuous
process. Hence, in particular, it belongs to L"[0, Г].
Example 1. If 0 < t, < t2 and П„ - {«, - *„, lf tn> tnn - t2} is a

5. ITO'S FORMULA 79
sequence of partitions of [tv t2] with mesh |П„|-»0 then, by Theorem 2.10,
, n-l
wit) dw(t) = hm У. w[t k)(u>[t k+1) — w[t k))
-* toD 2| {[(W(tn,k + 1)f-(w(tn,k)J]
~ (u>(tn.it+i)- w{tntk)J}
= iMOJ- iM*i)J- 2 Jlim  M*n.k+i) ~ ^n,k)f
where limn_>00 is taken as the limit in probability.
By Theorem 3.3.3, the last limit in probability is equal to t2— tv Hence
fhw(t) dw(t) = b{w(t2)J- H»(*i)J- i(*2 - '1), E-1)
or
d(w(t)f= dt + 2w(t) dw{t). E.2)
*n,*M*n,* + i)-«>(',.,*)] in probability.
Example 2.
Г td
Clearly
By
w{t)
fh
Theorem
n
w(t) dt =
2.10,
= 1
= lim
n—>схз
n-l
2
*=1
for all w for which w(t, w) is continuous. The sum of the right-hand sides is
equal to
n-l
„hm,, 2 Kk+i««(*n,fc+i) - tn-kw{tn_k)]= t2w(t2) - tMh)-
Hence
d{tw(t)) = w(t) dt + t dw(t). E.3)
Definition. Let ?(t) be as in the definition above and let f(t) be a function
in L«[0, T]. We define
f{t) dt(t) = f(t)a(t) dt + f(t)b(t) dw(t).
Example 3. f(t) d?(t) is a stochastic differential d-q, where

80 4 THE STOCHASTIC INTEGRAL
Theorem 5.1. If d^(t) = a^t) dt + b{(t) dw(t) (i = 1, 2), then
) = Ш di2{t) + Ш <*Ш + 4t)b2(t) dt. E.4)
The integrated form of E.4) asserts that, for any 0 < tY < t2 < T,
MM - €i(*i)fe(*i) = Г Ш^) dt + fh ШЪЛМ dw(t)
Jt. Jt.
f b(t)ai(t) & + f
b^b^t) dt. E.5)
Proof. Suppose first that aj5 Ъг are constants in the interval [tv t2). Then
E.5) follows from E.2), E.3). Next, if aj; Ьг are step functions in [tlt t2),
constants on successive intervals Iv I2, . . . , I;, then E.5) holds with tv t2
replaced by the end points of each interval I(.. Taking the sum we obtain
E.5).
Consider now the general case. Approximate aj; Ъг by nonanticipative step
functions au „, Ь( „ in such a way that
к„(*)-а,(*)|*-»0 a.s.,
С\\n(t) ~ k(t)f
-'o
*U a.s.
-'о
Let
кп(*) - Ф) + f a, „(a) ds + f bt „(a) du;(S).
•'о •'о
By Theorem 3.4,
SUP |C „(t) - Ф)\ Д 0 if
0< t< T
hence there is a subsequence i^ n- such that
?,,„'(*)-»?(*) uniformly in te[0, T], a.s. E.6)
Using E.6) and Theorem 2.7 it is easily seen that
i.n(t)bun(t) dw(t) Л j* ЬЩЦ) dw(t)
as n = n' —> oo. Clearly also

5. ITO'S FORMULA 81
a.s. Writing E.5) for af „, bin, ? n and taking n-»oo, the assertion E.5)
follows. Since tv t2 are arbitrary, the proof of the theorem is complete.
Theorem 5.2. Let d?(t) = a(t) dt + b(t) dw(t), and let f(x, t) be a con-
continuous function in (x, t)GRl X [0, oo) together with its derivatives fx, fxx,
ft. Then the process /(?(?), t) has a stochastic differential, given by
df(№, t) = [fMt), t) + /,({(«), t)a(t) + tfjt(t), t)b*(t)] dt
dw(t). E.7)
This is called Ito's formula. Notice that if w(t) were continuously
differentiable in t, then (by the standard calculus formula for total deriva-
derivatives) the term \fxxb* dt would not appear.
Proof. The proof will be divided into several steps.
Step 1. For any integer m > 2,
d(w{t))m= mWt))m4+|m(ffl - l)(w{t))m~2 dt. E.8)
Indeed, this follows by induction, using Theorem 5.1.
By linearity of the stochastic differential we then get
dQ(w{t)) = Q'{w(t)) dw(t) +iQ"{w{t)) dt E.9)
for any polynomial Q.
Step 2. Let G(x, t) = Q(x)g(t) where Q(x) is a polynomial and g(t) is
continuously differentiable for t > 0. By Theorem 5.1 and E.9),
dG(w(t), t) - f(w(t))dg(t) + g(t)df(w(t))
= [f(w(t))g'(t) + ig(t)f"(w(t))]dt + g(t)f'(w(t)) dw(t),
i.e., for any 0 < t1 < t2 < T,
G(w(t2), t2) - G(w(t1), h) = f 2 [Gt(w(t), t) + iGjw(t), t)] dt
Jh
+ С Gx(w(t),t)dw(t). E.10)
Step 3. Formula E.10) remains valid if
m
G{x, t) = 2 /,(x)&(*)
i = l
where ft(x) are polynomials and gi(t) are continuously differentiable. Now let
Gn (x, t) be polynomials in x and t such that
Gn(x, t)-»/(x, t),
-^ Gn(x, t)^fx(x, t), ^ Gn(x. 0-/„(х. О, -| Gn(x, t)-»/,(x, 0

82 4 THE STOCHASTIC INTEGRAL
uniformly on compact subsets of x, t) E R1 X [0, oo); see Problem 11 for the
proof of the existence of such a sequence. We have
С„М*2). h) - Gn(w(*i). *i) - J 17 G>('), t)
+ | |1с>@,*Iл
2 ЭГ J
+ /'2 -^ Gn(w(t), t)dw(t). E.11)
It is clear that
'')-/.(»(').')
2
0 a.s.
Hence, taking n—>oo in E.11), we get the relation
/(«(*?). *2) - /M*l). *l) = / [/,(»(*). 0 + */„(»(*). *)] dt
f2fx{w(t),t)dw(t). E.12)
Step 4. Formula E.12) extends to the process
where |l5 аъ b1 are random variables measurable with respect to ^t, i.e.,
[t'fx(i(t),t)b1dw(t) E.13)
where f(f) = ^ + a^ + h^t).
The proof of E.13) is a repetition of the proof of E.12) with obvious
changes resulting from the formula
\а, dt + b, dw(t)] + Ы™ - 1)A(*)Г~2Ь? dt,
E.14)

5. ITO'S FORMULA 83
which replaces E.8). The details are left to the reader.
Step 5. If a(t), b(t) are step functions, then
). h) ~ f{i{h), h) =?2 [ft(i(t), t) + fx(t(t), t)a(t)
+ fyx(Z(t),t)b(t)dw(t). E.15)
Indeed, denote by Ilt . . . , Ik the successive intervals in [tv t2] in which a, b
are constants. If we apply E.13) with tv t2 replaced by the end points of 1г,
and sum over I, the formula E.15) follows.
Step 6. Let аг, Ъг be nonanticipative step functions such that
Г |аД0 - а@| А-»0 a.s., E.16)
0
f \h(t) - b(t)\2 dt Д 0, E.17)
and let
= |@) + Г at(s) ds + f bt(s) dw(s).
Jo Jo
Then
sup \ф) - {@| - 0.
0<*<T
Hence, for a subsequence {i'},
sup |?(г) - |(f)|-»0 a.s. if i = i'—>oo. E.18)
0< t< T
This and E.17) imply that
), t)bt(t) - fMt), t)b(t)\2 dt 4.0 if i = i'-»oo.
It follows that
( 2Ш*)- *)bi@ Ae(t) Л Г */.(€(*)¦
It is clear from E.16)-E.18) that also
[Ш*).')+/.(€. ('MM')
P
M)+/«(€('). *)°(')
if <-<'-»».

84 4 THE STOCHASTIC INTEGRAL
Writing E.15) for a = at, b = bp ? = ^ and taking i = i'—»oo, the
formula E.15) follows for general a, b. This completes the proof of the
theorem.
Theorem 5.3. Let d?({t) = a{(t) dt + bt(t) d? A < i < m) and let
/(*!, . . . , xm, t) be a continuous function in (x, t) where x =
(xj xm)GJRm, t > 0, together with its first t-derivative and second
x-derivatives. Then f(?i(t), . . ¦ , ?,„(?), t) has a stochastic differential, given
Г
df(X(t),t)=\ft(X(t),t)+Z fXi(X(t),t)ai(t)
m
+ 1 2 /X|If(x(t),*)b1(t)b,(t)
i,; = l
+ 2 /X|(X(t),t)b,(t)<Mt). E.19)
inhere X(t) = &(*), . . . , ^(t)).
Formula E.19) is also called Ito's formula. It includes both Theorems 5.1
and 5.2.
The proof is left to the reader (see Problem 15).
Remark. Ito's formula E.7) asserts that the two processes f(?(t) t)
- /(|@), 0) and
JT' [fMs), *) + №'), Ms) + */„(€(«), s)b*(s)] ds
+ f /,(€(*). *)ь(*) «*»(*)
•'o
are stochastically equivalent. Since they are continuous, their sample paths
coincide a.s. Consequently
, 0) -J*[fM*), *) + fMt), t)a(t)
Г fMt), t)b(t) dw(t) E.20)
for any random variable т, 0 < т < Т.
If, in particular, т is a stopping time, when, taking the expectation and

6. APPLICATIONS OF ITO'S FORMULA 85
using Theorem 3.10, we find that
E/U(t), t) - Ef{№, 0) = E [T (L/)({(t), t) dt E.21)
•'o
where
provided
b(t)fMt,t)) belongs to МД0.Г],
), t) belongs to МДО, Г].
6. Applications of Ito's formula
Ito's formula will become a standard tool in the sequel. In the present
section we give a few straightforward applications. First we need two
lemmas.
Lemma 6.1. 7//GL^[a, /8] for some p > 1, then there exists a sequence
of step functions /„ in Ь?[а, /8] such that
Jim ffi\f(t)-fn(t)\pdt a.s.
a
Proof. The proof is similar to the proof of Lemma 1.1. Instead of A.5) we
now have
\f\vdt. F.1)
Indeed, from A.4) and Holder's inequality,
Y 4\f
Jf* \JJ\p dt<f* [ДрИ dzY 4\f^p(z)\f(t - z - e)\r dz\ dt
and F.1) follows. (If p = 1, we do not use Holder's inequality.)
Using F.1) we can proceed to show that
ffi\JJ-f\pdt-^O if €|0, F.2)
by the argument given the proof of Lemma 1.1. The rest of the proof is the
same as for Lemma 1.1.

86 4 THE STOCHASTIC INTEGRAL
Lemma 6.2. If feM?[a, /8] for some p > 1, then there exists a sequence
of bounded step functions fn in M^[a, /8] such that
E f\f(t) - fn(t)\v *-»0 if n^oo.
•'a
This follows by obvious changes in the proof of Lemma 1.2, exploiting
Lemma 6.1.
Theorem 6.3. Let/6M^[0, T] where m is a positive integer. Then
( /(*)<**>(*) <[тBт-1)]тГ*-^Е fjf*"(t)&\. F-3)
For m = 1, there is equality.
Proof. We apply Ito's formula with f(x, t) = x2m, ?(t) = /?/(s) du>(s), and
obtain
J m= 2m ? |JT'/W ] Ш
]
2m
+ mBm-l)l I f(\)dw(\)\ f(s)ds. F.4)
'o ['o J
Suppose f(t) is a bounded step function. From the definition of
Ssof(\) dw{\) and the fact that
E\w(t)\r < Crt (Cr>f constant)
for any t > 0, r > 0, we find that
\ff{X)dw(\)~\ f(s) belongs to M*[0, T\.
Hence, taking the expectation in F.4) and using Theorem 2.5, we get
г т 2iTf% _ _ 2 771 2
EjT/(e) dw(e) »mBm-l)JT?n'/W('«'W /2(s) ds-
F.5)
By Holder's inequality, the right-hand side is bounded by
F.6)
From F.5) we see that the function

6. APPLICATIONS OF ITO'S FORMULA 87
is monotone increasing in t. Hence
т Г s "l2m т Г т l2m
f E\ff(\)dw(\)\ ds<j E\J f(\)dw(\)\ ds.
Using this to estimate the first expression { • • • } in F.6), we then obtain
from F.5),
-,2m
f(s) dw(s)
{Г T l2m) (m~1Vm . . l/m
TE^ /(*) d«,(*)J | Ц ?/2mE)ds| ,
and F.3) follows.
To prove F.3) in general, we can take, by Lemma 6.2, a sequence of
bounded step functions fn such that
E [T\f(t)-fn(t)\2mdt-+O if n-»oo. F.7)
Then
fTfn(t)dw(t) Д fTf(t)dw(t).
•'o •'o
We may assume that the convergence is a.s., for otherwise we can take a
subsequence of the fn. Writing F.3) for fn, and using Fatou's lemma and
F.7), the assertion F.3) follows.
From Theorems 3.7 and 6.3 we obtain:
Corollary 6.4. If f E M,2 [0, T] where m is a positive integer, then
{ I rt 2m1 rr
E sup / /W<M*) < CJT-Ъ I \f(t)\2mdt
where Cm = [Am3/{2m - l)]m.
Theorem 6.5. Let /GL^[0, Г], and let a, /8 be ant/ positive numbers. Then
Froo/. Consider first the case where / is a bounded step function, and set
i(t)
Sit)

88 4 THE STOCHASTIC INTEGRAL
Applying Ito's formula to f(x) = e* and the process ?(t), we get
$(t) = $(s) + (* e«*>f(\) dwfr).
Since / is a bounded step function,
em < Ao П e^w{^
i = l
for some positive numbers At, tt. It follows that [exp(?(X))]/(X) belongs to
Л0О, Г]. Hence, by Theorem 2.8,
ЕШЮ-М, F-9)
i.e., f (t) is a martingale. Note also that ?f @) = 1.
Set
Ш = f fa) dw(\) - % f'f2(\)d\,
Ш = f' «/(Л) dw(X) - ? Г f(\)d\
By what we have already proved, expD,(?)) is a martingale with expecta-
expectation 1. Hence, by the martingale inequality,
P{ max ?,(*) > p) = P{ max [exp ?,(»)] > ea^) <
This proves F.8) in case / is a bounded step function.
Now let / be any function in L^, [0, Г] and take bounded step functions fn
such that
a.s.
Theorem 3.4 implies that
» 0 in probability.
By taking a subsequence, if necessary, we obtain convergence a.s. It follows
that a.s.
'Ш dw(\) - f f'ft
uniformly with respect to t, 0 < t < Г. Hence, by writing F.8) for each /n
and taking n—»oo, the assertion F.8) follows.
The inequality F.8) will be referred to as the exponential martingale
inequality.

7. INTEGRALS AND DIFFERENTIALS IN n DIMENSIONS 89
Corollary 6.6. If f?Ll [О, Г], then the process
S(t) = expf С /(Л) dw(X) - i С f(\) d\) F.10)
is a supermartingale, i.e., —?(?) is a submartingale.
Proof. Define
?,(*) = P/M dw(X) + ffn(\) dw(\) - i f f(A) ^ - i Cf'W d\,
J0 Js J0 Js
for s < t < Г, where the /n are step functions as in the preceding proof. By
the proof of F.9), ?n(t) is a martingale for s < t < Г. Hence, for any event
А6?„
f Ut)dP=[ U*)dP. F.11)
^A ^A
Notice that fn(s) = f (s) where f (*) is defined by F.10). If n = n'->oo ({n'}
a suitable subsequence of {«}), then fn(*)-*f @ a.s. Hence, taking n—»oo in
F.11), and using Fatou's lemma, we get
f
JA
JA
and the assertion follows.
7. Stochastic integrals and differentials In n dimensions
Let w(t) = (wx(t), . . . , wn(t)) be an n-dimensional Brownian motion. Let ?F(
(t > 0) be an increasing family of a-fields such that w(t) is '.'i( measurable
and <5(w(\ + t) - w(t), Л > 0) is independent of <$t, for any t > 0.
We shall say that a matrix of functions belongs to L?[a, ft] (or to
Mwl«, P]) if each of its elements belongs to L?[a, K] (or to M?[a, ?]).
Let b = (bj) be an m X n matrix that belongs to L^[a, ft]. The stochastic
integral J^b(?) dw(t) is an m-vector defined by
f\(t)dw(t)= ( 2
where each integral on the right is defined as in Section 2.
If we substitute a = ftfdwt, ft = J'^gdWf in the identity 4<x/3 =
(a + KJ - (a - /?J, and use Theorem 2.5, we find that
E f2f( t) dwl t) ( h g( t) dwt(t) = E f k f( t)g( t) dt G.1)
provided / and g belong toM^[t,, t2].

90 4 THE STOCHASTIC INTEGRAL
We also have
E f2f{t) dw{{t) f2 g(t) dw^t) = 0 if i Ф I, G.2)
because the integrals are independent and with zero expectation.
Using G.1), G.2) we easily see that if b = (by) is an m X n matrix in
Мш[^, ?2]> then
E P2 b(t) dw(t) 2 = E P2 |b(*)|2 dt G.3)
where
m n
|ь|2 =И (ь/
Definition. Let i(t) be an m-dimensional process for 0 < t < T, and
suppose that, for any 0 < tY < t2 < T,
a(t) dt + f2 b(t) dw(t)
J
where a = (a^ . . . , am) and the m X n matrix fe = {b^j belong to L^[0, T]
and L^[0, Г] respectively. Then we say that ?(*) bas a stochastic differential
d?(t) given by
d?(f) = e(*) dt + b(t) dw(t). G.4)
We shall now state Ito's formula.
Theorem 7.1. Let u(x, t) be a continuous function in (x, t)GRm X [0, oo)
together with its derivatives ut, ux , ux x . Let ?(t) be an m-dimensional
process having a stochastic differential
dZ(t) = a{t)dt + b(t)dw{t)
where a = (alt . . . , am) and b — (b^) A < i < m, 1 < \ < n) belong to
L^[0, T] and L?[0, Г] respectively. Then u(?(t), t) has a stochastic differen-
differential
r m
du{i{t), t) = ut(?(t), t) + 2J "x,(!(*). *)a«(*)
n m 
+ i 2 2 «,,,,(€(*), t)ba(t)b^t) dt
г— l«,/
n m
/-1i-i
We begin with

7. INTEGRALS AND DIFFERENTIALS IN n DIMENSIONS 91
Lemma 7.2. If w±, w2 are independent Brownian motions, then
d(w1w2) = u>i dw2 + w2dw1. G.6)
Proof. It is easily seen that w(t) = (u>j + w2)/^2 is a Brownian motion.
Hence, by E.2)
d(w2) = dt + 2w dw.
Since d(w2) and d(u>2) are also given by the same formula, d(wlw2) exists
and is equal to
d(w2 - \w\ - \w\) = dw2 - \ dw\ - | dw2 = w1 dw2 + w2 dwv
Lemma 7.3. If d^ = at(t) dt + 2"=1bj#@ dwf (i = 1, 2) where wx, . . . , wn
are independent Brownian motions, then
Proof. The proof is similar to the proof of Theorem 5.1. It is based on the
special cases G.6), E.2), E.3) and the approximation procedure employed in
the proof of Theorem 5.1.
Proof of Theorem 7.1. We begin with the special case of u(?. +
a.t + b.w(t), t) where tx < t < t2 and ?„ a., b. are random variables
measurable with respect to *$,. The special case where и = x* follows by
induction on m, using Lemma 7.3. The more general case where и
= xi1' ' ' x?r follows by using the previous special case and Lemma 7.3.
The case where и = g(t)x^ ¦ • ¦ x%" follows by again using Lemma 7.3.
Now approximate general и by linear combinations of u's of the last special
form.
Once Theorem 7.1 has been proved in the special case where
?(t) — ?, + a.t + b,w(t), we can obtain the general case by a proof similar
to that of Theorem 5.2.
Let
ц j
Lu = * ^ ач дх дх + ^ щ ~дх~ ~dt •
Then we can write Ito's formula G.5) in the form
du(?(t), t) - ЬиШ, t) dt + ихШ, t) • h(t) dw{t). G.7)

92 4 THE STOCHASTIC INTEGRAL
Let us define formally a multiplication table:
dw{ dt = 0,
dt dt = 0,
dw{ dwf = 0 if i ф /',
dwt diVj = dt,
so that
= 2 fej;fy Л. G.8)
Then Ito's formula G.5) takes the form
+ i 2 ч,, U(t), t) d^ d^. G.9)
From Ito's formula we obtain (cf. E.20))
«U(t), t) - «(«@), 0) = Г (Ьи)Ш, s) ds
+ Г ux(Z(s), *) ¦ Hs)dw(s) G.10)
for any random variable т, 0 < т < Г. If т is a stopping time and Lu,
ux • fe are in M^[0, Г] and M^[0, Г] respectively, then (cf. E.21))
Eu(?(t), t) - ?«U@), 0) = ? Г LU(|(S), e) ds. G.11)
•'o
The results of Sections 2-4 extend with obvious changes to stochastic
integrals in n dimensions. We shall give here the generalization of Theorem
4.5.
Theorem 7.4. Let /= (fv ...,/„) belong to L^[0, oo) and suppose that
So'\f\2dtss oo twtfe probability 1, and define
s; f°\f(\)\2d\=t].
Then the process
is a Brownian motion.
Proof. The proof is similar to the proof of Theorem 4.5 once we have

PROBLEMS 93
established that if / G M%[0, oo) and ?x, ?2 are bounded stopping times,
?! < ?2, then
a.s. The proof of these formulas is similar to the proof of Theorem 4.3. It
employs Theorem 2.8 which remains valid for n-dimensional stochastic
integrals.
We conclude this section with an extension of Theorem 6.5 to n di-
dimensions.
Theorem 7.5. Let /= (?, . . . ,/„) belong to L*[0, Г], and let a, /? be
positive numbers. Then
/(X) dw(\) - f Jf' |/(X)|2 d\] > /?} < e-*. G.12)
The proof is similar to the proof of Theorem 6.5. First we prove G.12) in
case f(t) is a step function, using the martingale inequality, and then proceed
to general / by approximation.
The inequality G.12) is referred to as the exponential martingale inequal-
ity.
Corollary 6.6 also extends to the present n-dimensional case, i.e.,
expf С fdw-l f \f\2ds
is a supermartingale.
PROBLEMS
1. Prove B.20).
2. Prove Theorem 3.9 [Hint: Apply Theorem 3.6 to ?f(t), ? bounded and
?Ta measurable.]
3. Suppose /?L*[0, oo) and ? is a stopping time such that
Ejif2(t)dt< oo. Prove that
2
E (Sf(t)dw{t)-0, E
JO
4. Let
с exp[l/ (|x|2 - 1)] if |x| < 1
(X) ' 0 if |x| > 1

94 4 THE STOCHASTIC INTEGRAL
for xGR", where с is a positive constant such that JRn p(x) dx — 1. If / is
a function locally integrable, then
U/)W - ? / p( ^)/(y) <%
is called a mollifier of/. Prove:
(i) /JisinC°°(fln);
(ii) If К is a compact set and fl a bounded open set containing K,
then
= f p(z)f(x- es)dz (
J\z\<l
provided e < dist(K, H"\fl).
(iii) If /G LP(fl) for some p > 1, then
/ \U\P dx) <{( |/|" dx .
(iv) If /G LP(fi) for some p > 1, then
f 1Л/ ~ /lp *f-»0 if e->0.
•'к
5. Let f(x) be a continuous function for a < x < ft, and let
Let 5 be any positive number. Prove that (Pj/)(x) -*f(x) uniformly in
xG[a + 5, K-5]asfc-»oo. [Hint: [/J(l - y2)* dy//J(l - у2)* <fy] -> 0
if k -* oo, for any e > 0.]
6. Let f(x) be a continuous function in an n-dimensional interval
I = {x; а( < x < /?j, 1 < i < n}, and let
/ xw ч JSi •¦•ft n? tf
(P»/)(x) =
Let Io be any subset lying in the interior of I. Prove that, as k—>oo,
(Pkf) (x) ~*fix) uniformly in xEl0.
Notice that P,J is a polynomial. It is called a polynomial moUifier of /.

PROBLEMS 95
7. If in the preceding problem / belongs to Cm(I) and / vanishes in a
neighborhood of the boundary of I, then
uniformly in x ? Io, for any (ix, . . . , in) such that 0 < ix + • • • + tn < m.
8. If fGCm(Rn), then there exists a sequence of polynomials (^ such
that, as
uniformly in x in compact subsets of R". [Hint: Approximate /by functions
with compact support, and apply Problem 7 to these functions.]
9. If in the previous problem it is assumed that /, / A < i < n) and /
B < i,j < n) are continuous in R" (instead of /еС"(й")), then
B < », / < n)
Эх, Эх- Эх, Эх.
uniformly on compact subsets of R".
10. Let f(x, t) be a continuous function in (x, t)?R" X [0, oo) together
with its derivatives ft, fx , fx x . Prove that there exists a function F con-
continuous in (x, t)GR" X fl1 together with its derivatives Ft, Fx , Fx x , such
that F(x, t) = f(x, t) if xGRn,t > 0.
11. Let f(x, t) = /(x1( . . . , xn, t) be a continuous function in
(x, ?)Gfl" X [0, oo) together with its derivatives ft, fx,fxx- Then there
exists a sequence of polynomials Qm{x, t) such that, as m—>oo,
о _>f i. n _»f -5-o-»f э2 .
m Э^ m Эх, m x<1 Эх, Эх, т x'xt
uniformly in compact subsets. [Hint: Combine Problems 9, 10.]
12. Prove E.8).
13. Prove E.14) and complete the proof of E.13).
14. Let /eL?[0, oo), |/| < К {К constant) and let d?(t) = f{t) dw{t),
|@) = 0 where w(t) is a Brownian motion. Prove:
(i) if/< 0,thenE|«t)l"< №
(ii) if / > a > 0, then E||(t)|2 > a2*.
15. Prove Theorem 5.3. [Hint: Proceed as in the proof of Theorem 5.2, but
with
Ф(и>(*), t) - /(?,„ + axt н

96 4 THE STOCHASTIC INTEGRAL
where |j0, at are random variables and the bt are random n-vectors; cf. Step
4.]
16. Let ?(t) = Jo^(') dw(t) where b is an n X n matrix belonging
toL2 [0, oo). Suppose that d^ d^ = 0 if i Ф j, d^ d^ = dt (see G.8) for the
definition of di^ djL), for all 1 < i, / < n. Prove that ?(*) is an n-dimensional
Brownian motion. [Hint: First proof: Use Theorem 3.6.2. Second proof:
Suppose the elements of b are bounded step functions and let
= exp[iy • ?(t) + у^/2]. By Ito's formula df = if у dw. By Theorem 2.8
Use Problem 2, Chapter 3.]
17. Let у > 0, a > 0, т = min{?; mj(*) = a} where w(t) is one-dimen-
one-dimensional Brownian motion. Prove that P{r < oo) = 1 and
Ее-!* = ехр(-Л/2у a).
[Hint: For any с > О,
P[ max w(s) > c] < p\ max (w(s) - ~ s) > ^1 < e-^2'
where a = c/t, K = c/2. Hence Р(т < oo) = 1. Since t/(f) = exp[yu>(^) —
y^/2] is a martingale, so is y(t л т). Hence
? exp[yu>(^ лт)" kl2{t Л т)] = 1.
Take 11 oo.]
18. Under the conditions of the previous problem
[Hint: Use the fact (see, for instance, Feller [1]) that if the Laplace transform
of two probability distributions concentrated on [0, oo) coincide, then the
probability distributions coincide.]
19. If w(t) is a Brownian motion and 0 < y, x < y, then
P(w(t) G dx, max mj(s) G dt/)
т^: Use Problem 12, Chapter 2 and Theorem 3.6.3 to deduce that
P[w(t) e dx, max w(s) > t/] = J P(t G ds)P[u>(^ - s) + у ?
where т = min{i; u>(t) = y), and apply the preceding problem.]

PROBLEMS 97
20. Let f(t) be a continuous process in L^[0, Г] and let Пп : tn 0 =
0 < tn j < • • • < tn n = T be a partition with mesh |ПП| -> 0 as n -» oo.
Define
i=0
*„.,<
Prove that for some subsequence {n'} of {n},
sup I f(s) dw(s) — gn-(t) —»0 a.s. if n' —> oo.
о < t < т Jo
21. Let a(a:, f) be a measurable function in (x, t)GRn such that
\a(x, t) - a(x, t)\ < tj(|x - x\), i){8) | 0 if 5 | 0,
and let/(^) be an n-dimensional continuous process in L^,[0, Г]. Let
o€(x,t) = ± Г Р(*-8-
where p(t) is defined as in Lemma 1.1 and a(x, s) = a(x, 0) if — 1 < s < 0.
Pe
0 /o la(x' *) ~ a€(x' ^)|2 dt ^>0 uniformly in x in bounded sets, as
(») /o k(/@. «) - *«(/(*). 01* * -* 0 a.s. as e -> 0.
(iii) supo<(<r|/^ a(f(s), s) dw{s) - f'o aCn (f(s), s) dw{s)\ -¦ 0 a.s. for
some sequence en J, 0.
[Ят^: for (i), use the uniform continuity in x of /a(x, f) dt and of
/o«(x, *) dt.]
Prove:
@
e-»0.

5
Stochastic Differential Equations
1. Existence and uniqueness
If a = (ai() is a matrix, we write \a\2 = 2^|a,-|2.
Let b(x, t) = (b^x, t),..., bn(x, t)), a\x, t) = (ац(х, t))",-i and suppose
the functions bt(x, t), ац(х, t) are measurable in (x, t)GR" X [О, Г]. If |(f)
@ < t < Г) is a stochastic process such that
di{t) = ьш, t) at + o{t{t), t) dw(t), (i.i)
1@) =|0 a.s., A.2)
then we say that |( t) satisfies the system of stochastic differential equations
A.1) and the initial condition A.2). Note that it is implicitly assumed that
b{i{t\ t) belongs to ЬЦО, Т] and o(?(t), t) belongs to L^O, T].
Theorem 1.1. Suppose b(x, t), a(x, t) are measurable in (x, t)GRn X [0, Г]
and
\b(x, t) - b(x, t)\ < K.\x - x\, \a(x, t) - a(x, t)\ < K.\x - x\,
\b(x, t)\ < K(l + |x|), \a(x, t)\ < K(l + |x|)
where К,, К are constants. Let |0 be any n-dimensional random vector
independent of <5{w(t), 0 < t < Г), such that E\Q2 < oo. Then there
exists a unique solution of A.1), A.2) in M^[0, T].
The assertion of uniqueness means that if ^(t), |2(f) are two solutions of
A.1), A.2) and if they belong to M2[0, T], then
НШ = Ш for all 0 < t < T) = 1.
Proof. To prove uniqueness, suppose ^(t) and |2(?) are two solutions
belonging to Л0О, Г]. Then
Ш - Ш = f'[b(«iW,*) - ь2(«2(*),*)]л
-'о
(s), s) dw(s). A.4)
-'O -'O
0Й
- f
-'o

Set/((s) - o(?,(s), s) and note that the stochastic integral Sofi(s) dw(s) is
defined with respect to an increasing family of o-fields ^ which may
depend on i. If/,(s) is a step function, for i = 1, 2, then using the definition
of the stochastic integral we get (cf. the proof of Lemma 4.2.2 and formula
D.7.3)) ^
Г'/iW dw(s) - С Us) dw(s) = E f'l/iW - /2MI2 ds. A.5)
-4) •'0 •'0
By approximation we find that A.5) is true for any pair fx, /2 of
nonanticipative functions with respect to ^\ and 'ff respectively, provided
Е/0|/,(*)|2*<юA = 1,2).
Taking the expectation of the squares of the absolute values on both sides
of A.4) and using A.3) and A.5) with f{(s) = a (?,(*), s), we find that
е\ш-Ш\2
< 2Kh f Е\Ш - Ш\2 ds + 2K.2 ( Е\Ш - Ш\2&.
Thus the function <j>(t) = E\^(t) - ?2(t)\2 satisfies
С Г <t>(s) ds,
ф{0) = 0,
where С is a positive constant. Therefore <p(t) = 0, and the assertion of
uniqueness is proved.
To prove the existence of a solution we introduce an increasing family of
o-fields % @ < t < T) such that ?0 is % measurable and w(t) is $,
measurable, and such that $(w(t + s) — w(t), 0 < s < T — t) is indepen-
independent of ^t, for all t > 0. We can take for instance ^t to be the a-field
generated by ?0 and ^(w(s), s < t); here we use the assumption that ?0 is
independent of f (w(s), 0 < s < Г).
Define ?n(t) = L and
I. A.6)
4L(S)> s)ds +
The inductive assumption is that ^, ? МДО, Г] and that
(* +
for 0 < к < m - 1, A.7)
where M is some positive constant (depending only on К, К,, Т).
Since Jo? ^o> Cn + i is We^ defined if m = 0. Further
2 . 2
- Sol2 < 2
I
-'o
, s)ds
+2
I
Taking the expectation and using A.3), we get

100 5 STOCHASTIC DIFFERENTIAL EQUATIONS
if M > 2K2(T + 1)A + E|?0|2). This implies that ^GM2[0, Г] and A.7)
holds for m =0.
We now make the inductive assumption for any m > 0 and prove it for
m + 1.
Since ^GM^O, Г] it follows, using A.3), that &(?„(?), t) and <*(?„(?), t)
belong to МДО, Г]. Thus the integrals on the right-hand side of A.6) are well
defined.
Next,
I ,t 2
Mo
I (ь 2
+ 2U0 La^"^'^ ~ ate>-iW'*)J «Ф) • A-8)
Taking the expectation and using A.3),
Thus,
•A)
if M > 2К2(Г + 1). Substituting A.7) with к = m - 1 into the right-hand
side, we get
- Ш\2 <мГ
(Mt)
m+\
о '»! (m + 1)!
Thus A.7) holds for к = m. Since this implies that ?„+1 G M2[0, Г], the
proof of the inductive assumption for m + 1 is complete.
From A.8) we also have
sup |е„+1(г)-Ш2<2ГК2
о < t < т Jo
+ 2 sup
o< t<г Ко
Taking the expectation and using Theorem 4.3.6 and A.7), we find that
E sup |^, + 1(t) - Ш12 < 21
0<t< Г

1. EXISTENCE AND UNIQUENESS- 101
where С = 2К.2Г2 + 8К.2Г. Hence
P\ SUP Ип + Лч - ^п\ч\ > o^ < 2 mC ——j .
Since Е[2т(МГ)т/т!] < oo, the Borel-Cantelli lemma implies that
f 1 )
P\ sup |t» + 1(t) - U')l > ^T i-°- = °-
Thus, for almost any и there is a positive integer m0 = mo(u) such that
p |^ +
o< t< т
It follows that the partial sums
к
are convergent uniformly in t G[0, Г]. Denote the limit by ?(?). Then ?(?) is a
continuous process. It is clearly also a nonanticipative function and it belongs
to L2[0, T]. Since for a.a. w,
Hini*)' *)^Н?(*)> *) uniformly in te[0,T],
o(i,,(t),t)-Ki(t(t), t) uniformly in te[0, Г],
and hence also
f
if we take m^oo in A.6) we obtain the relation
t{t) = ^0 + Г * Щ(з), s)ds+ft o{S(s), s) dw(s). A.9)
-'o Jo
Thus |(t) is a solution of A.1), A.2).
From A.6) we have
2
ftb(in(s),s)ds
+3E f
Jo
where С is some constant depending only on К, Т. By induction we then get
С + ch + C3 |j- + • • • + C-+2 ^-^ J[1 +
Therefore

102 5 STOCHASTIC DIFFERENTIAL EQUATIONS
Taking m | oo and using Fatou's lemma, we conclude that
ес*. A.10)
This implies that $(t) belongs to МДО, Т].
The above method used to prove the existence of a solution ?(t) is called
the method of successive approximations; it is modeled after the correspond-
corresponding proof for ordinary differential equations.
Remark. Very often we shall take the initial value ?0 to be a constant
function x, i.e., ?0(w) = x a.s. Notice that this random variable is indepen-
independent of ^(w(t), t > 0).
From A.9) we obtain
Г г
sup |?(*)|2< 3|?0|2 + 3 ( \b{S(s), s)\ ds
0<t<T \J0
+ 3 sup
rt
I o(?(s), s) dw(s)
2
0< t<T
Taking the expectation and using A.3) and Theorem 4.3.6, we get
E sup \mZ < C0(l + Efef) + Co (
0<t<T J0
where Co is a constant depending only on К, Т. Making use of A.10), we
obtain
Corollary 1.2. Under the assumptions of Theorem 1.1,
El sup |?@|2l < C*(l + E|?o|2) A.П)
Lo<t<r J
where C* is a constant depending only on К, Т.
2. Stronger uniqueness and existence theorems
Theorem 2.1. Suppose bt(x, t), ot(x, t) are measurable functions in
(x,t)&Rn X [0,T],for i = 1, 2, satisfying
\b,(x, t) - b,{x, t)\ < K.\x - x|, \at(x, t) - стДж, t)\ < K.\x - x\,
b,(x,-t)|<K(l + |x|), \o,(x,t)\ < K(l + \x\).
Let D be a domain in R" and suppose that
Ьг(х, t) = b2(x, t), a^x, t) -f o2(x, t) if xGD, 0 < t < T. B.1)

2. STRONGER UNIQUENESS AND EXISTENCE THEOREMS 103
Let ?(t) (i = 1, 2) be the solution of
in M^[0, Г] (with the same family of a-fields 5,) where E|?;0|2 < oo.
Assume finally that ?10 = ^ for а.а. и for which either ?10(w) G D or
?20(w) G D. Denote by т, the /irst tone ?(?) intersects R" \ D if such time
t < T exists, and т, = Г otherwise. Then
sup
= 0) =
Thus, if two stochastic equation have the same coefficients in a cylinder
Q = D X [О, Г] and if the initial conditions coincide in D, then the corres-
corresponding solutions agree until the first time they both leave D; they first leave
D at the same time. This is a local uniqueness theorem. It remains true (with
similar proof) for the general domains Q.
Proof. Let <j>{(t) = 1 if ?(s) GD for all 0 < s < t, and ф,(*) = О in all other
cases. Then
Ф1Ш10 ~ S20) = 0 a.s.
Hence
Ш - Ш] = фМ f WiW,
-'о
r*[«»i(€i(*), *) -
*i(*) Г k(€i(s), s) -
-'o
+ /2 + /3 + h-
If фх(^) = 1, then b^^s), s) = b2(?i(*)> s). Hence lx = 0. If t < rl5 then
ai(?i(s)> s) = a2(%i(s)> s)- Hence, by Theorem 4.4.7, /3 = 0 a.s. if t < rv
Since also ф!(*) = 0 when t > t1s we have /3 = 0 a.s. Thus
2
0
2
B.2)
I •'0

104
5 STOCHASTIC DIFFERENTIAL EQUATIONS
Since фх(?) < 4>i(s) if s < t,
• t
о
n(jT*
B.3)
Next, if /eL*[0, Г] and te[0, Г],
f'f(s)dw(s) < f ^(sjfis) dw{s
B.4)
Indeed, let A = {^(t) = 1}, В = (ф^*) = 0}. If wEB, then the left-hand
side of B.4) vanishes; hence B.4) is true. If wEA, then ^>l{s)f{s) = f(s) if
s < t; hence, by Lemma 4.2.11,
<f>i(s)f(s) dw(s) = I f(s) dw(s),
о Jo
and B.4) again follows.
Using B.3), B.4) in B.2), we find that
Ji-t
4>i(s)[oJ?As), s) - oJL(s),s)\ dw(s)
0
Using the Lipschitz continuity of b2(x, t), o2(x, t) in x, we get
E<t>i(t)\?i(t) - ?г(*)|2 < С f E<j>Js)\L{s) - L(s)\2 ds (C constant).
-'o
This implies that
Hence, by the continuity of ^(t),
sup
0<t<T
- 0.
= 0 a.s.
It follows that ^(t, (S) = |2(?, u) a.s. if 0 < t < t^w). Consequently,
P(t2 > Tj) == 1. Similarly Р(тг > т2) = 1, and the proof is complete.
We shall now use Theorem 2.1 in order to improve Theorem 1.1.
Theorem 2.2. Suppose b(x, t), a(x, t) are measurable in (x,t)GRn X [0, Г]
and
\b{x, t)
\a(x, t)\
x .
B.5)
Suppose further that for any N > 0 Лете is a positive constant KN such that
\b(x, t) - b(x, t)\ < KN\x - x\, \a(x, t) - a(x, t)\ < KN\x - x| B.6)

2. STRONGER UNIQUENESS AND EXISTENCE THEOREMS 105
if |*| < N, \x\ < N, 0 < t < Г. Le? ?0 be any n-dimensional random vector
independent of ^(w(t), 0 < t < Г) such tfurt E|?o|2 < oo. ITien there exists
a unique solution ?(t) of A.1), A.2) in Л0О, Т]. Further
E\ sup |?W|2] < C*(l + E|?o|2) B.7)
iti/iere C* is a constant depending only on T, K.
Thus, Theorem 1.1 and Corollary 1.2 remain true if the condition A.3) is
replaced by B.5), B.6).
Proof. Let
f fc(«) if |SoMl < N,
Ю if |€o(«)| > N,
[Ь(ж, t) if |ж| < N,
bN(x, t) = | b(x, 0B - (|x|/N)) if N < \x\ < 2N,
[0 if |ж| > 2N,
la(x,t) if |*| < N,
aN(x, t) = j „(x, t)B - {\x\/N)) if N < \x\ < 22V,
[0 if |*| > 2N.
By Theorem 1.1 there exists a unique solution ?#(<) in МДО, Г] of
dtN{t) = bMt), *) dt + oN(?N(t), t) dw{t),
Ш = W B-8)
For definiteness we take 5, to be the a-field generated by w(s), 0 < s < t
and |0. By Corollary 1.2,
E\ sup |^(t)|2l < С*A+?У2) B.9)
L 0< t<T J
where C* is a constant independent of N.
Let rN be the first time \?N(t)\ > N, if such a time t < T exists, and
Тд, = Г otherwise. If N' > N, then by Theorem 2.1 ?N(t) = ?N.(t) a.s. if
0 < t < Тдг. Therefore,
P{ sup \Ш - ZN'(t)\ > O) = P{ sup |^(t)| >
0<t< Г 0< «<Г
Notice that the set ?lN = {sup0<t<r |?#(*I > N} decreases when N
increases, and P(SlN) | 0 if TV | oo. Define
?{t, и) ш lim ?N(t, u) if wgE lim UN.
.V-.00

106
5 STOCHASTIC DIFFERENTIAL EQUATIONS
Then, ?(t, и) = ?N(t, и) if w?&N. It is clear that ?(t) is in L*[0, T].
By Lemma 4.2.11,
J-t r-t
o(?(s), s) dw(s) = I oN(?N(s),s) dw(s) a.s. if ug.tiN.
о Jo
From the stochastic differential equation for ?N(t) we then obtain
?(*) = ^o + Г Ь(?(*)» *) * + Г "(^(*), s) dw(s)
0 0
a.s. if u g Пд,. It follows that ?(t) is a solution of A.1), A.2).
Finally, from B.9) and Fatou's lemma we obtain B.7).
It remains to prove uniqueness. Let ?x(?), |2(*) be two solutions of A.1),
A.2) in МДО, Г]; ? is nonanticipative with respect to 5J. Let <j>(t) = 1 if
suPo<s<r l^(s)l < N for i = 1, 2 and ф(^) = 0 in all other cases. Then (cf.
the proof of B.2))
rt ft 2\
I o(ix(s), s) dw(s) — I o(tz(s), s) dw(s) )
It is clear that
К 2t
•'o
ds.
B.10)
B.11)
Next, if f{(s) are step functions, nonanticipative with respect to the
a-fields #{ and in МДО, Г], then (cf. the proof of B.4))
]
~ f Ш dw(s)
s) ds.
B.12)
By approximation, this inequality holds for any /ls /2 in МДО, Г]; /( being
nonanticipative with respect to ^\. Using B.12) with f{(s) = a(^(s), s)) and
using B.11), we obtain from B.10)
Consequently
i.e.,
Р{Ш
BГ + 2)KN f
•'о
Ш - Ш\2} = о,
sup |^(*)| > n] + P{ sup

2. STRONGER UNIQUENESS AND EXISTENCE THEOREMS 107
Since ?j(s) and ?2(s) are continuous process, the right-hand side converges to
0 as N->co. Therefore,
р{Ш ф Ш) = о.
Since both processes ?x(?) and ?2(t) are continuous,
= Ш f°r 0 < t< T) = 1.
We conclude this section with an estimate on the higher moments of the
solution ?(?).
Theorem 2.3. Let b(x, t), a(x, t) be as in Theorem 2.2, and let E|?0|2m < oo
for some positive integer m. Then
E|?(*)|2m < A + E|?0|2m)ec', B.13)
El sup |?(s) — ?ol } ^ C(l "*" ?|?ol )tm, B-14)
where С, С are constants depending only on К, Т, m.
Proof. By Ito's formula with f(x, t) = \x\2m, ?N(t),
\t (A\2m = It (C\)\2m + Г < 9mlf (v)\2m~H (ч) • h (? (<t) <t)
J0 \
n
?u I^JVV /I ii
" "I
+ 2j 2mBm - 2)|^(s)|2ma^(s)^(s) ds
J
B.15)
where (a^) = i oAr(ftv(s), s)(a/v(^(s), *))• (A* = transpose of A).
Since
Us) = ^@) + f bN(?N{s), s)ds+ f oN(ZN{s), s) dw(s)
and Е|?дг@)|2 < oo, while bN(x, t) and aN(x, t) are bounded, it follows (using
Theorem 4.6.3) that E\?N(s)\2m < CN for all 0 < s < T, where CN is a
constant. Hence the expectation of the stochastic integral on the right-hand
side of B.15) is equal to 0. We therefore get
Е\Ш\2т < Е1ШГ + L f E{[1 + КлШШ!*"-2} ds,
where L is a constant depending only on m, K. Since the integrand on the
right is < 1 + 2E|^(s)|2m, the inequality B.13) for ?N readily follows. Now
take N f oo to obtain B.13).

108 5 STOCHASTIC DIFFERENTIAL EQUATIONS
To prove B.14) notice that for any Хб[0, Г]
sup т~ы2т
0<«<X
Г x 1^ t
<n22m\ ( \b{U{s),8)\ds\ +2Г sup Г a(U{s),s)dw(s)
[J0 J 0<«<X J0
Taking the expectation and using Corollary 4.6.4 and B.13) we obtain
E sup |f(t) - |0|2m < LX—1 [\l + E\?(s)\2m]cb < L1Xm(l + E||0|2m),
where L, L1 are constants depending only on К, Т, m; thus B.14) holds.
Notice that B.14) implies that
E{ sup \Z(s)\4 < C*(l+?||0|2), B.16)
where C* is a constant depending only on К, Т, m.
Remark. The results proved in this and the previous sections extend
immediately to stochastic differential equations A.1) with initial condition
?(s) = ?s where sE[0, Г]. Here ?S is assumed to be independent of the
a-field f (w(t + s) - w(s), s < t < T - s).
3. The solution of a stochastic differential system as a Markov
process
We shall assume:
(A) The n-vector b(x, t) and the n X n matrix a(x, t) are measurable
functions for (x,t)ERn X [0, oo) and, for any Г > 0,
\b{x, t)\ < K(l + \x\), |a(x, t)| < K(l + |x|), C.1)
\b(x,t) ~ b(x,t)\< K\x- x\, \o{x,t) - a{x,t)\ < K\x- x\ C.2)
if 0 < t < T, i6fl", 3cЕй", where К is a constant depending on T.
By Theorem 1.1 there exists a unique solution in Mj[0, oo) of
dZ{t) - ЪШ, t) dt + аШ, t) dw(t), C.3)
1@) = So, C-4)
provided ?0 is independent of f (to(X), X > 0) and ?||0|2 < °°- Similarly (see
the remark at the end of Section 2), for any s > 0 there exists a unique
solution in M*[s, oo) of C.3) and
f (*) = i. C-5)
provided |, is independent of ^(w(\ + s) - w(s), X > 0) and E\?,\2 < oo.

3. THE SOLUTION AS A MARKOV PROCESS 109
If ?s = x a.s., where x is a point in fl", then we denote the solution of
C.3), C.5) by 4,s(t).
For any Borel set A in Л" and for any t > s, let
t)GA). C.6)
Theorem 3.1. Let (A) hold and let |0 be independent of ?r(w(t), t > 0),
E|?o|2 < oo. Denote by *$t the o-field spanned by ?0 and w(s), 0 < s < t.
Then the unique solution ?(f) of C.3), C.4) satisfies
P(f(t)eA|ff)-P(f(t)eA|f(S)) = p(S,f(S),t,A) a.s. C.7)
for all t > s and for any Borel set A. Further, p(s, x, t, A) is a transition
probability function.
Proof. We can write
g(t) = Ф) + С b(f(\), X) d\ + С o(f(X), X) dw(X).
By the method of successive approximations, if we set
Ф) = у (where у = f(s))
fc(t) = у + Г' Ь&.Д), X) dX + f о&.ЛХ), X)
then 4(t) —> i;(t) a.s. By induction one can show that each ?k(t) is measurable
with respect to the a-field spanned by 4F(w(u + s) — w(s), s < и < t) and
y. The same is therefore true of |(t). More specifically, using Problems 20, 21
of Chapter 4 and Theorem 4.3.4, one can approximate a.s. (and uniformly in
t) each ?k(t) by a sequence of functions
Fm(t, y, w(uml + s) - w(s), . . . , w{um ^ + s) - w(s))
where 0 < um , < t — s and Fm(t, x0, хг, . . . , x^J are Borel measurable
functions depending only on the functions b(x, X), o(x, X) in the interval
s < X < t. The same therefore holds for ?(?), i-e., a.s.
C.8)
with suitable Borel functions Fm and suitable um t. In particular,
$,,,(*) = mlimoFm(t, x, «;(uml + e) - uj(s), . . . , «з(ат>л> + s) - tc(s)).
C.9)
Now, for any bounded measurable function
x xk) = F0(x0)F1(x1> . . . , xk)

110 5 STOCHASTIC DIFFERENTIAL EQUATIONS
and for any u{ > 0,
), ю(щ + s)~ w(s), ...,w(uk+s)- w
(u1 + s) - w{s), ..., w(uk + s) - w(s))\%]
(a1 + s) - «j(e), . . . , u>(ut + s) - «,(«))]
since the random variables w(ut + s) — u>(s) are independent of <§s. Thus,
E[F(|(s), t^K + s) - w(s), ...,w(uk + s)- u>{s))\$t]
= {EF(x0, «j(Ml + s) - u>(«), . . . , u>(tifc + «) - «з(в))}^_в,). C.10)
By approximation, C.10) remains true for any bounded Borel measurable
function F(x0, xv . . . , xfc). Taking, in particular, F = f(Fm) where / is a
bounded continuous function and Fm are as in C.8), C.9), we conclude that
where ф(х) ^
Taking a sequence of/'s that increase to the indicator function of an open
set A in R n, we obtain the assertions C.6), C.7) for any open set A. Since the
class of Borel sets for which C.6), C.7) hold form a monotone class, and since
the open sets belong to this class, this class must contain the a-field
generated by the open sets, i.e., it includes all the Borel sets.
It remains to prove that p is a transition probability function.
From C.6) it is clear that p(s, x, t, A) is a probability measure in A, for
fixed s, x, t. Since ^ s(t) is Borel measurable in x (for the same is true of
each function in the sequence that converges to t^ s(t) by the method of
successive approximations), p(s, x, t, A) is also Borel measurable in x, for
fixed s, t, A. Thus it remains to verify the Chapman-Kolmogorov equation.
From C.6),
for any bounded Borel measurable function rf>(y). Taking \p(y) — p(t,
у, т, A) where t < т, we get
J p(s, x, t, dy)p(t, у, т, A) = J p{t, &_.(*), т, A) dP
= Ep(t, ?,,(*). r, A)
where C.7) has been used in the last equality. Since the expression on the
right is equal to
= P{s,x,t,A),
the Chapman-Kolmogorov equation holds for all x <E R", 0 < s < t < т.

3. THE SOLUTION AS A MARKOV PROCESS 111
Denote by Q the class of all continuous functions x( ¦ ) from [0, oo) into
R n. Denote by <DltJ the smallest a-field generated by the sets
{x( ¦ );x(u)EA], S < U < t
where A is any Borel set in Rn. Denote by 'DIL^ the smallest a-field
generated by {x( ¦ ); x(u)EA), и > s.
Define a continuous process X(t) = X(t, x( • )) by
X(t, x( ¦ )) = x(t). C.11)
Finally, let
Px,s{x( ¦ )GB} = P{<o;^s( -,<oNB} C.12)
where В is any set in ?)ltj. Notice that for each шбй, ^s(t) = ^jS(f, w) is a
continuous path. This path is the continuous function 4,s( ' > u) appearing
on the right-hand side of C.12).
It is easily seen that Px s is a probability measure on ?J1t*. We shall now
show that
PXtS{x(t + ЛNА|ЭД = p{t,X(t), t+h,A) a.s. C.13)
Since
P{i;S(* + h) e А\Ц^$(Х), \<t)}= p(t, ?,,(*), t + ft, A),
for any s < tY < t2 < • ¦ ¦ < tm < t and any Borel sets Av . . . , Am, we
have
h)eA, 4,,@еА„ ..., 4,5(t
pu.^.w.t + h.A)*»
_1[4,.(«|)ел|]
4..(*). t + h,A) dp.
Using C.11), C.12), we conclude that
px>,[x(t+h)eA,x(t1)eA1,...,x(OeAj
t, X(t), t+h,A)
= f p(t,X(t),t+h,A)dPxs.
This implies C.13).
We have proved:
Theorem 3.2. Let (A) hoM. Then {6, <Dlt, ^, X(t), PXjS} is a confinuous
n-dimensional Markov process with the transition probability function C.6).
We shall call this Markov process also the solution of the stochastic

112 5 STOCHASTIC DIFFERENTIAL EQUATIONS
differential system A.1). The process X(t) will often be denoted also by |(t).
Lemma 3.3. Let (A) hold. Then for any R > О, Г > 0,
E sup |i s(t) - ^T(t)\2 < C(|x - i/|2 + |s - t|) C.14)
T<t<T
if \x\ < Л, | y\ < Л, 0 < s < т < Т; С is a constant depending on R, T.
Proof. Clearly, if т < t,
t) - у +f\b&,(\), X) - b(^,T(\), X)] dX
(t)
By Theorem 2.3,
?|С.(т) ~ У\г < 2Е|^,(т) - x|2 + 2|x - ?/|2 < C0|t - s| + 2|x - ?/|2
C.16)
where Co is a constant. Taking the expectation of the supremum of the
squares of both side of C.15) and using C.16), C.2), we get
E sup |4if(t) - ^T(t)\2 < C^lx - y\2 +\s- t|)
T<t<t'
where Cj is a constant. This easily implies C.14).
Theorem 3.4. Let (A) hold. Then the solution {6, <Dlt, <!Ж|, X(t), Px s} of
the stochastic differential system A.1) satisfies the Feller property, and
therefore also the strong Markov property.
Proof. If / is a bounded continuous function, then, by Lemma 3.3 and the
Lebesgue bounded convergence tlieorem,
Also,
Thus, the function
(s, x)-* J p(s, x, t + s, dy)f( y)
is continuous, i.e., the Markov process {в, "DIL, <DltJ, X(f), Px s} satisfies the
Feller property. Since it is also a continuous process, Corollary 2.2.6 asserts

3. THE SOLUTION AS A MARKOV PROCESS 113
that it possesses the strong Markov property.
Consider a stochastic differential system
d?{t) = b(f(t), t) dt + o(?(t), t) dw(t) C.17)
where w(t) is another n-dimensional Brownian motion. According to
Theorem 3.3, there is a Markov process solution {B, <Dlt, <DltJ, X(t), Px s].
Definition. If for any Brownian motion w(t), Px s = Px s for all x, s, then
we say that there is uniqueness in the sense of probability law. By contrast,
the uniqueness asserted in Theorem 1.1 is called pathwise uniqueness.
Theorem 3.5. Let (A) hold. Then there is a unique solution of A.1) in the
sense of probability law.
Proof. From the proof of C.8) one sees that the Fm do not depend on the
particular Brownian motion w(t). Hence, for any s < tY < t2 < • • ¦
< tk < oo and for any bounded continuous function ф(х1г . . . , xk), if ?(f) is
a solution of C.17) with |(s) = ?(s) and if ?(s) = x, then
s) -
.... Fm(tk, x, w{um< j + s) - w(s), . .., w(um ^ + s) - w(s
,i + s)-w(s),. . . ,w(umilm+ s)-
Since ф is arbitrary, it follows that Px , = PXi s on cylinder sets in IL*.
Therefore also Px< s = Px_, on the whole of "DILJ.
The results of this section can be extended to the case where the
condition (A) is replaced by the weaker condition:
(A') b(x, t) and a(x, t) are measurable; for every Г > 0 the inequality
C.1) holds for 0 < t < T, x<ERn with К depending only on Г; for every
T > 0, R > 0,
\b{x, t) - b(x, t)\ < KR\x - x\, \a{x, t) - a(x, t)\ < KR\x - x\
if 0 < t < Г, |x| < R, \x\ < R where KR is a constant depending on T, Л.
Theorem 3.6. Theorems 3.1, 3.2, 3.4, 3.5 remain true if the condition (A) is
replaced by the weaker condition (A').
The proof is left to the reader.

114 5 STOCHASTIC DIFFERENTIAL EQUATIONS
4. Diffusion processes
A continuous n-dimensional Markov process with transition probability func-
function p(s, x, t, A) is called a diffusion process if:
(i) for any e > 0, t > 0, 16Й",
lim у f pit, x,t + h, du) = 0; D.1)
hio h ./|!,-X|>?rv *' v >
(ii) there exist an n-vector b(x, t) and an n X n matrix a(x, t) such that
for any e > 0, t > 0, x6 Rn,
(№ 4) p(, x, t + h, dy) = bt(x, t)
A < i < n), D.2)
x<)(?//- x/)P(*. x, t + h, dy) = af/(x, t)
A < i, / < n), D.3)
where Ь = (Ь1? . . . , bn), a = (aj.
The vector Ь is called the drift coefficient and the matrix a is called the
diffusion matrix; when n = 1 we call a the diffusion coefficient.
Lemma 4.1. The following conditions imply the conditions (i), (ii):
(i*) for some 8 > 0, t > 0, xE Rn,
lim j; f \x- y\2+sp(t, x, t+ К dy) - 0; D.4)
(ii*) /or ant/ i>0,x6R",
lim ^ Г (у,- xt)p(t, x,t+ h, dy) = bt(x, t) A < i < n),
D.5)
lim - Г (j/. - x,)( j/. - x/)p(t, x, f + ft, dy) = aJx, f)
h IО Л1 Уд п ' ' '
A < i,) < n). D.6)
Proof. Using D.4) we have
f p(t, x, t + h, dy) < -^ [ | у - x\2+sp(t, x,t+h, dy)
J\y-x\>€ e Jr*
if h-*0,

4. DIFFUSION PROCESSES 115
so that D.1) holds. By D.4) we also have, for /' = 1, 2,
j- f | у - x\'p{t, x,t+h, dy)
if h-*0. Consequently D.2), D.3) hold if and only if D.5), D.6) hold.
Theorem 4.2. Let (A') hold and let b(x, t), a(x, t) be continuous in
(x, t)?Rn X [0, oo). Then the solution of A.1) is a diffusion process with
drift b(x, i) and diffusion matrix a(x, t) = o(x, t)a*(x, t).
Here a* is the transpose of ст. Thus, if a = (aj(), then a{, = 2*=1ajita/it.
Proof. Since p(t, x, t + h, A) is the probability distribution of the random
variable ?,,*(* + ^)'
t(t + h) - x) = Г /( у - x)p(t, x,t+h, dy) D.7)
for any continuous function/(z) with \f(z)\ < K(l + |z|") for some К > 0,
a > 0; we use here the fact that E\^t(t + h)\a < oo for any a > 0 (see
Theorem 2.3).
In view of Lemma 4.1 it is therefore sufficient to prove that
where ?,' t is the
M-*)(CU«-
ith component
From Theorem 2.3,
this gives D.8).
{t + h) - x\4 <
-h)
+ h)
f fc)
of 4
-
-
2A
c|-
*]-
*0
¦b(x,
х)-*ач(х,
+
x\4)
if ^
«)
(K
if fc-»0,
if ft—»0,
constant);
D.8)
D.9)
D.10)
Next,
? \ ft+k(X), X) dX
hs),t + hs)ds.

116 5 STOCHASTIC DIFFERENTIAL EQUATIONS
Consider the random variables Xh(s, u) = b(?x t(t + hs), t + hs). Since
f1 E\Xh\2 ds < С f* ?[l + |4 t(t + hs)\2] ds < d
•'o •'o
where C, Cx are constants independent of h, the random variables Xh satisfy
the condition of uniform integrability, i.e.,
sup Г |ХЛ| dP-*0 if X-*oo. D.11)
h J\xh\>\
p
We also have that Xh(s, u) -* b(x, t) uniformly with respect to s. Hence,
by Lemma 1.3.6,
f1 EXh{s) ds-*b(x,t) as h-*0, D,12)
•'o
and the proof of D.9) is complete.
To prove D.10) we use Ito's formula D.7.11) with u(z, t) = ^zf:
^(кЛЧ Щ dK D.13)
where x = (xl5 . . . , xn).
Noting that
C2
where C2 is a constant independent of h, we can apply the argument used to
prove D.12) in order to deduce from D.13) that, as h—*0,
? Qt{t + h) - x,x,]~»x,b,(x, t) + xfb{{x, t) + a^x, t).
It follows that
lim { 1 E(?t(t +h)- x ,)(#,(* + h) - x,)}
= x,b,(x, t) + x,b,(x, t) + o,(x, t) - x,ton \e[U,M + h)-x,]
-Xijimo^E[^t(t+ h)~xt]
= Oqix, t),
where D.9) has been used in the last equality.
We shall use Theorem 4.2 in order to compute the infinitesimal genera-

5. EQUATIONS DEPENDING ON A PARAMETER 117
tors &s of the Markov process solution of A.1). By definition (see B.3.7)),
((^w = !?o h
where
(Ts,s+hf)(x)=fp(s,x,s + h,dy)f(y).
Thus,
(&sf)(x) = Urn \ /[/( y) - f(x))p(s, x,s + h, dy). D.14)
Suppose / is bounded and twice continuously differentiable. Then
1 = 1
1 П
Substituting this in the integral in D.14) for у in a small neighborhood of x,
then taking h I 0 and using D.1)-D.3), we find that
(<M(*) = \ 2 ф, *) -^4- + 1 k(x, s)-^-= (LJ)(x). D.15)
The operator Lsf is a partial differential operator. This same operator
arises also in Ito's formula:
и(ф), t) - иШ, s) = f'( |? +Lxu)(|(X), X) dX
Js \ о л /
+ I и (|(X), X)a(?(X), X) dw(\) D.16)
Л
where |(t) is any solution of A.1). We refer to Ls as the differential operator
corresponding to the stochastic system A.1). Using D.16) with t = т, one can
easily derive Dynkin's formula B.3.6) when / is twice continuously differen
tiable.
5. Equations depending on a parameter
Let A(x, t), B(x, t) be a random n-vector and n X n matrix respectively,
defined for (x, t)E.Rn X [s, oo) for some s > 0 and assume:
(i) A(x, t), B(x, t) are continuous in (x, t), for each ы;
(ii) A(x, t), B(x, t) are measurable in {x, t, w);
(iii) A(x, t), B(x, t) are ^t measurable for each (x, t), where '.'7, is an

118 5 STOCHASTIC DIFFERENTIAL EQUATIONS
increasing family of o-fields such that w(t) is §t measurable and ?F(u?(t +
X) - w(t), X > 0) is independent of % for all t > 0;
(iv) there is a constant К such that
|A(x, t)| < K(l + |x|), \B{x, t)\ < K(l + |*|) a.s., E.1)
|A(x, t) - A(x, t)| < K\x - x\, \B{x, t) - B{x, t)\ < K\x - x| a.s.
E.2)
Let <f>(t) be a function in M^°[s, Г]. Consider the equation
), X) dX + /" B(|(X), X) dic(X). E.3)
We refer to it as a system of stochastic differential equations with random
coefficients.
Theorem 5.1. I/(i)-(iv) hold and <f>GM^°[s, Г], then there exists a unique
solution ?(f) in M*[s, T]; further, ?(f) belongs to M™[s, T].
We outline the proof leaving the details to the reader.
One defines by induction
Ф@ + f Afo..^), X) dX + f BDm_1(X), X) d«;(X),
and shows that the ^ are well defined and
Since
Е\Ш - ?0(t)|2 < Ki f ?(l + |Ф(А)|2) d\
Lm{t - s)m
E\?m\4 ~~ ^m-l(*)l ^ i (¦k Constant).
Now we proceed as in the proof of Theorem 1.1 to show that lim ?„,(*) is a
solution. The proof of uniqueness is the same as for Theorem 1.1.
Theorem 5.2. Let Ao(x, t), Ba(x, t), <t>a(t) satisfy the assumptions of
Theorem 5.1 for any 0 < a < 1, with the constant К (in E.1), E.2))
independent of a, and with sups<-t<-TE|<f>a(t)|2 < С, С a constant indepen-

5. EQUATIONS DEPENDING ON A PARAMETER 119
dent of a. Suppose that for any N > 0, tG[s, T], e > 0,
lim p{ sup \Aa{x, t) - A0{x, t)\ > e} = 0, E.4)
«Ю K\x\<N '
lim P{ sup \Ba(x, t) - B0(x, t)\ > e} = 0. E.5)
«Ю 1|,|<2V ;
Suppose also that
lim sup E|<f>a(f)-<f>o(f)|2 = 0. E.6)
«io«<t< т
Consider the solutions ?,(*) o/ the equations
Ш = Ф„(*) + f Аа(Ш, X) ^Х + f Ва(Ш, V dw{\). E.7)
Then,
sup E|$,(t)-«0(t)|2-»0 i/ «10. E.8)
T
Proof. Take for simplicity s = 0. We can write
+ С[Ва(Ш,°) - ВаШ,*)] dw(s)
where
¦na(t) = Ф„(*) - Фо@ + Г'[АаAо(*)> *) - Ao(«o(e), «)]
•'о
+ f([Ba(U4*)-B0(|0(s),s
•'о
Using E.1), E.2) for Aa, Ba, we easily find that
Е\Ш - Ш2 < ЗЕ|ц,(*)|2 + С f'
•'о
If we prove that
sup E\i)a(t)\2^>0 as а-*0, E.10)
0<(<T
then the assertion E.8) follows from E.9).
Now,
2
Ia = e| J K(|0(s), e) - Ao(«o(*), «)]
< (E r'|Aa(|o(S),s)-Ao(|o(s),s)|
•'0

120 5 STOCHASTIC DIFFERENTIAL EQUATIONS
The last integrand is bounded by 2KA + |?0(s)|2), which is an integrable
function. In view of E.4), this integrand also converges to 0, in probability,
as a—>0. Hence, by the Lebesgue bounded convergence theorem, Ia—>0 if
a^O.
Next,
¦'о
By the previous argument, Ja-»0 if a—>0.
Finally, making use also of E.6), the assertion E.10) follows.
We shall apply Theorems 5.1, 5.2 in studying the behavior of the solution
uxt,(t) in the parameters x, s. Recall that
^s(t) = x +(' bDiS(X), X) dX+ (' oD>s(X), X) dw{X). E.11)
We shall use the notation
D* = ~hr = 9xf ¦ • • Эх,,"» ' l«l = «i + ¦ • ¦ + «n-
If f = f(x, t), then the vector (df/dxv . . . , Э//Эхп) is denoted briefly by
Definition. Let g(x) = g(xv . . . , xn), f(x) — f(xy, . . . , xn) be random
functions for x in some open set. If
1
^ [g(*i, • ¦ • , x<-i. x(+ h, xi + l, . . . , xn) - g(xv . . . , xn)]
2
-/(*!, . . . ,Xn)
as h—*0, then we say that g(x) has a derivative with respect to x. in
the L2(u) sense, and the derivative is equal to f(x). We write
(9/9xi)g(x) = f(x). Similarly one defines the derivative Dag(x) in the L2(fl)
sense, for any a = (ax, . . . , an).
We shall need the following condition:
db/dxt, Эо/Эх, exist and are continuous (l < i < n). E.12)
Theorem 5.3. If (A) and E.12) hold, then the derivatives d^s(t)/dx { exist
in the L2(fl) sense and the functions ^(t) = 3^iS(t)/dxt satisfy the

5. EQUATIONS DEPENDING ON A PARAMETER 121
stochastic differential system with random coefficients
Ш «« + /"' Ш • K&JX), X)dX+f MX) ¦ ох(^,(Х), X) dw(X)
E.13)
where e. is the vector with components 8{j.
Proof. Take for simplicity i = 1, and write h = (hu 0, . . . , 0) where
hx ф 0. Then
Г" k+h,s(t) ~ 4,.(')] = *i + f [bD + k». X) ~ Hi,M X)] dX
/" ). E-14)
By Theorem 5.1 there exists a unique solution fx of E.13) with i = 1. We
shall now complete the proof by invoking Theorem 5.2 with a = h1 if
Ц > 0 and a = — hx if hj < 0. First we note that
г
dk.
A similar formula holds for the stochastic integral on the right-hand side of
E.14). Therefore, the function
satisfies E.7) with
Aa(z, t) = z ¦ Г' bxD.,(t)
•'o
Ba(z, t) = 2- f 4(&,» + /i(^+h ,(t) - C.,(*)). 0 *
•'o
if a t^ 0, and ?0(t) = f1(t) satisfies the same equation with
A0(z, t) = z- Ьх(^,(г), t). B0{z, t) - z ¦ ax(^t(t), t).

Since, by Lemma 3.3,
we conclude, upon using E.12), that E.4), E.5) are satisfied. The assertion of
the theorem now follows from Theorem 5.2.
Remark. Notice that 3^ s(t)fdx { satisfies the stochastic differential system
with random coefficients obtained by differentiating formally the stochastic
differential system of j^,(t) with respect to xt.
Theorem 5.4. Let (A) hold and assume that D"b(x, t), D"a(x, t) exist and
are continuous if \a\ < 2, and
|D«b(x, t)\ + |D>(x, t)| < «„(I + |*|") (|«| < 2) E.15)
where Ko, /? are positive constants. Then the second derivatives
d\_,(*)/3x4 9x- exist in the L2(fl) sense, and they satisfy the stochastic
differential system with random coefficients obtained by applying formally
d2/dxt dXj to E.11).
The proof is left to the reader.
Theorem 5.5. Let f(x) be a function with two continuous derivatives,
satisfying
\DxJ(x)\ < C(l + |x|") (\a\ < 2) E.16)
where C, f$ are positive constants. Let the conditions of Theorem 5.4 hold,
and set
ф(х) = E/(€,, ,(*))• E-17)
Then ф(х) has two continuous derivatives; these derivatives can be computed
by differentiating the right-hand side of E.17) under the integral sign.
Finally,
|D>(x)| < C0(l + \x\y) if |a| < 2, E.18)
where Co, у are positive constants.
Proof. We shall prove that
Take for simplicity i = 1 and set h = (hv 0, . . . , 0). Then
ф(х + h) - ф(х) ]
*..@-€,.

Hence
E.20)
8L(() in
ax1
We claim that also
.(»)) in L2(fl) E.22)
as h-*0. Indeed, this follows from Lemma 1.3.6 if we observe that
probability and
where Cx is a constant independent of h; the assumption E.16) is used here.
From E.20), E.21), and E.22) we see that
ф(х + h) - ф(х)
Having proved E.19), one can now check that the right-hand side of
E.19) is a continuous function in x and is bounded by C0(l + \x\y) for some
positive constants Co, y. The same is then true of Зф/3зс;. The proof that
<p(x) has second derivatives, that these derivatives are obtained by
differentiating under the integral sign of E.17), and that they are continuous
and satisfy E.18), is similar to the proof for the first derivatives; this is left to
the reader.
Theorems 5.4, 5.5 extend without difficulty to higher derivatives.
6. The Kolmogorov equation
Consider the function
Notice that
where {6, <Ж, С!Я1', ?(t), P,,} is the Markov process that solves the system
of stochastic differentia) equations A.1).

124 5 STOCHASTIC DIFFERENTIAL EQUATIONS
Theorem 6.1. If f and b, a satisfy the conditions of Theorem 5.5, then uv
ux, u^ are continuous functions in (i,()?fl"X [О, Г) and satisfy
u(x, *)-*/(*) if ЦТ. F.2)
Equation F.2) is called the Kolmogorov equation, or the backward
parabolic equation.
Proof. Let g(x) be a twice continuously differentiable function satisfying
|Deg(x)| < C(l + \x\fi) if \a\ < 2, where C, /3 are positive constants. By
Ito's formula (see D.16))
-h(t)) ~ g(x) = E С (L.dO, t.h(s), s) ds (A>0) F.3)
where Ls is the partial differential operator defined in D.15). The argument
used to derive D.12) can be applied also here to deduce that
\E f
« Jt-
(fg)Ditfc(),)(,g)(,) as A->0.
-h
Hence, if we divide both sides of F.3) by h and let h—>0, we get the formula
ton i ?[g(i, ,_h(t)) - g(«)l = (^g)U. »)• F-4)
Next, by the Markov property,
«(x, t-h)- ЕМ_^€(Т)) = E,,t_hEx,t_h(/(«(^)I^Lrfc)
- ?Xif_h??((U/(|(T)) = EXit_h«(«(t), t) = Е«D.«-ь(»). 0.
so that
u(x, t) - u{x, t - h) Eu(^t_h{t),t) - u(x,t)
h = h ¦ F'5)
In view of Theorem 5.5, the function g(x) = u(x, t) has two continuous
ж-derivatives, bounded by C0(l + |xrlY), for some positive constants Co, y.
Consequently the formula F.4) can be applied. From F.5) we then conclude
that
u(x, t) - u(x, t-h) . . , ч
lim ^^ ^ = - Ltu{x, t). F.6)
From F.3) one easily deduces that
?|?g&. ,-*(»))-g(*)l< с
where С is a constant independent of h. Applying this to the function

PROBLEMS 125
g(x) = м(х, t) and then using F.5), we get
|m(x, t) - u(x, t - h)\ < Ch.
It follows that, for fixed x, u(x, t) is absolutely continuous in t. Therefore,
Зм(x, s)/ds exists for almost all s and
г* Эм(х, s)
mx, t) = mx, 0 + I \ ds.
Jo ds
In view of F.6),
m(x, t) = m(x, 0) — I Lsu(x, s) ds.
Jo
But since Lsu(x, s) is a continuous function, it follows that Эм(х, t)/dt exists
everywhere and F.1) holds.
Next,
м(х, t) - f(x) =
Using Lemma 3.3 and Lemma 1.3.6, we find that the right-hand side
converges to 0 if 11 Г. This proves F.2).
PROBLEMS
1. In Theorem 2.2, replace B.5) by the inequalities
x ¦ b(x, t) < K(l + |x|2), \a(x)\ < K,
and prove that the assertions of Theorem 2.2 are still valid. [Hint: Apply
Ito's formula to ?^(i). Prove that
E sup |Ш|2 < C,
0< «< T
С independent of N.]
2. Prove that if (A') holds, then for every xG Л", s > 0,
where 7j(r)->0 if r->0. [Hint: Use B.7).]
3. Prove Theorem 3.6.
4. If / G L?[0, Г] and т is a stopping time, 0 < т < Г, then
fTf(t)dw(t)=(T~Tf(T+s)dw(s)
-4 •'O
where w(s) — w(s + т) - w(t).
5. If {Я, 4$, Щ, i(t), Px s) is a Markov process with transition probability
p(s, x, t, A) and if g(x, () is a continuous function in (x, (jGfi1 X [0, oo),
strictly monotone in x, then {Я, 9F, ?FJ, g(|(<), *)> ^"x, s) is a Markov process
with the transition probability p(.s, x, ?, A) = p(.s, g~l(x, s), t, g~1(A, t))

126 5 STOCHASTIC DIFFERENTIAL EQUATIONS
where g~x(x, t) is the inverse function to g(x, t).
6. If the Markov process in the preceding problem is a diffusion process
with drift b(x,t) and diffusion coefficient a(x, t) and if gt, gx, g^ are
continuous and gx(x, t) ф 0 for all (x, t), then the process (Я, ff, <3=j,
g(|(f), ?), Px s} is also a diffusion process with drift and diffusion coefficients
given by:
?(*• *) = Uy. *) + b(y, t)gx(y, t) + \a(y, t)gxx(y, t),
a(x,t) = a(y,t)(gx(y,t)f, y = g-\x,t).
7. Complete the details of the proof of Theorem 5.1.
8. Prove Theorem 5.4.
9. Complete the proof of Theorem 5.5.
10. Consider a linear stochastic differential equation
di{t) - [a(t) + РШШ dt + [y{t) + »(*)€(*)] dw(t) F.7)
where the functions a, /?, y, 8 are bounded and measurable. Prove:
(i) If a = 0, у = 0, then the solution ? = ?0(t) is given by
Ш = Ф) exp( C[(i(s) - iS2(s)] ds+f 8(s) dw(s)}.
(ii) Setting |(f) = ?0(f)?(t) show that |(f) is a solution of F.7) if and only
?@) + /"fc,(*)«W - y(s)8(s)]ds+ С y(s)Us) ds.
if
Thus the solution of F.7) is ?„(*)?(*) with |@) = ?0@)J @).
11. If an equation of the form d? = b(x, f) df + a(x, f) dti>(f) can be
reduced by a transformation tj(() = /(|(f), *) to an equation of the form
dq(t) = P(t) dt + 8{t) dw(t), then
Эх { [ a2 dx \ a 1 ^JJ
12. Let g(x, t), h(x, t) be continuous functions in (x, f)G Д1 X [0, Г] to-
together with their first two x-derivatives, and assume that
|D«g(x, t)| + |D>(x, t)| < K0(l + |x|") if \a\ < 2,
where Ko, fi are positive constants. Let the conditions of Theorem 5.5 also
hold, and consider the function
u(x, t) = E exp {iX /Tgfe», e) ds + щ /Tfc(€,,t(e), *) dic(s)

PROBLEMS 127
Prove that
-r—I-Lit + щЪ,их + (iXg — ^fi2h2)u = 0 if зсЕЛ1, 0<(<Г,
u(x, t)-»/(x) if 11 Г.
13. Verify that all the results of this chapter remain true for a system of n
stochastic differential equations
di = 2 o</(?> ') d»j + Hi t)dt A < i < n)
, = i
where (wv . . . , wt) is an ^-dimensional Brownian motion.

6
Elliptic and Parabolic Partial
Differential Equations and
Their Relations to Stochastic
Differential Equations
1. Square root of a nonnegative definite matrix
As is well known, if a is an n X n nonnegative definite matrix, then it has a
unique nonnegative square root a, i.e., there is a unique nonnegative definite
matrix a such that aa = a. If a depends on a parameter x, say a = a(x), then
also a = a(x). We shall be concerned here with the smoothness of a(x) in x,
assuming that a(x) varies smoothly with x.
Set a(x) = (a^(x)), o(x) = (o^(x)). The point x varies in an open set G of
R".
We shall denote by Cm(G) the class of all functions f(x) having con-
continuous derivatives up to order m in G. If the mth derivatives of f(x) satisfy a
Holder condition with exponent a in compact subsets of G, then we say that
/belongs to Cm+a(G); here 0 < a < 1.
Lemma 1.1. If a(x) is positive definite for all i?G and if efyE Cm+a(G)
for all i, \ (or if пц Е Cm(G) for all i, /), then the elements of a(x) belong to
Cm+a(G) (or to Cm(G)).
Proof. Let Go be an open bounded set with closure in G. Let Г be a
smooth contour lying in the half-plane Re z > 0 of the complex plane and
containing all the eigenvalues of a(x), xE. Go. We claim that if i6G0, then
°(x) = 7Г- f V^ («(*) ~ zI)~1 <& (J = identity matrix), (l.l)
2iir /p
To prove it, let Г' be another smooth contour in Re z > 0 whose interior
128

1. SQUARE ROOT OF A NONNEGATIVE DEFINITE MATRIX 129
contains Г. Then
a~(x
4w2 Л" I Л*
(o(x) -SI) X ~ (a(x) - zl)
-l
dz) d$
VI (g(
by Cauchy's theorem. Changing the order of integration and using Cauchy's
theorem, we get
dz
Now modify Г into the disk Гл = {z; \z\ = R }, R large. Since
\ — 1 / »t \ — 1 и ^ ^
dis
a(x)-ziy1-(zl)
dz.
1 + |z
It follows that
1 + Л2
if \z\ = Я,
•0 if R -»oo.
9 / ^
a2 x,
T / 1 Г / N dz 1 / \
= hm \ —— I a(x) — ? = a(x).
From A.1) it is obvious that the elements o(x) are as smooth in Go as the
elements of a(x). Since Go is arbitrary, the proof of the lemma is complete.
Theorem 1.2. If a(x) is nonnegative definite for all x&R" and if the aif (x)
belong to C2(R"), then the o^(x) are Lipschitz continuous in compact
subsets. If, further,
d\(x)
sup sup
dxk
M,
then
- y\.
A.2)
A.3)
Proof. Suppose first that a(x) is positive definite and A.2) holds. Consider
the function <a(x)?, ?> where < , > denotes the scalar product. By Taylor's

130 6 ELLIPTIC AND PARABOLIC EQUATIONS
formula,
0 < (a(xv . .., xk_v xk + h, xk+v . . . , x,)t, О
2а(
for a suitable point x. Using A.2) we get
Since the right-hand side is a quadratic polynomial in h taking only
nonnegative values,
<a(x)«, 0. A.4)
Denoting by ' the derivative with respect to xk and differentiating
o2(x) = a(x) with respect to xk, we obtain
a'(x) = a(x)a'(x) + a'(x)a(x). A.5)
Let T(x) be an orthogonal matrix such that T(x)a(x)T~1(x) is a diagonal
matrix A(x). Multiplying A.5) by T(x) on the left and by T~\x) on the right,
we get
a'(x) = Tixfa'ixlT-^x) = Y(x)A(x) + A(x)Y(x) A.6)
where Y(x) = (y^(x)) = T(x)a'(x)T~1(x). Denoting by \(x) the eigenvalues
of a(x), we obtain from A.6),
Уц=Х^ГТ where s'(x) - (a$(x)). A.7)
Using the estimate A.4) we find that
А% О-
A.8)
Taking ? = ({lf . . . , ?„) with ^ = 0 if; Ф i, ? = 1, we get
|5;(x)|2 < 2vM\2(x). A.9)
Next, taking in A.8) ? = (Zlt ...,?,) with |fc = 0 if к ф i, к ф / (where
г т^ /) and |j = ^ = 1, we find that
4vM{\*(x) + A«(*)).

1. SQUARE ROOT OF A NONNEGATIVE DEFINITE MATRIX
131
Hence
аЛх
\ + A, T A. + A;
< 4vM.
Recalling A.9) we get
A.10)
A.11)
A.12)
Suppose now that a(x) is nonnegative definite and A.2) holds. Then the
matrix a(x) + el (e > 0) is positive definite and satisfies A.2) with M
replaced by a constant Me, Me | M if e | 0. By what we have already proved,
the square root o€(x) of a(x) + el satisfies
Substituting A.9), A.10) into A.7), we obtain
| ya(x)\ < 2V^M if 1 < i, I < v.
Since (a^x)) = T-\x)Y(x)T(x) and T(x) = (Г.Дх)) is orthogonal,
We can apply the Ascoli-Arzela lemma to conclude that, for some
sequence em | 0, afm(x)—» a(x) uniformly on compact sets. This gives A.3).
We have thus completed the proof of the theorem in case A.2) holds.
If the ay belong to C2(R") but A.2) is not assumed to hold, then we can
slightly modify the previous proof, making use of the fact that for any R > 0
sup sup
||Я kl
дхк Эх,
< M(R) (М(Я) constant).
Remark 1. If a(x) is nonnegative definite in an open set G and a{j G C2(G),
then the square root a(x) is Lipschitz continuous in compact subsets of G.
Indeed, this follows from the proof of Theorem 1.2.
Remark 2. If n = 1 and an(x) = |x|\ then an(x) = \x\K/2. If Л = 2, then
an is in C2 and ou is Lipschitz continuous, but if Л = 2 — e for any small
e > 0, then an is in C2~c and on is not Lipschitz continuous. This example
shows that assumption that a^G C2 in Theorem 1.2 is sharp.

132 6 ELLIPTIC AND PARABOLIC EQUATIONS
Remark 3. If a(x) is nonnegative definite in a closed domain G with
smooth boundary, and if ai;.eC°°(G), there does not exist in general a
nonnegative definite matrix a in G that is Lipschitz continuous and satisfies
a2 = a in G. For example, if n = 1 and G = {x; 0 < x < 1}, au(x) = x,
then an(x) = Vx is not Lipschitz continuous.
2. The maximum principle for elliptic equations
Consider the linear partial differential operator
L«^S ф) -^г- + 2 b,(x) f- +c(x)u B.1)
with real coefficients defined in an n-dimensional domain D. L is said to be
of elliptic type (or elliptic) at a point x° if the matrix (а„(х0)) is positive
definite, i.e., for any real vector ? ф О, 2а(;(х°)?^ > 0.
Lemma 2.1. Let (aJx)) be a nonnegative definite matrix for xED, and let
c(x) < 0. If u(x) is in C2(D) and if it attains a positive maximum in D at a
point x°, then Lu(x°) < 0.
Proof. If A and В are nonnegative definite matrices, then tr(AB) > 0.
Indeed, if X is an eigenvalue of AB, then ABx = Xx for some x Ф- 0. Taking
the scalar product with Bx we see that X > 0. Since tr(AB) is equal to the
sum of the eigenvalues of AB, it follows that tr(AB) > 0.
Applying the previous result to A = ЦАх0)), В = —(uxx (x0)) we con-
conclude that 2а(/(х°)ы*,*((*°) < °- Since also M*,(*°) = °> ФЧЦх0) < 0, the
assertion Lu(x°) < 0 follows.
The weak maximum principle is the following theorem:
Theorem 2.2. Let (atj(x)) be nonnegative definite matrix for all x in a
bounded domain D, and assume that c(x) < 0 and
|ап(х)Л2 + Ьг(х)Х > 0 for some Л > 0 and for all xGD.
I/u?C2(D)П C°(D) and Lu > 0 in D, and if max^ u(x) > 0, then
l.u.b.u(x) < maxu(x)
where 3D is the boundary of D.
Thus, if и takes positive values in D, then the maximum of и in D is
attained on 3D; it may also be attained at points of D.
Proof. If the assertion is not true, then there is a point xED such that

2. THE MAXIMUM PRINCIPLE FOR ELLIPTIC EQUATIONS 133
u{x) > max3D u{x). Consider the function
k(x) = ехр(Лх°) - exp^x^
where x° is a real number such that хг < xj for all x = (xlt . . . , xn) in D. It
is clear that k(x) > 0 and Lk(x) < 0 in D. If t is positive and sufficiently
small, then the function v = и — ek satisfies: v(x) > max3D v(x). Hence v
assumes a positive maximum in D at some point i° 6 D. By Lemma 2.1,
Lv{x°) < 0, i.e., Lu(x°) - eLk(x°) < 0. Since Lk(x°) < 0, we arrive at the
inequality Lu(x°) < 0; this contradicts the assumption that Lu > 0 in D.
Remark. If we apply the weak maximum principle to — u, we obtain the
following assertion: If the coefficients of L satisfy the conditions imposed in
Theorem 2.2, and if u?C2(D)nC°(D), Lu < 0 in D and и assumes a
negative minimum in D, then
g.l.b. u(x) > min u(x).
D *e3D
Definition. If there is a positive constant ju, such that
4=1
for all x E D, | G Л ", then we say that L is uniformly elliptic in D.
The strong maximum principle for elliptic operators is the following
theorem.
Theorem 2.3. Let L be a uniformly elliptic operator with bounded
coefficients in compact subsets of D, and assume that c(x) < 0 in D. If
«E C2(D) and Lu > 0 in D, and if u(x) ^ const, then и cannot attain a
positive maximum in D.
Applying this result to — и we conclude that if Lu < 0 in D and
u(x) ^ const, then и cannot attain a negative minimum in D.
Proof. We shall assume that и takes a positive maximum M at some point
x°ED, and derive a contradiction. Since и ^ const, there is a point i'eD
such that u{xl) < M. Connect xl to x° by a continuous curve у lying in D.
Then there is a first point x2 along у such that u(x) < M if x lies on у
between x1 and x2, and ы(х2) = M (x2 may coincide with x°). Take a point
x* on у between x1 and x2 such that the ball {x; |x — x*| < |x2 - x*|} lies
in D. Let x** be the nearest point of the set {xED; u(x) = M} to x*.
Clearly, the closure of the ball B* - {x; |x - x*| < |x** - x*|} lies in D,
u(x) < M for all xGB*, and u(x**) - M. Take a ball В contained in B*
whose boundary dB touches the boundary of B* at x**. Then u(x) < M for
all xE В и ЭВ, х * x**. Denote the tenter of В by x.

134 6 ELLIPTIC AND PARABOLIC EQUATIONS
Denote by E a closed ball with center x** and radius < \x** — x\ lying in
D. Its boundary consists of a part Ex lying in В and a part E2 lying outside B,
On Ev и < M — 8 for some 8 > 0.
Set R = |x** — x|, and consider the function
h(x) = exp[-a|x - x\2] - exp[-afl2].
It satisfies: h > 0 in B, h < 0 outside B, /i = 0 on 3B, and Lh > 0 in E if a
is sufficiently large. Consider the function v = и + eh (e > 0). If e is
sufficiently small, then v < M on Ev v < M on E2, and Lv > 0 in E. By
Lemma 2.1, и cannot take a positive maximum in the interior of E. Since
however v(x**) = M > 0, we get a contradiction.
Consider the problem of finding a solution и of
Lu(x) = /(x) in Д B.2)
u(x) = ф(х) on 3D. B.3)
This is called-the Dirichlet problem or \he first boundary value problem.
A barrier wy(x) at the point t/E 3D is a continuous nonnegative function
in D that vanishes only at the point у and for which Lw (x) < — 1.
If there exists a closed ball К such that К П D - 0, К n D = { y}, then
u>w(x) = fc{|x° - y\~p - |* - УГР} (*° = center of K) B.4)
is a barrier at t/ provided к and p are sufficiently large.
Theorem 2.4. Assume that L is uniformly elliptic in D, that c(x) < 0, and
that a.-, bt, c, f are uniformly Holder continuous (exponent a) in D. If every
point of 3D has a barrier and if ф is a continuous function on 3D, then there
exists a unique solution и in C2(D)n C°(D) of the Dirichlet problem B.2),
B.3).
For the proof of existence the reader is referred to Friedman [1, 2]. The
uniqueness follows from the maximum principle.
3. The maximum principle for parabolic equations
Consider the partial differential operator
Mu^\ 2 ф, t) ?L- + ± bt(x, t) %+c(x. t)u - % C.1)
л i,;-l axi ax1 i-1 axi at
with real coefficients defined in an (n + l)-dimensional domain Q. If
2a(/(x0, t%§ > 0 for all ?ЕЯ", ? Ф 0, then we say that M is of parabolic
type at (x°, t°). M is uniformly parabolic in Q if there is a positive constant /i

3. THE MAXIMUM PRINCIPLE FOR PARABOLIC EQUATIONS 135
such that
s Щ №f for ^ (*. *) G Q> fG я"-
Suppose Q lies in the strip 0 < t < T, and define
DT = {(x, Г); (x,T - 8) e @ for all 0 < 5 < 80, 80 depending on x},
<?o = Q U DT, dQ = boundary of Q, d0Q =dQ\ (int DT).
The weak maximum principle is the following theorem:
Theorem 3.1. Let (atj(x, t)) be nonnegative definite matrixjor all (x, t) in a
hounded domain Q, and assume that c(x, t) < 0. If и E C°(Q) and ux , ux x ,
ut belong to C°(Q0), and if Mu > 0 in Qo and maxg и > 0, then
l.u.b. u(x, t) < max u(x, t).
()sQ ()eaQ
Proof. If the assertion is false, then, for some e > 0 sufficiently small, the
function v = и — et takes positive maximum in Q at a point (x°, t°) in
QUDT. If t° - Г, then vt(x°, t°) < 0; and if t° < T, then vt(x°, t°) = 0.
Employing Lemma 2.1 we conclude that Mv(x°, t°) < 0. Since however
Mu > 0 in @0 and M(-et) = -cet + e > 0, we have Mv(x°, t°) > 0, a
contradiction.
For any point P° = (x°, t°) in Q, denote by S (P°) the set of all points P in
Q that can be connected to P° by a simple continuous curve in Q along
which the t-coordinate is nondecreasing from P to P°. We denote by С (Р°)
the component, in ( = t°, of Q П {t = t0} that contains P°. Notice that
C(P°) с S(P°).
The strong maximum principle for parabolic equations states the follow-
following.
Theorem 3.2. Let M be uniformly parabolic in a domain Q, with bounded
coefficients, and assume that c(x, t) < 0. If u, ux ,uX)X, ut are continuous in
Q and Mu > 0 in Q, and if и takes a positive maximum in Q at a point
P° = (x°, t°), then u(P) = u(P°) for all P e S(P°).
We need a few lemmas.
Lemma 3.3. Let Mu > 0 in Q and suppose that и takes a positive
maximum К in Q. Suppose that Q contains a closed solid ellipsoid E:
2 \(*( - **J+\0(t - t*)\ R2 (A,>0, Д>0),
и < К in the interior of E, and u(x, t)sxK for some point P •= Cc, t) on
the boundary Э? of E. Then x ™ x* where x* = (x*, . . . , x*).

136 6 ELLIPTIC AND PARABOLIC EQUATIONS
Proof. We may suppose that P is the only point on 3? where и = К since
otherwise we confine ourselves to a smaller ellipsoid lying in E and touching
3E at P only. Suppose x Ф x* and let С be an (n + l)-dimensional closed
ball contained in Q with center P and radius < \x —x*\. Then
\x - x*\ > const > 0 for all (i,()eC. C.2)
The boundary of С is composed of a part C2 lying in E and a part C2 lying
outside E. Clearly
и < К — 8 on Cj, for some constant 8 > 0.
Consider the function
= expj - J ?
h(x, t) = expj - J ? Л.(х. - x*J + yt - f Jj j - ехр{-«Л2}
where a is a positive constant. Clearly h > 0 in the interior of E, h = 0 on
ЭЕ, and /i < 0 outside E.
Using C.2) one easily verifies that Mh > 0 in С if a is sufficiently large.
Consider now the function v = и + eh in C, where t > 0. If e is sufficiently
small, then и < К on both Cj and C2. Since o(P) = K, it follows that и takes
its positive maximum in С at an interior point P = (x,t). By Lemma 2.1,
Mv < 0 at P. Since, however, Mv = Mu + Mh > 0 in C, we get a contra-
contradiction.
Lemma 3.4. If Mu > 0 in Q and if и takes a positive maximum in Q at a
point P° = (x°, t°), then u(P) = u(P°) for all P G C(P°).
Proof. If the assertion is not true, then there exists a point P1 = (x1, t°) in
C(P°) such that u{Pl) < u(P°). Connect P1 to P° by a simple continuous
curve у lying in C(P°). Then there exists apoint P* = (x*, t°) on у such that
u(P*) = u(P°) and m(P) < u(P°) for all P on у between P1 and P*. Take
one such point P = (ж, t°) whose distance to the boundary of Q is
>2|P-P*L
Since u(P) < u(P*), there exists a sufficiently small interval a = {(x, t);
t° - t < t < t° + e} such that
u(P) < u(P*) for all PEa. C.3)
Consider the family of ellipsoids ?x:
\x~x\2 + X(t- t°f< R2 (Л>0).
Take Л 2 = Ле2 so that the end points of a lie on the boundary of ?x. If Л—»0,
E^-^kj. As Л increases, Exn {( = ?0} increases; for sufficiently large Л it will
contain the point P*. Hence, because of C.3), there is a first value of Л, say
Л = Ло, such that и < u(P*) in the interior of EXo and м = u(P*) at some

3. THE MAXIMUM PRINCIPLE FOR PARABOLIC EQUATIONS 137
boundary point P = (y,T) of Ex<). Since и < u{P*) on a, it is clear that
у =? x. This contradicts Lemma 3.3.
Lemma 3.5. Let R be a rectangle
x? - at < x{ < x° + at (i = 1, . . . , n), t° - a0 < t < t°
contained in Q, and let Mu > 0 in Q. If и takes a positive maximum in R at
the point P° = (x°, t°), where x° = (x?, . . . , x°), then u(P) = u(P°) for all
РЕЛ.
Proof. If the assertion is not true, then there is a point P in Л such that
u(P) < u(P°). Since then и < u(P°) in a neighborhood of P, we may
assume that P does not lie on t = t°. On the straight segment у connecting P
to P° there exists a rx>int P1 such that u(Pl) = u(P°) and м(?) < м(Р2) for
all P on у between Pjind Pl. Without loss of generality we may assume that
P1 = P° and that P lies on t = t° — a0, for otherwise we can confine
ourselves to a smaller rectangle.
Denote by Ro the subset of R consisting of all points with ( < t°. Since,
for "every Pl in Ro, C(P1) contains a point of y, and since и < u(P°) on y,
Lemma 3.4 implies that «(P1) < u(P°).
Consider the function
h(x, t) = t°- t- A\x- x°\2 (A > 0).
Clearly h = 0 on the paraboloid П: t° - t = A\x - x°|2, h < 0 above П,
and h > 0 below П. Also, as easily seen, Mh > 0 in R if 4А2ай < 1 in Д
and if the dimensions of Л are sufficiently small.
П divides Л into two regions. Denote by Л' the lower region. Its upper
boundary B' touches f = (° at the point P° only. Hence on the comple-
complementary boundary B" of Л' we have и < ы(Р°) — S for some 8 > 0. It
follows that if e > 0 is sufficiently small, then the function v = и + eh
satisfies v < u(P°) on B". Next, v = и < u(P°) at all the points of B' other
than P°. Since Mv = Mu + eMh > 0 in Л', v cannot take a positive
maximum in Л' at an interior point (by Lemma 2.1). We thus conclude that
the maximum of v in the closure of R' is attained at just one point, namely,
at P°. This implies that 3u(P°)/3( > 0. Since 3/i(P°)/3( = -1, we con-
conclude that
3M(P°)/3t > 0. C.4)
Since however и takes a positive maximum at P°, Lemma 2.1 implies that
Mu < -ди/dt at P°. But since we have assumed that Mu > 0 in Q,
du(P°)/dt < 0. This contradicts C.4).
Proof of Theorem 3.2. Suppose и Ш u(P°) in S(P°). Then there exists a

138 6 ELLIPTIC AND PARABOLIC EQUATIONS
point P in S(P°) such that u(P) < u(P°). Connect P to P° by a simple
continuous curve у lying in S(P°) such that the ^-coordinate is nondecreasing
from P to P°. There exists a point P1 on у such that u(Pl) = u(P°) and
u(P) < «(P1) for all P on у lying between P and P1. Construct a rectangle
Й:
x{ - a < x{ < X* + a (t = 1, . . . , n), f1 - a < t < f1
where P1 = (xj, . . . , x^, t1) with a sufficiently small a so that R is contained
in Q. Applying Lemma 3.5 we conclude that и = u(P1) in R. This contra-
contradicts the definition of P1.
Let Q be a bounded domain in the (n + l)-dimensional spaceof variables
(x, t). Assume that Q lies in the strip 0 < t < T and that В = Q П {t = 0},
BT = Q П {t = T) are nonempty. Let BT = interior of Вт, В = interior of
B. Denote by So the boundary of Q lying in the strip 0 < t < T, and let
S = So \ BT. The set d0Q = В U S is called the normal (or parabolic)
boundary of Q.
The first initial-boundary value problem consists of finding a solution и of
Mu(x, t) = f(x, t) in Q\jBT C.5)
u{x, 0) = ф(х) on В, C.6)
u(x, t) = g(x, t) on S, C.7)
where /, ф, g are given functions. We refer to C.6) as the initial condition
and to C.7) as the boundary condition. If g = ф on В n S, then the solution и
is always understood to be continuous in Q.
A function wR (P) (й Е В U S) is called a bonier at the point R if u;H (P) is
continuous in PE@, u;H(P) > 0 if PGQ, P =? R, wR(R) = 0, and
MwR < -1 in Q и Вт.
Suppose Q is a cylinder В X @, Г), and assume that there exists an
n-dimensional closed ball К with center x such that KnB = 0, К Г\ В
= {x0}. Then there exists a barrier at each point H = (x°, t°) of S @ < t°
< T), namely,
where у > c(x, t), Ro = |x° - S|, R = [|x - x\2 + (t - tf]1/2, and к, р are
appropriate positive numbers.
Theorem 3.6. Assume that M is uniformly parabolic in Q, that a^, bt, c, f
are uniformly^ Holder continuous in Q, and that g, <j> are continuous
functions on B, S respectively and g = ф on В П S. Assume also that there
exists a barrier at every point of S. Then there exists a unique solution и of
the initial-boundary value problem C.5)-C.7).

4, THE CAUCHY PROBLEM AND FUNDAMENTAL SOLUTIONS 139
The solution и has Holder continuous derivatives ur , u, , , u..
For the proof of existence we refer the reader to Friedman [1]. The
uniqueness follows from the inequality
тах|и| < еаТтахЫ if Ми = 0 and c(x, t) < a in p. C.8)
Q Bus
This inequality follows from the weak maximum principle applied to
v = ue~yt.
4. The Cauchy problem and fundamental
solutions for parabolic equations
Let
L^H ф, t) -?±- + ? bt(x, t) |« + c(x, t)u D.1)
be an elliptic operator in R" for each tE[0, T]. Consider the parabolic
equation
9«(x, t)
Mu = Lu(x,t) -?~ =f(x, t) in R"X@,T\ D.2)
with the initial condition
и{х,0) = ф{х) on Я". D.3)
The problem of solving D.2), D.3), for given /, ф, is called the Cauchy
problem. The solution is understood to be continuous in R " X [О, Г] and to
have continuous derivatives ux , ux x , щ in R" X @, T].
We first prove uniqueness.
Theorem 4.1. Let (а^(х, t)) be nonnegative definite and let
\ait{x, t)\ < C, |b,(x, t)| < C(|x| + 1), c(x, t) < C{\x\2 + l) D.4)
(C cotwtont). If Mu < 0 in R" X @, Г] and
u(x, *) > -Вехр[^|х|2] in Rnx[0,T] D.5)
/or some positive constants B, /3, and if u(x, 0) > 0 on R", then u(x, t) > 0
in«" X [0, Г].
Proof. Consider the function

140 6 ELLIPTIC AND PARABOLIC EQUATIONS
One easily checks that, for every к > 0, if ju and v are sufficiently large
positive constants, then MH < 0. Consider the function v = u/H and take
к > /3 and ft, v such that MH < 0. From D.5) it follows that
lim inf o(x, t) > 0 D.6)
|x|-»oo
uniformly in t, 0 < t < l/2ju. Further, v satisfies
MV=2 24 1^ +2** ^+-
where / = (Mu)/H < 0 and
By D.6), for any e > 0, v(x, t) + i > 0 if |x| = R, 0 < t < l/Bjn) pro-
provided Й is sufficiently large. Also M(v + e) < ce < 0 if 0 < t < l/Bju) and
v(x, 0) + e > 0 if |x| < R. By the maximum principle, o(x, <) + e > 0 if
|x| < Й, 0 < t < l/2ju. Taking R -^ oo and then e | 0 it follows that
o(x, t) > 0 if 0 < t < l/2ju. Hence u(x, *) > 0 if 0 < t < l/2/i. We can
now proceed step-by-step to prove the nonnegativity of и in the strip
0 < t < T.
Corollary 4.2. If (aJx, t)) is nonnegative definite and D.4) holds, then
there exists at most one solution и of the Cauchy problem D.2), D.3)
satisfying
\u{x,t)\ < Bexp[0|x|2]
where B, /3 are some positive constants.
The next result on uniqueness has different growth conditions on the
coefficients of L.
Theorem 4.3. Let (ajx, t)) be nonnegative definite and let
k/x, t)\ < C(|x|2 + 1), |M*> t)\ < C(|x| + 1),
c{x,t)<C (C constant). D.7)
I/Ми < 0 in R" X @, Г] and
u{x,t) > -M№ + 1) in й"х[0, Г] D.8)
where N, q are positive constants, and ifu(x, 0) > 0 on R", then u(x, t) > 0
inR" X [0, T].
Proof. For any p > 0, the function
w(x, t) = (|x|2 + Ktfeat

4. THE CAUCHY PROBLEM AND FUNDAMENTAL SOLUTIONS 141
satisfies Mw < 0 in R" X [О, Г] provided K, a are appropriate positive
constants. Take 2p > q and consider the function v = и + ew for any
e > 0. Then Mv < 0. Since v(x, 0) > 0 and (by D.8)) e(x, t) > 0 if |x| = R
(R large), 0 < t < T, the maximum principle can be applied (to ve~a) to
yield the inequality o(x, *) > 0 if |x| < R, 0 < t < T. Taking first й^оо
and then e—»0, the assertion follows.
Corollary 4.4. Let (о,Дх, ?)) be nonnegafttie definite and let D.7) ЬоИ. Then
there exists at most one solution of the Cauchy problem D.2), D.3) satisfying
|«(x, t)\ < N{1 + \x\4)
where N, q are some positive constants.
Definition. A fundamental solution of the parabolic operator L — д/dt in
Й" X [0, Г] is a function Г(х, t; 4, т) defined for all (x, t) and (^, т) in
R " X [0, Г], t > r, satisfying the following condition:
For any continuous function f(x) with compact support, the function
Г(х, t; €, t)/({) d€ D.9)
satisfies
Lu-du/dt = 0 if x?Rn, т < t < T, D.10)
u{x,t)^f(x) if (|r. D.11)
We shall need the conditions:
(Ax) There is a positive constant ju such that
2( ) ft|4|2 for all (x, t)SRn x[0, Г],
(A2) The coefficients of L are bounded continuous functions in
R" X [0, Г], and the coefficients atj(x, t) are continuous in t, uniformly with
respect to (x, t) in R" X [0, T].
(A3) The coefficients of L are Holder continuous (exponent a) in x,
uniformly with respect to (x, t) in compact subsets of Й" X [0, Г]; further-
furthermore, the a«(x, t) are Holder continuous (exponent a) in x uniformly with
respect to (x, t) in R" X [0, T].
Theorem 4.5. Let (AX)-(A3) hold. Then there exists a fundamental solution
T(x, t, ?, t) for L — Э/Э* satisfying the inequalities
Ю-Г(х. «s i т)| < C(t - T)-<« + H)/2 exp| _ c ]^J
for \m\ «¦ 0, 1, where С, с are positive constants. The functions
О,тГ(х, t; i, t) @ < \m\ < 2) «nd О,Г(х, (; |, т) are соп(тиои5 in

142 6 ELLIPTIC AND PARABOLIC EQUATIONS
(x, t, €, t) G Л" X [О, Г] X К" X [0, Г], t > т, and LT -дТ/dt = 0 as a
function in (x, t). Finally, for any continuous bounded function f(x),
f
JR"
if «|t. D.13)
The construction of Г can be given by the parametrix method; for details
regarding the construction of this Г and the proof of the other assertions of
Theorem 4.5, see Friedman [1; Chapter 9]. (The assertion D.13) follows from
Friedman [1, p. 247, formula B.29); p. 252, formula D.4), and the estimate
on Г - Z].)
The following theorem is also proved in Friedman [1]:
Theorem 4.6. Let (Aj)-(A3) hold. Let f(x, t) be a continuous function in
R" X [О, Г]), Holder continuous in x uniformly with respect to (x, t) in
compact subsets, and let ф(х) be a continuous function in R". Assume also
that
|/(x, t)\ < A exp(o|x|2) in Rn X[O, T], D.14)
|ф(х)| < Aexp(o|x|2) in R" D.15)
where A, a are positive constants. Then there exists a solution of the Cauchy
problem D.2), D.3) in the strip 0 < t < Г* where T* = min{T, с /a) and с
is a constant depending only on the coefficients of L, and
\u(x,t)\ < A' exp(a'\xf) in RnX[0,T*] D.16)
for some positive constants A', a'. The solution is given by
u{x, t)= f Г(х, t; i 0)ф(?) di- \ f T(x, t; i, r)f{i t) d? dr. D.17)
JR" J0 JR"
The formal adjoint operator M* of M = L — 9/ 9t, where L is given by
D.1), is given by
M*v = L*v +dv/dt,
L*v = \ § ф> *) "Й- + 2 b*(x- ') P- +c*(x> t)v DЛ8)
2 4i._j dxt dxi ij = 1 dxt
where
2 ф, дх{ Эх,.
It is assumed here that Эа^/Э^, Э^/Эх, dxjt ЭЬ(/9х^ exist and are bounded
functions.

4. THE CAUCHY PROBLEM AND FUNDAMENTAL SOLUTIONS 143
Notice that
M*v = - 2 Э2 (dyv) - 2 — (V) + cu + — . D.20)
i,j = 1 i j t= 1 »
Using this form of M*v, one can easily verify Green's identity
n n ( /)n
uMm — им*v = >. -r— — >. vau — май — «c^r
i = i <Ц 2 ,_ x \ *' dx# ' dx# о;
--^(uo). D.21)
Thus, if u, v have compact support in a domain G, then
ЛГ (vMu - uM*v) dx dt = 0. D.22)
G
Definition. A fundamental solution of the operator L* +Э/Э* in
K" X [0, Г] is a function Г*(х, t; ?, т) defined for all (x, t) and (?, т) in
R" X [0, Г], t < т, satisfying the following condition:
For any continuous function g(x) with compact support, the function
o(x, *) = f Г*(х, «; €¦ T)g(€) d?
JR"
satisfies
L*v +dv/dt = 0 if хЕЙ", 0 < t < т,
We shall need the following condition:
(A4) The functions
— Э2 , Э
are bounded functions, and the coefficients of L* satisfy the conditions (A2),
(Аз)-
Theorem 4.7. If (AX)-(A4) hold, then there exists a fundamental solution
T*(x,t;t,r)qfL* +d/dt,and
Г(х, *;?, т) = Г*(?,т;х, *) (t > t). D.23)
Proof. The construction of Г* can be carried out in the same way as for Г.
Further, Г* satisfies inequalities similar to D.12) and L*T* + dT*/dt = 0 as
a function of ({, t). Thus it remains to prove D.23). Consider the functions
u( y, a) - Г( i/, a; {, т), о( у, ст) - Г*(«/, a; x, t)

144 6 ELLIPTIC AND PARABOLIC EQUATIONS
for i/ей", т < a < t. Integrating Green's identity D.21) over the domain
| y\ < R, T + e<a<t-e (R > 0) and using the relations Mu = 0,
M*v = 0, we obtain
f u(y,t - e)v(y,t- e)dy - f u{ у, т + е)о( у, т + e) dy = It „
where
<, я = Z I I
2 , = \ '
' 9У/ 9 /
cos(?', i/j) dS da;
>> is the outwardly directed normal to | t/j = R and dS is the surface element
on | y\ = Й. Using D.12) and the corresponding inequalities for Г* we find
that Ie я^0 if й^оо. Hence
f u(y,t- t)T*{ y,t-t; x, t)dy = ( v( у, т + е)Г( у, т + e; 4, т) dt/.
Уяп JRn
D.24)
Taking e | 0 the assertion D.23) follows; cf. Problem 10.
5. Stochastic representation of solutions
of partial differential equations
Let L be an elliptic operator in a bounded domain D given by B.1). Assume
that L is uniformly elliptic in D, i.e.,
2 а^хЩ > (л\е if xED, ?e«n (fi>0). E.1)
Assume also that
atj, b{ are uniformly Lipschitz continuous in D, E.2)
с < 0, с uniformly Holder continuous in D. E.3)
Assume finally that the boundary 3D of D is in C2, so that barriers exist at
all the points of 3D (see Problem 2).
Then, by Theorem 2.4, the Dirichlet problem B.2), B.3) has a unique
solution и for any given functions /, ф satisfying:
/ is uniformly Holder continuous in D, E.4)
ф is continuous on 3D. E.5)
We shall now represent и in terms of a solution of a stochastic differential
system.

5. STOCHASTIC REPRESENTATION OF SOLUTIONS 145
By Lemma 1.1 and A.1) it is clear that the nonnegative definite square
root o(x) = (<r^(x)) of the matrix a(x) = (а^(х)) is Lipschitz continuous in D.
Extend o(x) and b(x) = (b1(ac) bn(x)) into the whole space R" so that
\a(x)-a(y)\ < C\x-y\, \b(x) - b( y)\ < C\x - y\. E.6)
Consider the system of stochastic differential equations
d?(*) = аШ) dw(t) + b{t(t)) dt. E.7)
Denote by Vt the closed 6-neighborhood of 3D and let D( = D\V€. Let v
be a function in C2(R") that coincides with the solution и of B.2), B.3) in
De/2, and let т be any Markov time with respect to the time-homogeneous
Markov process solution of E.7). By Ito's formula,
-v(x)
Г [Lc(€(t))] expf Г c(H(s)) ds] dt.
¦'o L ¦'o J
E.8)
Take x G Д and т = те л Т where те is the hitting time of V€. Then
о(?(*)) = "(К*)) for all 0 < t < т? л Т. Hence E.8) holds for v = u. Taking
e —»0 and using the Lebesgue bounded convergence theorem, we get
u(x) = Ехи(?(т л Г)) ехр[|оТЛГс(^*)) ds\
1
(€(*)) Л А E-9)
where т is the exit time from D.
Theorem 5.1. Let E.1)-E.5) hold and let 3D belong to C2. Then the
unique solution и of the Dirichlet problem B.2), B.3) is given by
u(x) =
-Ex f'Mt)) exp[jT'c(€W) d»] A E.10)
where т is Ле exit time /rom D.
Proof. If we prove that
Exr < oo E.11)
then, by taking T f oo in E.9) and using the Lebesgue bounded convergence
theorem, we get the assertion E.9).
To prove E.11) consider the function
h(x) - -Л***'

146 6 ELLIPTIC AND PARABOLIC EQUATIONS
for all x = (xv . . . , xn) in D. If A,X are sufficiently large (A depending on
\), then
2V44+2V>4< -1 inD-
By Ito's formula,
ЕхЦ?(т A T)) - h(x) < -Ех(т а T).
Since \h(x)\ < К in D (K constant), we then have Ех(т A T) < 2K. Taking
Г | oo and using the monotone convergence theorem we get Ехт < 2K; this
proves E.11).
Remark. The proof of E.11) requires only that an(x) > 0 in D. Thus the
right-hand side of E.10) makes sense even if 3D is not in C2, L is not elliptic
but has bounded coefficients with c(x) < 0, an(x) > 0 in D, a = oo*, and o,
b are Lipschitz continuous in D. Recall that, by Theorem 1.2 and Remark 1
following it, if (a,,) is nonnegative definite and in C2 in some neighborhood
of D then о is Lipschitz continuous in D.
Consider next the initial-boundary value problem (written for t replaced
by T - t)
Lu + Эм/dt = f{x,t) in Q = В X [О, Г),
и(х,Т)=ф(х) on В, E.12)
u(x, t) = g(x, t) on S,
where В is a bounded domain with C2 boundary dB, S = dB X [О, Г), and L
is defined by D.1).
Consider also the system of stochastic differential equations
ОД = аШ, t) dw(t) + b(€(«), t) dt E.13)
where (o(x, t)J = a(x, t) in Q. The coefficients o, b here are extensions of
o(x, t), b(x, t), originally defined in Q, such that
\o(x,t) - o{y,s)\ < C(\x- y\ + \t- s\),
\b(x,t)-b(y,s)\< C(\x - y\ + \t - s\).
We shall assume:
, f*|l|2 if {x,t)EQ, | G R" (ft > 0),
atp Ь( are uniformly Lipschitz continuous in (x, t)E Q,
с is uniformly Holder continuous in (x, t) G Q,
f is uniformly Holder continuous in (x, t)G.Q,
g is continuous on S, ф is continuous on В and
g(*. Т)-ф(х) if жЕЭВ.

By Theorem 3.6 there exists a unique solution и of E.12).
Theorem 5.2. Let ЭВ belong to C2 and let E.14) hold. Then the unique
solution и of the initial-boundary value problem E.12) is given by
!s E.15)
where т is the first time X G [t, T) that ?(\) leaves В if such a time exists
and т = Г otherwise.
The proof of Theorem 5.2 is similar to the proof of Theorem 5.1. Here one
applies Ito's formula to
,\)exp\f\(Z(s),s)ds]. E.16)
Consider next the Cauchy problem
Lu + ди/dt = f(x, t) in K" X [0, T),
E.17 )
u(x, Т) = ф(х) in«"
where L is given by D.1).
We shall assume:
(i) The functions a^, bt are bounded in R " X [0, Г] and uniformly
Lipschitz continuous in (x, t) in compact subsets of R " X [0, T].
(ii) The functions a^ are Holder continuous in x, uniformly with
respect to (x,t)inR"X [0, T].
(B2) The function с is bounded in R" X [0, Г] and uniformly Holder
continuous in (x, t) in compact subsets of R" X [0, Г].
We shall also assume:
f(x, t) is continuous in R" X [0, T\, Holder continuous in x uniformly
with respect to (x, t) G R" X [0, T], and
|/(x, t)| < A(l + |x|°) in R" x[0, Г], E.18)
ф(х) is continuous in Л" and |ф(х)| < A(l + |x|°) E.19)
where A, a are positive constants.
Under the conditions (A,), (B,), (B2), E.18), and E.19), there exists a
unique solution и of the Cauchy problem E.17) satisfying
|u(*. «)| < const(l + |x|e). E.20)

Indeed, uniqueness follows from Corollary 4.2. The existence of и follows
from Theorem 4.6; the estimate E.20) on u(x, t) is an easy consequence of
the estimate D.12) (with m = 0) on Г and the assumptions E.18), E.19).
Using D.12) with \m\ = 1 we also get
\ux(x, t)\ < const(l + |x|°). E.21)
By Lemma 1.1 and A.1), the nonnegative definite square root a(x, t) of
a(x, t) is Lipschitz continuous in (x, t), uniformly in compact subsets of
R" X [0, Г]. It is also clear that a(x, t) is uniformly bounded in R" X [0, Г].
Thus (by results in Chapter 5) the stochastic system E.13) has a unique
solution for any initial value |@) = x.
Theorem 5.3. If (A^, (БД (В2) and E.18), E.19) hold, then the solution of
the Cauchy problem E.17), E.20) is given by
u(x, t) = ?ХЛФ(|(Г)) ехркГс(!E), s) ds
-EX;t ftTf(Z(s), s) explf c(€(X), \) d\\ ds. E.22)
The proof follows by applying Ito's formula to the function in E.16) and
the process E.13), and then taking the expectation. Since, by E.21),
\ux(x, t)a(x, t)\ < const(l + |x|°)
and since
sup ?Xit||(s)r < oo for any m > 0,
t<s< T
the stochastic integral occurring in Ito's formula has zero expectation. The
assumptions E.18), E.19) ensure that the expectations on the right-hand side
of E.22) exist.
Let
bo = | ? а,(х, t) -^ + ? b,(x, t) -JL E.23)
and denote by T$(x, s; y, t) the fundamental solution of Lo + 3/3s (s < t).
Taking / = 0, с = 0 in E.22), we get
u(x,t) = EXt${Z{T)). E.24)
By Theorem 4.6 (with t replaced by T — i) we can also represent и in the
form
u{x, t)-[ T*(x, t; у, Т)ф( у) dy.
JRn
Comparing this with E.24) we conclude that
Г TS(x,f,y, T)<!>(y)dy-EXtMZ(T)).
JR"
Notice that this relation holds for any function ф satisfying E.19).

5. STOCHASTIC REPRESENTATION OF SOLUTIONS 149
Similarly, for any 0 < s < t < T,
f T*(x,s;y,t)^(y)dy=Ex^(t))=f *(y)PXit(€(t)edy). E.25)
Since <J> is arbitrary function satisfying E.19), we conclude that the
transition probability function
p(s, x, t,A) = Px>f(€(t)GA) = P(^,(t)GA)
of the Markov process solution of E.13) has density, and that the density
function is Tq(s, x, t, y). We sum up:
Theorem 5.4. // (Ax), (Bx) hold, then the transition probability function of
the solution of the stochastic differential system E.13) has density, i.e.,
Px>,(€(t)GA) = f T*(x, s; y, t) dy (s < t) E.26)
•'A
for any Borel set A, and Tq(x, s; y, t) is the fundamental solution of
Lo +d/dt constructed in Section 4.
The density function of the transition probability function is called the
transition density function. From the results of Section 4 we conclude that
the transition density function Т?(х, s; y, t) of the solution ?(?) of E.13)
satisfies in (x, s) the backward parabolic equation
-jjj rj(x, S; y, t) + | 2 Ощ(х, s) ^ ^ T*(x, S; y, t)
+ 2 b,(x, s) -^- T*(x, s; y, t) = 0. E.27)
i=i axi
Under the assumptions of Theorem 4.7, Г„ (x, s; y, t) also satisfies in ( y, t)
the forward parabolic equation
- Yt r*(*'s; у- *) + | Д j^. Ыу> f)To(x' *; y. *)]
-ii^[bi(y.Oro*(X,5;y,0]=0. E.28)
Notice that under the conditions (Ax), (Bx) the backward equation E.27) is
equivalent to the Kolmogorov equation E.6.1).
If the coefficients a{j, b{ are independent of t, then Го(х, s;
y, t) = Г*(х, 0; y,t - s) = T(x, t - s, y); T(x, t, y) satisfies in (x, t) the
parabolic equation
ЭГ(«,*,у) j » ч Э2Г(х, t, у)
ЭГ(х, t, у)
Эу - «¦ E-29)

150 6 ELLIPTIC AND PARABOLIC EQUATIONS
Under the conditions of Theorem 4.7, it also satisfies in (y, t) the
parabolic equation
Щ*.*.у) . l А э2 r , ,r/ , .,
{5Щ
PROBLEMS
1. Verify that the function in B.4) is a barrier for L.
2. If D is a bounded domain with C2 boundary 3D, then for any у ? 3D,
there exists a closed ball К such that KnD = 0 and К nD = { y); thus, a
barrier exists.
3. Let L be as in Theorem 2.2 and let и be a solution of B.2), B.3) where
D is a bounded domain. Prove that
max|u| < max|<J>| + A max|/|
0 D С
where A is a constant independent of /, <p.
4. Let Lu = yV^ - ZxyUfy + xhiyy — xux — yuy. Verify that any C2
function и = f(r) (where r =\x2 + y2 ) satisfies Lu = 0. Hence the weak
maximum principle does not hold in any domain e2 < x2 + y2 < R 2. Notice
that (а„) is nonnegative definite, an + a^ > e2, but neither au nor a22 are
positive throughout the domain.
5. Let {пц(х, t)) be nonnegative definite, c(x, t) < a and
an(x, t)Xz + Ьх(х)Л > 1 in a cylinder Q = D_x [0, T]. Denote the diameter
of D by d. Show that if и is continuous in Q and Mu is bounded in Q (M
given by C.1)), then
max|u| < eaT{ max|u| + (eXd - l)max|Mu|)
Q K э0<? Q '
where d0Q is the normal boundary of Q.
6. Let the coefficients of M satisfy the conditions of either Theorem 4.1 or
4.3. Prove that a fundamental solution Г is uniquely determined by the
requirements:
(i) T(x, t; |, t) is continuous in ?, for fixed x, t, т;
(ii) for any continuous function / with compact support, the function
u(x, t) given by D.9) satisfies: \u(x, t)\ < C(l + \x\<>) in Й" X [0, Г], where
C, q are some positive constants (depending on /).
7. Let the coefficients of M satisfy the conditions of either Theorem 4.1 or
4.3 and let Г be a fundamental solution satisfying (i), (ii) of the preceding
problem. Prove that T(x, t; ?, т) > О.

PROBLEMS 151
8. The operator Д-Э/Э* (where Д is the Laplacian operator
S7_i32/3xf) is called the heat operator. Prove that
is a fundamental solution for the heat operator.
9. Let Г be the fundamental solution constructed in Theorem 4.5. Prove
that, for any e > 0,
f Г(х, t; fc t) d?->0 if t It,
Г(х, t; fc t) dx^O if t \, т.
f
J\x~
10. Let f(x, t) be a continuous bounded function in R " X [0, T], and let Г
be the fundamental solution constructed in Theorem 4.5. Prove that
Г Г(х, t; fc r)/(fc t) d^f(x, t) if ||т,
Уд»
Г Г(х, t; fc т)/(х, т) &->/(? т) if Uf.
•/Д"
[ffint: Use the preceding problem.]
11. Let Г(х, t; fc t) be the fundamental solution constructed in Theorem
4.5. Prove that
Г(х, t; fc т) - Г Г(х, t; у, a)T( у, a; fc т) dy (т < а < t).
JR"
12. Give another proof of Theorem 4.1, by applying the maximum principle
to и + ev in 0 < t < I/a, where
o(x, t) = exp{ C{\x\2 + l)eat}
with suitable constants a, /J.
13. Let M* be an operator of the form M*v = 2aj;uxx +2/8,0^
+ yv + Svt where aif, /3{, y, S are continuous functions. Prove that if D.22)
holds for any u, о inC00 with compact support in G, then M* is given by
D.20), i.e., M* is the formal adjoint of M.
14. Prove Theorem 5.2.
15. Consider two stochastic differential systems of n equations
d€ = a(fc t) dw + b(fc t) dt, dfc = a'(fc, t) dw' + b'(fc, t) dt
where w and uj' are n-dimensional Brownian motions. Suppose that the
coefficients are uniformly Lipschitz continuous and that aa* is uniformly
positive definite. Prove that if aa* = a'(a')*, b = b', then the two processes
have the same joint distribution functions, given that {@), fc@) have the
same distribution functions.

7
The Cameron-Martin-Girsanov
Theorem
1. A class of absolutely continuous probabilities
Any nonnegative random variable g defines an absolutely continuous mea-
measure Pg whose Radon-Nikodym derivative is g, i.e., PJ<A) =/a§ dP for any
measurable set A. In this section we show that the random variable
g = exp{ fTf(s) dw(s) - * fT\f(s)\2 ds
defines a probability Pg, i.e., Eg = 1, provided / belongs to L^[0, Г] and
satisfies a certain growth condition. More precisely:
Theorem 1.1. Let f = (fy, • • ¦ ,/„) belong to L%[Q, T] and assume that
there exist positive numbers ц, С such that
Eexp[ju|/(f)|2] < С for 0 < t < T. A.1)
Then
E exp{ (hf(s) dw(s) - | fV(*)l2 ds)=l if 0 < tt < t2 < T.
A.2)
We first prove two lemmas.
Lemma 1.2. If in Theorem 1.1 the condition A.1) is replaced by the
condition
\( \f(s)\2ds \< oo for some Л > 1, A.3)
-'о )
then the assertion A.2) is valid.
152

1. A CLASS OF ABSOLUTELY CONTINUOUS PROBABILITIES
153
Proof. By Lemma 4.1.1 (see D.1.5)) there exists a sequence of continuous
functions fm(t) = (/ml(t), . . . ,fmn(t)) in L>, Г] such that
A.4)
A.5)
We may assume that each fm is bounded, i.e., \fm(t, w)| < Cm a.s. in
(t, w), for otherwise we replace fm by fm N with components <j>N {fm^j, where
<j>N(r) = r if \r\ < N and ^N(r) = Nr/\r\ if \r\ > N, and Nm | oo if m | oo.
We claim that
E exp 2 / fm(s) dw(s) < Q, for all 0 < ^ < f2 < Г, A.6)
where C^ is a constant depending on m.
To prove it notice that
Eey\Mt)\=f ey\x\
,-\xf/2t
dx =
A.7)
Let П, = {s[, . . . , Sj?} be a sequence of partitions of [tv t2] with mesh
|П,|^0. Then
ton 2 /.
< Cjim 2k(s,'+1) - w{sl)\
where С is a constant (depending on m). By Fatou's lemma,
? exp
2 ft2fm(s) dw(s)
< lim E
l-KX>
|«j(s/+1) - Ws/)|
= lim П E exp[2C|u>(s/+1) - ic(s/)|].
By A.7), the right-hand side is equal to
= Jim П exp[2C2(s/+1 - s/)] = exp[2C2(f2 - tx)].
Thus A.6) follows.
Applying Ito's formula to и = ех and the process
//.(•) Ms) - k f *\Ш\2 da.

154 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
we obtain
f2 \fm(s)\*ds) - 1
. A.8)
Denote the last integral by j\ h(s) dw(s). By A.6) and the fact that
|/m(t)i < Cm it follows that
Ej \h{t)\2dt< oo.
Hence Ej J* h(t) dw(t) = 0. Taking the expectation in A.8) we then get
E exp{ f2fm(s) dw(s) - * Г'' |/m(s)|2 ds ) = 1. A.9)
We shall prove that
t t \ 1+t
I = E exp{ Г 2 L(s) dw(s) - k ( 21 L(s)l2 ds I < К A.10)
for some e > 0, where К is a constant independent of m. This would imply
that the sequence of random variables
Xm = expjj^/js) Ms) ~ к ?* \L(s)\2 ds]
is uniformly integrable. Since, by A.4),
fhfm(s)dw(s)I>ft2f(s)dw(s) as m^oo,
P i f'2 ri f*2 io \
•X,,,-» exp{ I f[s) dw{s) - \ I \f(s)\2ds}.
Hence, taking m->oo in A.19), the assertion A.2) follows.
It remains to prove A.10). By Holder's inequality,
I = E I expf A + t)ft'2fm(s) dw(s) - {-^- f*2\fm(sf
ds
exp
1/A + e)
:«pl(i
E exp
€JB+?) f.
fh\fm(s)\2ds
€/(! + €)

1. A CLASS OF ABSOLUTELY CONTINUOUS PROBABILITIES 155
By A.9) the first factor on the right is equal to 1. Using A.5) we get
I < < E exp
da\) = K.
If e is sufficiently small so that A + «JB + e) < 2Л, then, by A.3), К is a
finite number. This completes the proof of A.10).
Lemma 1.3. Under the assumptions of Lemma 1.2,
ft^t}j=l a.s. A.11)
for any 0 < tx < t2 <T.
Proof. Taking the conditional expectation of both sides of A.8) with
respect to *§г, we obtain the assertion A.11) for/ = fm. Using A.10) one can
easily justify passage to the limit m—»oo in the formula A.11) for f — fm.
Proof of Theorem 1.1. Let Л > 1. Since eXx is a convex function, Jensen's
inequality (see Problem 7, Chapter 1) gives
ds] = exp[ y^-j j'\(t" - t')\f(s)\*
Hence, if X(t" - t') < jm, then A.1) implies that
|/(s)|2ds| < oo.
j
Consequently, by Lemma 1.3,
f(s)dw(s) -\^ |/E)|2*}|^| = 1 a.s. A.12)
Now let t1 = st < s2 < ¦ • • < sm = t2 where X(sj + 1 — st) < jn. Then

156 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
by A.12). Proceeding similarly step-by-step, we arrive at the formula
?/ Ау1Л I I T" i С 1 /14Л I С I —— * I I T I Cll /If1 I I VJ" \ ^— 1 ОС / I I 4 1
\ CAIJI I I l о I iJUJU I о I о I I / \ /1 '*" I 1 i_7 ^ / "~ X # ct*3a \ * 'L*-' /
V I • 1 ^1 J )
Taking the expectation, A.2) follows.
Corollary 1.4. Under the assumptions of Theorem 1.1, the relation A.13)
holds for any 0 < tt < t2 < T.
Example. If /(*) = w(t), then condition A.1) is satisfied with jn < 1/2Г.
Corollary 1.5. Foranyf=(fl,..., /J in L^[0, T],
< 1, A-14)
< 1 a.s. A.15)
EexV\ft2f(s)dw(s)-$ft2\f(s)\2ds
dw(s) - \ f'2 \f(s
f
for any 0 < t x < f2 < T.
To prove A.14) we take m -^ oo in A.9) and use Fatou's lemma. To prove
A.15) we first note (by taking the conditional expectation in A.8)) that A.15)
holds (with " = ") for / = fm. Now take m -^ oo and use Fatou's lemma.
2. Transformation of Brownian motion
A process {w(t), 0 < t < T] that satisfies all the conditions imposed on an
n-dimensional Brownian motion (including continuity) in the interval
0 < t < Г is called an n-dimensional Brownian motion in the interval
[О, Г].
Let w(t) be an n-dimensional Brownian motion in an interval [0, Г]. Let
<^t be an increasing family of a-fields such that <3(w(A), X < i) is a subset of
C5t and f {w(\ + t) - w(t), 0 < Л < Г - t) is independent of f,, for all
*G[0, Г]. As in Chapter 3 (where w(t) and % were defined for all t > 0),
we can define L^[0, T] and stochastic integrals / *0 f dw @ < t < T) with
respect to the present family ^.
If P is a measure on (fi, ^) given by
P(A)=ffdP (Aef),
JA
then we write dP(w) = f(u>) dP(w).
Theorem 2.1. Let w(t) be an n-dimensional Brownian motion in [0, Г] and

2. TRANSFORMATION OF BROWNIAN MOTION 157
let ф = (фх, . . . ,фп)Ъе a function in L2 [0, T]. Define
Vs (Ф) = Гф(и) dw(u) - i С |<J>(u)|2 du, B.1)
w(t) = w(t) - f <|>(s) ds, B.2)
4 •'o
i [. B.3)
>)
P(S) = 1, B.4)
then w(t), 0 < f < T is an n-dimensional Brownian motion in the probabil-
probability space (fi, <3, P).
Recall that in Section 1 we have established sufficient conditions for B.4)
to hold. We have also proved (Corollary 1.5) that Р(Я) < 1 for any
2
Theorem 2.1 is due to Girsanov [1]. Cameron and Martin [1] have
previously established results of the same nature on nonlinear transforma-
transformations of Brownian motion. We shall refer to Theorem 2.1 as the Cameron—
Martin-Girsanov theorem.
We begin with some lemmas.
Lemma 2.2. If феЬ%,[0, T] and if \$(t)\ < с a.s., then, for any a > 1,
E expK' (ф)] < exp[ ^f^ (t - s)c2]. B.5)
Proof. By Corollary 1.5, E ехр[?/(аф)] < 1. Hence
E expK' (ф)] = E exp[f/ (a<J>)] exp[ ^f^ jT |ф(и)|2 du j
< E exp[^ (аф)] exp[ ^f^- (t - ,)c2] < exp[ ^f^ {t - ,)c2].
Denote by E the expectation with respect to P.
Lemma 2.3. Let <J> e L^[0, T]. Then for any nonnegative and % measur-
measurable random variable tj,
EV<E{V етр[Й(ф)]}, 0 < t < T. B.6)
// ф satisfies B.4), tfien
1, B.7)
1 а.5. B.8)

158 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
for all 0 < s < t < T, and
Er, = E {г, ехр[$*(ф)]} B.9)
for any ^t measurable random, variable tj for which Er, exists.
Proof. The assertion B.6) follows from
Er, = Er, ехр[{ог(ф)] = EE{r,
< Ет,
where A.15) has been used.
Assume now that B.4) holds. Using A.15) we have
1 = E ехр[?ог(ф)] = Е ехр[«(ф)]Е {exp[f/ [
< E exp[fosD»)]? {exp[f/ (ф)]| ff,}. B.10)
Denote by Bm (m > 0) the set on which
Since, by A.15), E {ехр[?(ф)]| f,} < 1 a.s., we conclude from B.10) that
j
But since
Г exp[fos(<#.)] dF< 1 and exp Ц(ф) > 0 a.s.,
•'a
it follows that P(Bm) = 0. Since m is arbitrary, the assertion B.8) follows.
The validity of B.7) is now obvious, and B.9) follows from
Er, = Er, ехр[Цф)]Е{ехр[{Дф)]|^} = Er,
Lemma 2.4. Let ф be as in Theorem 2.1 and let r, be any 4>st measurable
random variable for which Er, exists. Then
BЛ1)
Proof. Set yY = exp Ц(ф), y2 = exp &(ф). Let Л be any bounded and $,
measurable random variable. Then
Ё(ЛЧ) = Ё\?(чШ B.12)
We also have
E(\r,) = ?(Лт,у1У2) = ЕЛУ1Е(чуа|^). B.13)
If у is $, measurable, then, by B.8),
Ey - EyE(y2|^,) - EE[(rf№.] - Eyyt.

2. TRANSFORMATION OF BROWNIAN MOTION 159
Applying this to у = Лух? [(т/у2)| SJ, we get from B.13)
E(Aij) = Е\у1У2Е[(г,у2)\%] =
Comparing this with B.12), we get
Since Л is arbitrary bounded random variable that is <SS measurable, B.11)
follows.
Lemma 2.5. Let 4> be as in Theorem 2.1 and let {<j>N} be a sequence of
bounded functions in L^[0, Г] such that ?/(<#> iV)-> ?/(<#>) in probability as
N-*co. Then
f |exp SJ (O - exp s: (Ф)| dP -> 0 if N-+co. B.14)
•'a
Proof. By Theorem 1.1,
f exp Si (ф») dP= 1. B.15)
•'n
Denote by й^ € the set where
By assumption, Р(йдг_,) > 1 - 8N(i), SN(i) -*0iiN-^oo. Since exp ?/(ф) is
integrable,
f exp[f/ (Ф)] dP- 1 - Г exp[f/ (ф)] dF> 1 - e B.16)
if 2V is sufficiently large. Hence
From B.15) it then follows that
Г exp[C(
Since ф satisfies B.7), B.16) implies that
Г «р[?(ф)]?ЙЧе. B.18)
Ja\aN,€
Making use of B.17), B.18), we find that
Г |ехр^(ф)-ехрС(О1^
•'a
<f \<*p s: (*) - «*p $: {*")] &
+ Г [exp ? (ф) + exp tf (фw) ]
if iV is sufficiently large, and the proof of B.14) is complete.

In proving Theorem 2.1 we shall rely upon Theorem 3.6.2. Since w(t) is a
continuous process, it will suffice to prove that, for any 0 < s < t < Г,
E[w,(t) - wt(s)]\V. =0 a.s., B.19)
E[wt(t) - щ(*)Щ(г) - «5,(S)]|ff = 8,(t - s) a.s. B.20)
Lemma 2.6. The assertion of Theorem 2.1 is true if ф(*) is bounded.
Proof. We shall verify B.19), B.20).
We need the following special cases of the integrated form of Ito's
formula dtfiSJ = Si #2 + S2 dSi + d^ d?2:
Г /(*) dw(s) f g(s) dw(s) = f f(s) ¦ g(s) ds
Jt0 •/t0 Jt0
+ l'o(iy(u)dw(u)))f(s)dw(s)
+ f(rf(u)dw(u))g(s)dw(s), B.21)
ff(s) dw(s)f h(s) ds =f(f Ни) du)f(s) dw(s)
+ С ( Г'f(u) dw(u))h(s) ds. B.22)
Here / and g are n-dimensional functions in L%[t0, t] and h is a scalar
function in L^[t0, t].
We shall also need A.8) with fm = ф, i.e.,
By Lemma 2.4,
E[w,(t) - щ
E[w{(t) - w{(,)]\l +f*e*p[S; (Ф)]Ф(и) dw(u)]\V.
С(ф)] du\ %s} - ?(Jf' *(«) *«) «р[Г»]| ^ • B.23)
In the last equation we have used B.21) and also the fact that
?fj"h(tt)dto(tt) p, =0
160

if
Ef \h(u)\2 du<<* (ft = (hb . . . , ftj) B.24)
where, for one stochastic integral,
ht(u) = Гехр[ШШ dw(v), ft, = 0 if ;*<
and, for another stochastic integral,
h(u) = (wt(u) - wt(s))
Using the fact that |ф| К с (с constant) and the estimate B.5), one can easily
see that these two functions h(u) indeed satisfy B.24).
In view of B.8) the right-hand side of B.23) is equal to zero. Thus B.19)
holds.
In order to prove B.20), set
i(t) = w(t) - w(s), ф) = exp Ц(ф).
Then
[&,(*) - U>,(8)][wf(t) - Wt(8)]4(t)
l+f\(uL>(u) *»(«)] -[jT' ф,(и
- [ Jf' ф,(«) du i(t)V(t) + [ f* ф,(и) d
where /( (i = 2, 3, 4) denotes the (i + l)th term on the right.
Denote by as equality moduli stochastic integrals, i.e., A as В if A
= В + S/Jft^M) dwf(u). Then, using B.21), B.22), we find that
h = ?(*)jf \j" ЧШ°) dw(v)} dw^u) + Щ)?$(и)ч(и)ф(и) dw(u)
Next,
/2
161

162 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
Notice that due to the boundedness of ф and Lemma 2.2, all the
stochastic integrals f's h(u) dw(u) which were dropped in the final expres-
expressions for Jv J2, /3 are such that B.24) holds. Consequently,
E[wt(t) - щ(>)][й>^) - ю,(8)]\$, - d^t ~ s)
, +EJ4\$t. B.25)
-Ef Jf'[
wt(t) - wf
The first term on the right-hand side is equal to
't
W B.26)
The first term in B.26) is equal to
(«) - щ(и) ф
- щ(и)] \fu) du}\$s = 0
by B.8) and B.19). Hence the first term on the right-hand side of B.25) is
equal to the second term in B.26).
Treating the second term on the right-hand side of B.25) similarly, we
conclude that
E[w{(t) - t3,(*)][tS,(t) - йу(*)]|^ - 8(i(t - s)
by the definition of /4. This completes the proof of B.20).
Proof of Theorem 2.1. Let $N = (<|>f, . . . , ф^) be bounded functions in
L^[0, Г] such that
( \<j>N(t) - <l>(t)\2 dt-*O a.s. as
•'n

2. TRANSFORMATION OF BROWNIAN MOTION 163
Define wN(t) = (w?(t), ..., w?(t)) by
wN(t) = w(t) - f ф»(и) du.
Notice that
wN(t)-+w(t) a.s. B.27)
In view of Lemma 2.6, wN(t) is a Brownian motion in the interval [О, Г],
with respect to the probability space (Й, 'S, PN) where PN is defined by B.3)
with ф = <j>N. Consequently, for any 0 < t0 < tY < • • • < tk < T and
XY, . . . , \k real,
E exp| V3J 2 МщЫ(Ь) ~ Wit,-!)]} exp #(ф")
-expf-^ iV(^-^-i)}. B-28)
and for any 0 < t, s < T and Л, /i real,
E exp{ V^T [АйДО + м^(*)]} exp Гог(ф^)
= ехр[-(Л2/2)(]ехр[-(/12/2)«] if i Ф ]. B.29)
We shall need the following fact:
If aN-+a a.s., \aN\ < с (c const), and ?| fiN — j8|-»0,
then E |aNPN - a/3 \-^0. B.30)
To prove B.30) write
E\aN/3N - аЦ\ < E\ait\\pN-p\ + E\aN- a\\fi\
< cE\PN- 0\ + E\aN - a\ \ 0\.
The first integral on the right converges to zero by assumption, and the
second integral converges to zero by the Lebesgue bounded convergence
theorem. Thus B.30) follows.
Taking aN to be the first exponent on the left-hand side of B.28) and
taking /3N = exp Ho(<l>N), and recalling B.27) and Lemma 2.5, we see that the
assumptions in B.30) are satisfied. Hence, letting N-+oo in B.28) we obtain
E exp
j V^T 2 Л,ВД - ?,(*,_!)]} = expj - ?
This implies (see Problem 2, Chapter 3) that u>,(t), 0 < t < Г is a Brownian
motion in the space (fi, 3r, F).
If we let N-*co in B.29) and apply B.30) (again using B.27) and Lemma
2.5), we obtain
(() + ,,?,(,)]} - ехр[-(А2/2)(]ехр[-(М2/2L

164 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
This proves that Я{wt(t), 0 < t < T] is independent of ${w^t),
0 < t < Г} if i Ф /'. Thus w(t) is an n-dimensional Brownian motion in the
space (fi, f, P).
Remark. Since ш^ satisfies B.19), B.20), the remark at the end of Section
3.5 implies that, if 0 < s < t < T,
Eexp[iX ¦ (wN(t) - u>N(sj)]\$t = exp[- ||Л|2(* - «)] a.s. B.31)
If a№ ^N are as in B.30) then E(aN/3N - a/?)^ -н> О in probability, as
N-* oo. Using this fact we get, as N-* oo in B.31),
? exp[i\ ¦ (w(t) - tt>(*))]|f, = exp[- ^|Л|2(( - .)] a.s. B.32)
Now, if X has n-dimensional normal distribution, then ?|X|m < cm+1m\
for some с > 0. Hence the series
2 im(x-x)m
m!
7/» « V/
is absolutely convergent in L1(S2) if |Л| < 1/c. It follows that
im(X ¦ X)'
if г-^оо. B.33)
(For if ?|ttiV - a|-^0, then ?(ttiV - a)|fs-^0.) If
?ел'х|^ =ехр[
then B.33) implies that
B.34)
In view of B.32), B.34) can be applied to X = w(t) - w(s). Comparing
coefficients for m = 1, 2 we deduce:
E~(w(t) - w(s))\^s =0 if 0<*<*<Г, B.35)
i[w,(t) - щ(в)][щ(Г) - wt(s)]\$s =8{j(t- s) if 0 < s < t < T.
B.36)
3. Girsanov's formula
Let w(t) be an n-dimensional Brownian motion in the interval 0 < t < Г. In
Chapter 4 we have defined the concept of the stochastic integral
flf(s) dw(s) for functions / in l?[0, Г], where L2[0, Г] is defined with

3. GIRSANOVS FORMULA 165
respect to an increasing family of a-fields ^ satisfying:
(i) $(и>(Х), Л < t) is in ff, for any 0 < t < T;
(ii) ^(w(t + X) - w(t),0 < X < T - t)is independent of %, for any
0 < t < T.
(More precisely, we have assumed that the 5, are defined for all t > 0
and satisfy (i) for all t > 0, and (ii) with 0 < Л < oo and for any t > 0.
However we have actually made use only of (i), (ii).)
Suppose we replace (ii) by the weaker condition:
(ii') For all 0 < s < t < T,
E[w(t) - w(s)]\$s =0,
E[Wi(t) - wt(s)][wf(t) - w,(s)]\Vt = 8M(t - s).
Then we can still define the stochastic integral and derive all the formulas
and estimates as in Chapter 4. In fact we first define the stochastic integral
for any step function, next derive Lemmas 4.2.2, 4.2.3 (see Problem 1), and
then approximate any / in L^ [0, Г] by bounded step functions and use the
procedure of Section 4.2. All the results of Chapters 4,5 and of Sections 1, 2
of this chapter remain valid, without any change, for this slightly more
general notion of the stochastic integral.
Definition. From now on we shall assume only that the family %t satisfies
(i) and (ii'). We shall say that %t is adapted to w(t).
Let <j>(t) be as in Theorem 2.1. In view of B.35), B.36), % is adapted to
w(t). If/GL*[0, Г],Шеп
T\f(s, w)\2ds < oo j = 1.
Since P is absolutely continuous with respect to F,
1-
Hence, if / S L2[0, Г], with any ff adapted to w(t), then / G L|[0, Г]
with the same 5,.
Let fk be a sequence of step functions in L^ [0, Г] such that
С l
, Г] and
a.s. in P.
С !/*(•) -f(s)\2ds-*0 a.s. in P.
•4)

166 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
Consequently
ffk(s)dw(s)^ff(s)dw(s),
•'o
Hence, for a subsequence {fc'},
fk'(s) dw(s)-+ I f(s) dw(s) a.s. in F,
;. , C-D
/ fk,(s) dw(s)-> f(s)dw(s) a.s. in F.
From B.2) we easily see that
f'fk,(s) dw(s) = ('fk,(s) dw(s) + f'/*<(*)*(*) ds. C.2)
•'0 •'0 •'0
It is also clear that
(*)<#>(*) * a-s- in p-
Hence, taking fc'-»oo in C.2) and using C.1), we get
f f(s) dw(s) - f f(s) dw{s) + С f{s)<S>{s) ds if /6L^[0, Г], C.3)
a.s. in F (or in F).
Definition. A stochastic process x(t) @ < t < T) is called an ltd process
with respect to {w(t), P, l§rt} (where ^ is adapted to w(t)) relative to the
pair a(t), b(t) if
x(t) = x@) + С b(s) ds + f* a(s) dw(s) for 0 < t < T C.4)
•'O •'0
where b = (b1; . . . , bn) is in L^[0, Г] and a = (<fy)",_i is in L^[0, Г].
Theorem 3.1. Let x(t) @ < f < Г) be an Ito process with respect to
{w(t), P, ^t) relative to the matrix a(t) and the vector b(t). Let
Ф(') = (Н*), • • • , <#>„(')) belong to Z4[0, Г] and set
?/(Ф)= f Ф(и) dw(u) - % Г \ф(и)\* du, C.5)
w(t) = w(t) - [' ф(.) ds, C.6)
•4)
[]. C.7)
Assume that
P(Q) = 1. C.8)

3. GIRSANOVS FORMULA 167
Then w(t) is an n-dimensional Broumian motion in the probability space
(fi, ^, P), ^t is adapted to w(t), and the process x(t) is an Ito process with
respect to {w(t), P, %t) relative to the matrix a(t) and the vector
b(t) = b(t) + o(t)<j>(t). C.9)
This follows by combining Theorem 2.1 with B.35), B.36), and C.3).
Denote by G% the space of continuous functions x(t) from 0 < t < T into
R". Denote by ?Ilr the o-algebra generated by the sets {x(t) G A} where
0 < t < Г and A is any Borel set in R".
A continuous n-dimensional process ?(t) @ < t < T) in (й, C, Р) induces
a probability jUj, on @J., 9ILr) as follows:
Denote by ?M the sample path t -* ?(t, со) @ < t < T), and denote by
QnT (I) the set of all elements ?ы, со G Й. For any set В G <D1lr define
Мр(В) = Р{со;?иСВ}. (ЗЛО)
It is clear that цР is a probability, and цР(В) = цР[В П в?-(?)]. In
particular
цР{х; х(ь)еА1г ..., x(tk)GAk) = Р{со; фь o>)(EAv ..., fa, o>)eAk}.
C.11)
Suppose now that Q is another probability on (Й, %) given by
dQ(w) = p(co) dF(co); C.12)
thus /p(co) dP(co) = 1. Introduce the probability /Xg on {&T, 91Lr) corre-
corresponding to Q:
Lemma 3.2 Under the foregoing assumptions,
Jj^ (O = p(«). (злз)
i.e., for any В e <Dltr, Bc6"r(I),
jUp(B) = Г p(co) dnP(i;J. C.14)
•'в
Proo/ It suffices to verify C.14) for sets of the form
Let
В - {w;fa,w)GA1 fa, co)EAt}.

170 7 THE CAMERON-MARTIN-GIRSANOV THEOREM
|/,| > k, then
Jjgkdwlfjfdw if /EL2[0, Г],
т It* t
l& - /I2 *-»0. ? / &*»-[ fdw _>0 if /ЕМДО, Г],
о I •'о •'о
(с) Use (a), (b) in order to prove Lemmas 4.2.2, 4.2.3 for any step
function / in L*[0, Г].
2. Consider a system of n stochastic functional differential equations
x(t, «) = x@, «) + Г a(x( •, w), s) du)(s, w) + f' b(x( •, «), s) ds C.23)
•'о Л)
where b = (b, bn), о = (o^,,,, and b,(x( ¦ ), t), оц(х( ¦ ), t) are
measurable functions on 0J. X [0, Г], measurable with respect to 91L,, for
each t (where <Dlt, is the a-algebra generated by the sets {x(s) E A),
0 < s < *, A a Borel set in Д"). Assume that
for/= ott and/= bj, where ||x( • )|| = maxo<t<T|x(*)|.
(i) Prove by the method of successive approximations that for any
jc(O, w) in L2(S2) which is independent of ^(w(s), 0 < s < Г) there exists a
unique solution of C.23) satisfying E\x(t, w)|2 < const @ < t < Г).
(ii) Prove uniqueness in the sense of probability law.
3. Extend Girsanov's formula to stochastic functional differential equations.
4. Let b(x) be uniformly Lipschitz continuous in xER1. Consider the
stochastic differential equation d?(t) = dw(t) + b(?(*)) dt. Prove that
f Ф(,, x,«, у) Г (у-хJ]
A y2(t - s) L v ; J
where
Ф(в, x, t,y)-E exp
and ^s(t) = x + u>(f) - w(s).
5. Let w(t) be an n-dimensional Brownian motion. The processes w(t) and
2w(t) induce probabflity measures in F^, 9HT), (i^ and f^,,, respectively.
Prove that nw and f^ are mutually singular.
6. Let у — f(x) be a local diffeomorphism in Я". A given stochastic system
dx = a(x) duj + fei(x) dt is then transformed, byjmeans of Ito's formula, into
ty = o(y) dw + b(y) dt. Denote by L and L the differential operators

PROBLEMS 171
corresponding to the first and second stochastic systems, respectively. If
u(x) = M(y) where у = f(x), prove that Lu(x) = Lu(y); thus the relation
between a stochastic differential system and its differential operator is
invariant under diffeomorphism.

8
Asymptotic Estimates for Solutions
In this chapter (except for Section 1) we consider the behavior, as t-»oo, of
solutions of a stochastic differential system in case the diffusion matrix is
nondegenerate for all (i,t)eR" X [0, oo).
1. Unboundedness of solutions
Consider a system of n stochastic differential equations
with initial condition
where ?0 is independent of <${w(t), t > 0} and E|?0|2 < oo. Set
ft
b = (bv . . . , fej, a = (aj;.)"' ._v a = (aj;-), a,,- = 2 <
If С is the matrix (cA we write
1/2
We shall need the following conditions:
(Ao) a(x, t) and Ь(зс, t) are measurable functions in R" X [0, oo) and, for
any T > 0, Я > 0, there are positive constants CT, CT R such that
Их, t)\ + \b(x, t)\ < Cj(l + \x\)
if xeR", 0 < t < T,
\a{x, t) - a(x, t)\ + \b(x, t) - b(x, t)\ < CTR\x - ?|
if \x\< R, |x| < K, 0 < t < T.
172

1. UNBOUNDEDNESS OF SOLUTIONS 173
(Ax) For any Я > 0 there exist positive constants yR, TR such that
°n(*> f)TR + fei(*> t) > yR if |x| < K, t > 0.
This condition is satisfied, for example, if an(x, t) > X, b^x, t) < С
where X, С are positive constants.
Theorem 1.1. Suppose (Ao) and (A,) hold. Let ?(t) be the solution of A.1),
A.2). Then
limsup||(t)| = oo a.s. A.3)
t-»00
Proof. Since ?(*) is a continuous process, the assertion A.3) is equivalent to
the assertion that
sup|€(t)| = oo a.s. A.4)
t>0
Consider first the case where the range of ?0 lies in a bounded set K. Let
В be any open ball containing К and denote by т(В) the exit time from B,
i.e.,
т(В) = first i such that ?(*) ? В if such t exists,
т(В) = oo if no such i exist.
For any Г > 0 let rT = r(B) л T. Set BT = В X {t = T}, ST -
dB X @ < t < T} where dB is the boundary of B.
Suppose ф(зс, t) is a smooth function satisfying
ЭФ А 32Ф Д Эф
4^ ^
or <,,-i oxi ож/ i-i or A.5)
in В X [0, J], ф > 0 on BT U Sj.
If we apply Ito's formula to ф(?(*), t) and substitute t = тт, we get after
taking the expectation (cf.> Theorem 6.5.2)
Еф(?@), О) > Етт, A.6)
provided the range of $@) is in К.
If 1 + xl < зс° for all (xj, x2, . . . , зс„) ёВ, then we can take
ф(х, t) = A[exp(ax?) - exp(ax1)],
where a, A are suitable positive constants. The condition (Ax) is used here in
verifying A.5). Since ф < С where С is a constant independent of T, A.6)
yields Ett < C.
Now, тт f t(B) if T | oo. Using the monotone convergence theorem we
conclude that r(B) satisfies
Er(B) < С A.7)

174 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
In particular,
r(B) < oo a.s. A.8)
Take a strictly increasing sequence of open balls Bm with center 0 and
radius Rra—»oo. Let
00
sup |€(t)\>Rn], n*= n am.
Denote by х„ the indicator function of Bm and let ^{t) be the solution of
A.1) with the initial condition ?„,(()) = Xm?@). Denote by rm(Bm) the exit
time from Bm of the solution ^(t). By Theorem 5.2.1, rm(Bm) = r(Bm) a.s.
and ^(t) - €(t) if 0 < * < t(BJ. Hence,
P{sup|€(t)| > Pjj = P{sup|U*)l > Я,»}
V t>0 t>0
and the right-hand side is equal to 1, by A.8). We conclude that P(Qm) — 1.
Since the sequence fiOT is monotone decreasing to П* = C] m-i^m'
Now, if ы€ЕЙ*, then wGfim for all m, so that
sup \?(t)\> R,» for all m, i.e., sup|?(t)|- oo.
t>0 t>0
This completes the proof.
From the proof of A.7) we have:
Corollary 1.2. Let ?(t) be a solution of A.1) with |?@)| < Я a.s. Suppose
au(x, t) > X, b^x, t) < С if \x\ < Я, t > 0, where X, С are positive con-
constants. Then
?tr < oo
where tr is the exit time from the ball \x\ < R.
2. Auxiliary estimates
In the sequel we shall assume that a(x, t), b(x, t) are measurable and ?(*) is
any solution of A.1), A.2) such that
sup E\?{t)\2 < oo for any Г > 0.
o<t<r
If the condition (Ao) holds, then, of course, there exists a unique solution.
Set

2. AUXILIARY ESTIMATES 175
Theorem 2.1. Assume that
n n
V a (v t\ < с V h I r t\ <: с
»-i (-1
«;/iere С is a positive constant. Then
Е\ф)\2 < Kt+ К' for all t>0, B.1)
where К, К' are positive constants.
Proof. Using Ito's formula with u(x) = |зс|2 and ?(t), and taking the expec-
expectation, we get
J/* t
0
where
Since, by our assumptions, g(x, t) < CY where Cx is a positive constant, B.1)
follows.
Notation. We shall denote the eigenvalues of a(x, t) by \(x, t) where
\x{x, t) < X2(x, t) < ¦ ¦ ¦ < \{x, t).
Lemma 2.2. Assume that
for all xeR", *>0 (C const); B.2)
4
for any Я > 0 there is a positive constant [i(R) such that
^aJx, Щ > [i(R)\Z\2 if \x\ < K, t > 0, JGH", B.3)
>.;
A + |x|J|b,(x, t)l < ?(l*l) for xGRn, t > 0,
i
«Леге «(r)^0 if r-»oo; B.4)
\,(x, t) > у > 0 /or iGfi", t>0 (yconet)- B-5)
Let ф(х, t) be a bounded measurable function such that ф(х, t) — 0 if xg. G
where G is a compact set. Then, for any i\ > 0,
[* ф(€(в), s)ds
•'o
K, + K2*A+"'/2 B.6)
where Klt K2 are positive constants. If further,
\n_l(x,t)>y'>0 for x(ER", t>0 (y'const), B.7)
then B.6) /wWs for some -1 < 17 < 0.

176 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
Proof. Let r = \x — x°\ where x° is a point lying outside G. For simplicity
we shall assume in what follows that x° = 0. It is clear that there exists a
continuous function \p(r) such that
\ф(х, t)\ < *(r) B.8)
and %j/(r) ~ 0 if r < r0 or if r > rl for some 0 < r0 <'rx < oo. We shall
construct a function F(r) such that the function f(x) — F(\x\) is in С2(Я")
and
(Lf)(x, t) > *(|*|). B.9)
One can easily verify that
F'tr)
2Lf(r) = A(x, t)F"(r) + —^- [B(x, t) - A(x, t) + C{x, t)] B.1.0)
where
1 n П
A(x, t) = — 2 aij(x' t)xiXp B(x> *) = S au(x> f)>
Let в{r) be a continuous function, vanishing for r < ro/2 and satisfying,
for r > r0,
(l + 0(r))A(x, t) < B(x, t) + C{x, t) B.11)
Since, for any R > 0, A(x, t) > fi(R) > 0 ii t > 0, \x\ < R, and since
B(x, t) > 0 and \C(x, t)\ is bounded, one can certainly construct such a
function 9(r). Noting that
Xx{x, t) < A{x, t) < \,(x, t), B{x, t) - X,(x, «) + •••+ \,(зс, t)
and using B.5) and B.4), we can construct 9{r) such that
в (r) = - 7] if r is sufficiently large (for any ¦») > 0). B.12)
If B.7) holds, then we can take
9(r) = -tj if r is sufficiently large (for some -1 < 77 < 0); B.13)
in fact, we can take any i\ such that
— 7) < y'/8' where 5' = lim sup I sup\,(:jc, t) \.
|x|-»oo t>0
Introduce the functions
I(s) = —— du,
J'o u
B.14)
F(r) = J Г e~m ds • f eI(u)^(«) d«
Л ^0 ^o

2. AUXILIARY ESTIMATES
177
for some X > 0, r = |зс|. Since F is in C2 and F(r) = 0 if r < r0, f(x) is in
С2(Я"). One can easily check that
B.15)
and that F'C) > 0- Using B.10), B.11), we then get
Г
2Lf(x) >\F"(r
x, t)
if 2X = /i(r,) (since A(x, t) > /i(r,) if |x| < rv t > 0).
Having proved B.9), we now use Ito's formula to get
Г ф(€(в), s) ds < ? f *(?(*)) ds<E Г
•'о •'о •'о
), e) ds
Noting that F(r) < Cr1+", we obtain
), *) ds
for suitable constants С Using Theorem 2.1, the assertion B.6) follows.
We shall now extend Lemma 2.2 to the case where ф(х, t) does not
necessarily vanish if |jc| is large, but either
С
or
|ф(х, t) - Ф| «
where Ф is a constant.
\ф(х, *) — Ф| <
«A*1)
\x\T
(« > 0),
B.16)
(а > 0, е(г)-»О if r-*0), B.17)
Lemma 2.3. Let the conditions B.2)-B.5) hold and let ф(х, t) be a bounded
measurable function satisfying B.16). Then
(* ф(€(в), s)ds-<!>t = o(ta+ri)/2) + Oit1-"/2) B.18)
•'o
for any tj > 0; if B.7) also holds, then B.18) holds for some -1 < tj < 0.1/
tfie function ф(х, t) satisfies B.17) instead of B.16),
Г' ф(€(в), в) ds - Ф* = О(*A+ч)/2)
Л)
B.19)
the same 17 as before.
Proof. Suppose first that B.17) holds. Then, for any « > 0, there exists an

178 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
R{ such that
|ф(х, t) - Ф| < ¦? tf М>Я<> *><>•
Let xt(x) = 1 if |x| < Rt, xf(x) = 0 if |зс| > R,, and write
ф(х, t) - Ф -[ф(х, t) - Ф]Х,(х) +[ф(х, t) - Ф][1 - Xt(x)]
= ф,(х, t) + Ф2(х, t). B.20)
By Lemma 2.2,
Е /%,(?(*),*)& = O(tA+")/2). B.21)
•'о
Next, let \blr) be a continuous function such that
for some 0 < r0 < R?. We shall slightly modify the definition of 0(r) by
replacing the condition B.11) by the stricter condition
A + 9{r))A{x, t) < PB{x, t) + C{x, t) B.23)
where p is any positive number < 1. We can still satisfy B.12) for any
V = ^(р) > 0 provided p is sufficiently close to 1; if B.7) holds, then we can
even satisfy B.13) if p is sufficiently close to 1.
Let/(x) be defined by B.14) with X > 0 to be determined later. Then, by
B.10), B.23),.
9{r) I F'lr)
^ F'(r) A(x t) + ^
Г 9{r) I F'l
2L/(x) >^F"(r) + -^ F'(r) A(x, t) + -
x, t)
X fi+«
'MfetMda B.24)
where с is a positive constant < A - р)В(х, t). Since eI(r) ~ г ч if r -> oo,
we find that
Lf(x) >^~l> |Фа(х, t)| if |x| > R? B.25)
where cx is a positive constant independent of e, provided 2Л < cv If
|x| < Rt, then clearly Lf(x) > 0 ¦ |ф2(х, t)\. '

2. AUXILIARY ESTIMATES 179
From Ito's formula and B.25) we obtain
Г'фа(€(в),в)Л <E Г* |фа(?(в), e)| &
•'о •'о
< E
¦'о
- EF(\t(t)\) ~ ЕРШ) < ЩШ\)- B.26)
Since
F(r)< С€ГГ"+С lf 2-«>1 + ^
I Qr1+" +C if 2-o<l + ij,
and since we may assume, without loss of generality, that a + rj ф 1, we
deduce, after using Holder's inequality and Theorem 2.1,
f Cc*1-"/2 + С if a + т) < 1,
Substituting this into B.26) and combining the resulting inequality with
B.21), B.20), we get
•t
E\ / ф(€(в), в) ds - Ф*
< <yA+")/2+ C^1-"/2 B.27)
where Cl is a constant independent of €. Since € is arbitrary, B.27) implies
the assertion B.19) with tj replaced by ¦»)', for any -q' > tj. But this clearly
completes the proof of B.19). The proof of B.18) (for ф satisfying B.16)) is
similar.
We proceed to evaluate (under additional conditions) more precisely the
number i\ occurring in Lemma 2.3. We shall assume that, for 1 < i, / < n,
а,ц{х, t)-*at, as |jc|—»oo, uniformly with respect to t, B.28)
where cL are constants. Let
d = number of positive eigenvalues of a — (aj(). B.29)
Observe that if B.3) holds, then B.5) holds if and only if d > 1, and B.7)
holds if and only if d > 2.
Note that the assumptions and assertions of Lemma 2.3 remain unchanged
if we perform a nonsingular linear transformation эс—»Аэс. Such a transforma-
transformation changes a(x, t) into Aa(x, t)A*. We can therefore choose A such that
aif = 0 if i ф /,
au = 1 if i = 1, 2, . . . , d,
att ш 0 if i = d + 1, . . . , n;
d ™ 0 means that a,, ™ 0 for all i, and d — n means that au = I for all t.

If d > 2, then we can take, in the proof of Lemmas 2.2, 2.3, в(г) = v for
any v < 1 provided r is sufficiently large. This leads to the assertions of
Lemmas 2.2, 2.3 with any -1 < tj < 0.
If d > 3, then we can take 9(r) = § provided r is sufficiently large. This
leads to the assertion of Lemma 2.2 with tj = — 1. If ф satisfies B.16), then,
instead of B.18), we have
e\ ? ф{й(8), s) ds - *t
= O(l) + O(tl-a/2) if а Ф 2. B.30)
We sum up:
Lemma 2.4. (a) Let the conditions B.2)-B.4), B.28) hold, and let d > 2.
Let ф(х, t) be a bounded measurable function. If ф satisfies B.16), then
B.18) holds for any -1< tj < 0, and if ф satisfies B.17), then B.19) holds
for any — 1 < Tj < 0.
(b) Let the conditions B.2)-B.4), B.28) hold and let d > 3. //ф(х, t) is
a bounded measurable function satisfying B.16), then B.30)
3. Asymptotic estimates
Theorem 3.1. Let the conditions B.2)-B.5) hold. Then
Е\ф)\2 > Kt~ К' for all t > 0 C.1)
where К, К' are positive constants.
Proof.
L\x\2 = 2^(*. t) + 22*A(*. t) > у - Ф(х)
where ф(х) is a bounded measurable and nonnegative function having
compact support. By Ito's formula,
E\Z(t)\2 ~ ВД2 - E f (L\x\2)(U(s), s)ds>yt-E /%(*(«)) ds.
Since, by Lemma 2.2,
E ('фЦ(8)) ds = O(t)
Л)
the inequality C.1) follows.
Remark. Lemma 2.2 and Theorem 3.1 remain true if the condition B.4) is
replaced by the weaker condition
(l+|x|J|b,(*.*)l<«o C.2)
provided c0 is a sufficiently small positive constant.

3. ASYMPTOTIC ESTIMATES
181
We shall need the following condition:
l^(x, *)j < c(|x|), e(r)->0 if г->0, 8 > 0.C.3)
Theorem 3.2. Let B.2), B.3), B.28), C.3) hold and let d > 1. Then
Е\ф)\2 = (tr a)t + O(tA+4)/2) + o(t1-»/2) C.4)
where tr a = "E{ati and tj is any positive number; if d > 2, then tj is any
number > — 1.
Proof. Notice that
|L|x|2-tra|<e'(|x|)/(l + |*|e)
where e'(r)-»0 if r-»oo. Hence, by Lemmas 2.3 and 2.4(a) with
<J>(x, t)=» L|x|2, Ф = tro,
=E Г (L|
•'o
-(trS)t+
), s) ds
where -q is any positive number if d > 1 and any number > — 1 if d > 2.
This yields C.4).
The special case 8 = 0 gives
Corollary 3.3. Let B.2)-B.4), B.28) hold and let d > 1. Then
We can now state a key lemma needed for deriving the main results of
this section.
Lemma 3.4. Let the conditions B.2), B.3), and C.3) hold with 0 < 8 < 1,
and let d > 2. Then, for any \ = 1, 2, . . . , n,
2
), 4') ds
= oft1"»).
C.6)
Proo/. For any e > 0 we can write
where g are bounded measurable functions; gx(x) has compact support,
g2(r) = e/(l + rI + 8 if f>rlf C.7)
and g2(r) = 0 if r < r, for some 0 < r, < oo. Clearly
\ [' b,U(s), s)ds
I •'0
2E
), «) ds
2E
C.8)

182
8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
Let x° be a point outside the support of gx and let p = \x — x°|.
Let F(p) be the function constructed in the proof of Lemma 2.2 for
ф = gx. Then, for any -1 < -q < 0,
F(r) < cr1+ri, F'(r) < cr"> (с positive constant) C.9)
provided r is sufficiently large. By Ito's formula,
I С &(*(«)) ds< С (LE)U(s), «) ds
ко •'о
< F(|€(t) - x°|) - ('dxF ¦ оШ, s) dw(s).
Consequently
A (' gi(i(«)) * < «1+
КО
Next, for large r,
\DXF • a\2 < Cr2*1 for any - 1 < r, < 0.
Hence, by Lemma 2.4(a) with a = — 2tj,
E С \DXF ¦ a\2 ds < Ctl+7> + C,
Jo
and C.10) then yields
С + 2? Г' \DXF • a\2 ds. C.10)
•'О
C.11)
C.12)
Ctl+V + C.
C.13)
To evaluate the integral corresponding to g2, let F0(r) be the function F(r)
constructed in Lemma 2.3 when <J>2 = g2. We can take — 1 < tj < — 8.
Then
F0(r) < Cer1'6 + Ce, F^(r) < Cer~s + Ce C.14)
where С is a positive constant independent of c. Analogously to C.10), C.11)
we get
2
El f' fe(l?(*)l) * < Cet1'6 + С + 2? (' \DXFO ¦ a\2 ds
ко •'о
\DXFO ¦ a\ < СеУг28.
By Lemma 2.4(a),
? C\DXFO • a\2ds < QA+">/2 +
^0
+ с
Hence
Cet
l~s
С
C.15)
C.16)
C.17)
C.18)
Combining C.18) with C.13) and C.8), and recalling that r/ < -8, the
assertion C.6) follows.

3. ASYMPTOTIC ESTIMATES 183
The following lemma is a variant of Lemma 3.4 in case d > 3.
Lemma 3.5. Let the conditions B.2), B.3), B.28) hold, let d > 3 and let
C.3) hold with some 8 > 1. Then, for any j = 1, . . . , n,
E
&,(*(«), 5)
= 0A).
C.19)
Proof. The function F(r) occurring in C.9) is now given by
F(r) = Г e~m ds Г eI{u)y(u) du if r > p0,
where y(r) = 0 if r > pv for some 0 < p0 < px < oo. We can take ij(r) = f
for large r, so that /(r) ~ log r3/2. Therefore
F(r) < C, F'(r) < C/r3/2. C.20)
Hence C.11) is replaced by
\DXF ¦ a\2 < C/r3. C.21)
Lemma 2.4(b) with a = 3 gives
E С \DXF ¦ a\2 ds = O(l), C.22)
•'o
and this estimate is to replace C.12). We conclude that (instead of C.13))
< С
C.23)
Let 0 < r0 < rx and let g2(r) be a continuous function satisfying
+ rI + e if r>r0,
if r<ro/2.
Take
F0(r) = С ГГ е-ад Г eIMg2(u) du (C > 0).
•'o •'o
Since 8 > 1, Fo satisfies (instead of C.14))
F0(r) < C, F^(r) < C/r* к = minE, f).
Using Ito's formula we find that
<C + E [' \DXFO ¦ a\2 ds.
Since, by Lemma 2.4(b) with a = 2k,
E f \DXFQ -a\2ds< C,
the right-hand side of C.25) is bounded by a constant. Combining this with
C.23), C.8), the assertion C.19) follows.
C.24)
C.25)

Remark. If in Lemma 3.4 we replace the condition C.3) by
i
then
We shall need the following conditions:
2 ki,(*, t) - 5if\ < € s , 8 > 0, e(r)->0 if r->oo, C.28)
where aif are constants. Note that
a = да*, tr a = |a|2.
We shall also need the condition
2 \oit(x, t) - дц\ < C/ A + |x|)8, 6 > 0. C.29)
We shall now state the main results of this section.
Theorem 3.6. (a) Let B.2), B.3), C.3), C.28) hold with 0 < 5 < 1, and let
d > 2. Then
E\?(t) - aw(t)\2 = oit1-6). C.30)
// C.3), C.28) are replaced by C.26), C.29) and 0 < 8 < 1, tfien
(b) Lef B.2), B.3), C.26), C.29) hold for some 8 > 1, and fet d > 3.
Then
?|?(t) - aw(t)\2 = O(l). C.32)
Proo/. Let B.2), B.3), C.3), C.28) hold with 0 < 5 < 1, and let d > 2.
Consider the expression
\ ft 2 ft
? I (a - a) dw(s) = ? I \o - a\2 ds.
ко Л)
Since |a(x, t) - a| < e2(|x|)/(l + |x|JS, Lemma 2.4(a) implies that
? f\o{u(s),s)- d\2 ds = oit1'6).
0
Writing
- aw(t) = ^0 + f' [a(i(s), s) - a] dw(s) + С b{i(s), s) ds
0 ^0
and using the previous estimate and Lemma 3.4, the assertion C.30) follows.
The proofs of C.31), C.32) are similar. In proving C.31) we make use of
the remark following Lemma 3.5. In proving C.32) we employ Lemma 3.5.

4. APPLICATIONS OF THE ASYMPTOTIC ESTIMATES IN
4. Application* of the asymptotic estimate*
We shall need the following conditions:
lim aJx, t) = d{. uniformly with respect to t, 1 < i, / < n, D.1)
|x|->oo
2 \k(x, t)\ < С for all x e R", t>0 (C const),
" D 2)
lim A + |x|) 2 \h(x' t)\= ° uniformly with respect to t. v " '
Theorem 3.6(a) with 8 = 0 states:
If B.2), B.3), D.1), D.2) hold and if d > 2, then
E\U(t) - aw(t)\2 = o(t). D.3)
From D.3) we see that, as t-»oo,
—^ a -^0 in L2;
consequently also in probability. This immediately yields the following
theorem on convergence in distribution of ?(?):
Theorem 4.1. Let the conditions B.2), B.3), D.1), D.2) hold, let a be
nonsingular matrix, and let n > 2. Then
where a is the inverse matrix to a.
Suppose next that B.2), B.3) hold and that
ч
for some 0 < 8 < 1. Suppose a is nonsingular and n > 2. Denote by a the
inverse of a. Then
V2t log log t V2t log log t
/' (€(e), «) ~ a]
V2* log log t °
+ ° = J^t) + J2(t) + J3(t) = J(t). D.7)
V2t log log t

1M 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
We shall denote various positive constants by the same symbol C. Let
tm = m\ Л = 4/8, m a positive integer. By the proof of Theorem 3.6(a),
P { sup \h(t)\ > ? } < p I _?_ J^1 |bD(t), .)| A > ^ ]
m
Cm2 ^ a-e^ С
Next, by the martingale inequality and the proof of Theorem 3.6(a),
sup \h * I > -
< P —— sup I С [o(?(s), s) - a]dw(s)
m
. Cm2 ,. a-« , С
Since, finally,
we conclude that
P\ sup
Applying the Borel-Cantelli lemma we deduce
consequently
P{ lim |/@| = 0} = 1. D.8)
From D.7) and the law of the iterated logarithm for w(t) (Theorem 3.3.1,
Corollary 3.3.2 and Theorem 3.6.1) we obtain:
Theorem 4.2. Let the conditions B.2), B.3), D.5), D.6) hold for some
S > 0, let a be nonsingular, and let n > 2. Then, for any i = 1, . . . , n,
lim — ' '' — = 1 a.s., D.9)
*^°° V2t log log t
V2 Hog log t

4. APPLICATIONS OF THE ASYMPTOTIC ESTIMATES 117
Further,
Ш - = 1 a.s. D.11)
f^°° V2Hoglogt
The next application is to the Cauchy problem:
К%а^^+Жш^-^'° if XGRn' f>0'
D.12)
u(x,0)=f{x) if iGfl". D.13)
We shall assume:
(i) o(t(x), b((x) are bounded functions in Я", uniformly Lipschitz
continuous, and the matrix (atAx)) is uniformly positive definite.
(ii) For all x, x in R",
\f{x)-f(x)\<C\x-x\y @<y<2, 0 0).
By Sections 6.4, 6.5, there is a unique solution u(x, t) of D.12), D.13)
bounded by O(|x|Y) uniformly in t in bounded intervals, and it is given by
u(x, t) = Efcit)) D-14)
where ^x(t) is the solution of A.1) with ?@) = x.
Let us further assume that
2КЫ-5(,|<С/A + |х|)в, D.15)
Z\k(x)\<C/(l + \x\I+S D.16)
i
for some 8 > 0.
If a^ = 8y and b{ = 0, then the solution й is given either in the form
Ef(w(t) + x), or in terms of the fundamental solution for the heat equation,
namely
^ t ( ^)<tl7)
From D.14) and (ii) we get
\u(x, t) - й(х, t)\ < C{E|^(t) - w(t) - x\2}y/2.
We can now apply Theorems 3.6(a), 3.6(b) to estimate the right-hand side. A
careful review of the proof of C.31) and C.32) for ?(t) = (^(t) shows that
where С is a constant independent of the initial condition ?@) = x. Hence:

188 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
0 < 5 < 1, then for all x?fi", t > 0,
\u(x, t) - fi(x, t)\ < C(l + t + |x|2)A"S)Y/2 (C const). D.18)
Ifn>3 and 8 > 1, then for all xGH", t > 0,
\u(x, t) - u(x, t)\ < С (С const). D.19)
5. The one-dlmenslonal case
In proving Theorem 3.6(a) or even in deriving D.3), we have assumed that
the matrix a has at least two positive eigenvalues. To see what may happen
when a has only one positive eigenvalue, we resort to the case n = 1. For
simplicity, consider first the equation
dt(t) = dw{t) + b(?(t)) dt. E.1)
We assume that b(x) is measurable and
f°° |b(x)|dr < oo. E.2)
•'-00
One can construct comparison functions explicitly by solving differential
equations of the form
L/ = if(*) + b(x)/'(xW(x). E.3)
Setting A (x) = 2/х_00Ь(у) dy, the general solution of E.3) is given by
/(x) = Г VA« dz\2 Г V<»>*( y) dy + Cx| + C2
•'o L ^o J
where Cv C2 are constants. Using this solution in the proof of Theorem 3.6,
one can derive the estimate
E\i(t) - w(t)\2 = o(t) E.4)
provided A(+ oo) = 0, i.e., provided
f°° b(x)dx = Q; E.5)
•' — 00
if further
|b(x)| < C/ A + \x\I+S @ < 5 < 1) E.6)
then
E\i(t) - w(t)\> - Oit1-8); E.7)
the details are left to the reader.
Consider next the more general stochastic differential equation
di(t) = аШ) dt + Щф dt. E.8)

5. THE ONE-DIMENSIONAL CASE
189
We assume that a(x) > 0 for all x, and set
dy
h(z) = Г
Л)
g inverse
to h.
o(y) '
As easily verified, the process -q(t) = h(?(t)) satisfies the equation
d-q(t) = dw(t) + b{r)(t)) dt
where
hi ) = b(g^)) _ i ,/ / s)
°\g(x))
The condition E.5) for the equation E.9) becomes
b(x) - ho(x)a'(x) j n
If
E.9)
E.10)
E.11)
* ^-t-^ E > 0) E.12)
A +1*1)
and if E.11) holds, then, by the assertion E.7) applied to the solution -q(t) of
E.9),
E\r,(t) - w(t)\2 < Ctl~s + С
If we further assume that
\ah(x) - x\ < C(l + И1"*1) @ < ju < 1, a > 0)
for some constant a, then we easily conclude that
E\?(t) - aw(t)\2 < Ktl~" + K' for all t > 0
where v = min(8, ju) and К, К' are positive constants.
Observe that if
a[x - a
A
then E.13) holds and, further, E.12) is equivalent to
С
-ko'(x)
(i + \X\Y
E.13)
E.14)
E.15)
E.16)
We can therefore state:
Theorem 5.1. // E.15), E.16), and E.11) hold, then for any solution ?(t) of
E.8) the estimate E.14) holds with v - minE, ц).

190 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
If E.15) is replaced by
a = lim a(x) exists, а ф 0,
|x|-«o
if the left-hand side of E.12) is in Lx(- oo, oo), and if E.11) holds, then, by
applying the assertion E.4) to the solution i\(i) of E.9), we easily obtain
E\?(t) - ow(t)\2 = o(t). E.17)
The condition E.11) is essential for the validity of Theorem 5.1. Taking
for simplicity the case a = 1, where the condition E.11) reduces to the
condition E.5), we shall prove:
Theorem 5.2. Let b(x) satisfy E.2). If t-(t) is a solution of E.1), then the
estimate E.4) holds if and only if E.5) holds.
Proof. We only have to prove that if
f°° b{x)dxl=0 E.18)
¦'-oo
then E.4) does not hold. We shall assume that E.18) and E.4) hold, and
derive a contradiction.
Clearly
??(*) = E?o + E /"&(€(*)) ds. E.19)
-'o
A particular solution of E.3) for \[> = b is
f{x) = Г e'Mz)[eMz) - l]dz.
•'o
Since A(- oo) = 0 and (by E.18)) A( + oo) ф О,
f{x) = Xx+ +o(\x\) as |x|->oo E.20)
where X is a nonvanishing constant.
By Ito's formula
E f ЬШ) ds = EflZit)) - EfiQ.
Substituting this into E.19), we get
Еф) = E?o - EfiQ +¦ Е/Ш. E.21)
Since fix) satisfies a uniform Lipschitz condition,
\EfiZit)) - Efiwit))\ < CE\Zit) - wit)\ (Cconst),
and, in view of E.4),
) » Efiwit)) + oit1'2). E.22)

6. COUNTEREXAMPLE 191
From E.20) we have, for any e > 0,
|/(x) - Xx+| < e|x| + C(e) (C(e) const).
Combining this (when x = w(t)) with E.22), we find that
|JE/(€(*)) - E[Xw+ (t) | < eE\w(t)\ + C(e) + o(t1/*) < Zct1'*
if ? is sufficiently large. Upon substituting this into E.21), we obtain
E€(t) = MEw+ (t) + fct1/* + С (С const, |0| < 2).
But, by E.4),
Еф) = Ew(t) + oft1'2) = o(t1/8).
Consequently,
|X|Eu; + (t) < 3ef1/2 if Hs sufficiently large.
Since, however, Ew+ (t) = (t/2<7rI^2, we get a contradiction if 18we2 < X2.
There is an intuitive reason why for n = 1, a = 1 the assertion E.4)
cannot hold unless E.5) is satisfied. In order for the distribution of ?(f)/V7
to approximate the normal distribution as t—юо the particles represented by
?(*), or !-(t, w), must be able to move without significant "resistance" from
intervals (a, /}) near +oo to intervals ( — /J, — a). Since in performing this
move they must cross the interval ( — a, a), they are subject to the influence
of the drift term b(x). This drift coefficient will resist the motion if
S -oo^(x) dx > 0; thus this inequality cannot take place. Similarly, the re-
reverse inequality cannot take place.
If n > 2 and a is nonsingular, then t-(t) may move from one n-dimensional
interval in a neighborhood G of infinity to another without leaving G. If the
drift coefficient b(x, t) is "small" in G (i.e., if \b(x, t)\ < C(l + \x\)~1~>l,
ц > 0), then there will be negligible resistance by the drift coefficient to the
movement of ?(?) ш G. Thus, no condition analogous to E.5) is required in
the case n > 2.
6. Counterexample
We shall give an example of a system of n equations (n > 1)
t (ел)
for which
b(x) = of
\
if |x|^oo F.2)
such that the estimate
E\?(t)\2 < Kt + K' for all t>0 (К, К 'positive constants) F.3)
is false.

192 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
Let /(*) be a function in C2(H") such that
f{x) = r2/log r if r = |x| > 2. F.4)
If bt{x) = XfPir) for |x| > 2, then
n 9f(x)
i A/W + 2 b,(x) -^
i ™ + 2 1 i 1 I , 2r Г
О 1 / 2 W
log г [ 2 log r (iOg rJ j log r [ 2 log r
Take
log r
2r2
where y(r) is defined so that
i Д/(*) + 2 Ь,(х) -j^- = 1 if |x| > 2. F.5)
It is easily seen that y(r) = O(l). We now define Ъ{(х) in Я" such that
b,(*)-*,[bg|*| + y(|x|)]/2|x|2 if |x|>2; F.6)
we can make the bt{x) as smooth in Я" as we wish.
By Ito's formula and F.5),
EMt)) = EfiQ + E [' ciUs)) ds F.7)
•'o
where c(x) - 1 = 0 if |x| > 2.
We wish to apply the proof of Lemma 2.2 to <J>(x, t) = c(x) - 1. Here the
ЬДх, t) = ЬДх) are not bounded by e(|x|)/(l + |x|) where e(r)—*0 if г—юо.
Nevertheless B.11) takes the form
«
where r = |x — e\ and e = (els . . . , en) is a point outside the support of
c(x) - 1, i.e., \e\ > 2. In view of F.6) and the boundedness of y(r) as r—xx>,
we can actually construct a continuous function в(г) satisfying F.8) such
that, for any A > 0, limr_>00^(r) = A. Hence, by the proof of Lemma 2.2,
E Г'[с(€(*))-1]Л-ОA) as t-»oo.
Combining this with F.7) we get
E\№t))-t\<C0 for t>0 (Co const). F.9)
For any e > 0 there is a constant В such that
|x|2/log |x| < ф|2 + В if |x| > 2.

PROBLEMS 193
Hence
/(x) < e|x|2 + С for all x<ERn,
where С is a constant depending on e. This implies that
Ef(i(t)) < *E\i(t)\2 + С F.10)
Now, if F.3) holds, then from F.9), F.10) we obtain
t < eKt + tK' + С + Co for all t > 0.
But this is impossible if e < l/K.
If the" function log |x| occurring in F.2) is replaced by other functions that
increase to infinity more mildly, as |x|—»oo, such as log log |x|, then one can
again show by the above method that F.3) cannot hold. If however
b(x) = O(l/|x|), then F.3) does hold (by Theorem 2.1). Recall that if C.2)
holds, then the estimate C.1) is also valid.
PROBLEMS
1. All the results of Sections 2, 3 remain valid if the condition B.4) is
replaced by the condition that
A +1*1J M*. 01 <« fora11 *екп, t>o
i
provided e is sufficiently small. Check this in the case of Theorem 3.6(a).
2. The methods of Sections 2, 3 can be used to estimate E |?(t) - Sw(t)\4,
provided 0 < 5 < \. Show that under the assumptions B.2), B.3), C.26),
C.29) with 0 < 5 < \,
3. Prove that if E.2) and E.5) hold, then any solution of E.1) satisfies the
estimate E.4).
4. Prove that if E.6) and E.5) hold, then the estimate E.7) is valid for the
solution of E.1).
5. If
2a«(*. 0 < C(l + Ixl2/1, |2*Л(*> 01 < c(l
for some constants C>0, 0</i< 1, then
E|?@|2'1 < Kt + K' for all t > 0,
where К, К' are positive constants.
6. Let b(x, t) = b(x, t)/(l + |х|У, S(x, t) = a(x,
a — 55*. Let the assumptions of Lemma 2.2 hold for a, b replaced by a, b.
Let ф(х) be any bounded measurable function with compact support. Prove
that if ?(t) is a solution of A.1), then
!*>) for all < > 0,

194 8 ASYMPTOTIC ESTIMATES FOR SOLUTIONS
where Kx is a constant and 17 is as in Lemma 2.2. [Hint: Construct
f(x) = F(r) satisfying Lf(x) > A + \х\2)Щ\х\).] _
7. Let B.2)-B.5) hold with a, b replaced by a, b (as in Problem 6), and let
0 < /1 < 1. Prove that
\2-2" > Kt- К' for all t> 0,
where К, К' are positive constants.
8. Extend Theorem 5.2 to the case of Eq. E.8).
9. Let t-(t) be a solution of A.1) and assume that \o(x, t)\ < C,
\b(x, t)\ < C, \b(x, t) — b\ -> 0 if t —» 00, uniformly with respect to x. Prove
Aat .. €@ r
hm = b a.s.
10. Consider one-dimensional equations
di(t) = dw(t) + Ъ,{ф), *) (i - 1^ 2).
Assume that b^x, t) < b2(x, t) for all iGfl1, t > 0. Prove that if
€i@) = €8@) then ^(t) < ?2(<) for all t > 0.
11. Let i-(t) be a solution of a one-dimensional equation
d€(«) - «MO + b(€(t), t) Л, 1@) - x
and assume that
b(x, t) > p(t) for all 16Я1,
lim С 0{s) ds > 1.
*" V og log t ¦'o
V2* log log
Prove that Нт^^^,^^) = oo a.s.
12. Consider a stochastic differential equation
d€(t) = e(€(t)) dw(t) + b(€(t)) dt
and assume that a(x) > 0 in the interval a < x < b. Denote by тх(а, b) the
exit time from the interval (a, b), given ?@) = x. Denote by po(x; b) the
probability that ?(^@, b)) = a. Prove:
(a) If ?a(x)u"(x) + b(x)u'(x) =-1 in a<x<b, and u(a)
= u(b) = 0 then Етж(а, b) = u(x).
(b) If Ф(х) = exp{ - fZBb(z)/o2(z)) dz}, then
u(x) = -/Х2Ф(у) Г —^- dy + y /Ь2Ф(у) Г ^ d
•'а •'а а2(г)ФB) •'а •'о
у = f Ф(г) dz/ Г Ф(г) dz.
а а2(г)ФB) •'а •'о а2(г)
b

PROBLEMS 195
(с) If \ o2(x)w"(x) + b(x)w'(x) = 0 in a < x < b, then
pa(x;b) =[w(x) - w(b)]/[w(a) - w(b)].
13. If B.2), B.3), C.3) hold with 5 > 0, and if B.5) is replaced by
\,(x, t) > c/(l + |x|)s (c > 0), then the assertion B.6) of Lemma 2.2 is
valid for any 17 > 0.
14. If B.2), B.3), B.5), and C.3) hold with 5 > 0, then B.6) holds with
17 = 0. [Hint: Take в(г) = —D/r& for large r, where ft is any positive
constant < 8 and D is a positive constant.]
15. Suppose, in Lemma 2.3, ф satisfies B.16) and the condition B.4) is
replaced by the stricter condition C.3) with 8 > 0. Prove that
for any P > 0. [Hint: Take 1 - p = D/Brfi), 0(r) = -D/rfi. In B.14)
replace \p(u) by \p(u)ufi; this gives B.24) with a replaced by a - /?.]
16. Under the same conditions as in Problem 13, if <J> satisfies B.16), then
for any 17 > 0. [Hint: Replace, in B.14), \f/(u) by ip(u)us.]
17. Under the assumptions of Theorem 3.2,
where e is any unit vector and < , ) denotes the scalar product.

9
Recurrent and Transient Solutions
Consider a stochastic differential system of n equations
d€(t) = o(№, t) dw(t) + Ъ(ф), t) dt @.1)
Let G be an open set in Я". Suppose that for any xG G and for any open
subset V of G,
Px{i(tm) ?V for a sequence of finite random times tm
increasing to 00} = 1. @.2)
Then we say that the solution of @.1) is a recurrent process in G. If G = R",
then we say simply that the solution of @.1) is a recurrent process.
If, for any x ? G,
«>}-1, @.3)
then we say that the solution of @.1) is a transient process in G. If G = R",
we simply say that the solution of @.1) is a transient process.
The condition @.2) says that, given ?@) = *> the solution ?(f) "visits" any
open subset V of G at a sequence of times increasing to infinity. The
condition @.3) says that, given ?@) = *. ?(*) "wanders out to infinity" with
probability one.
It is well known (see Ito and McKean [1]) that an n-dimensional Brownian
motion is recurrent if n < 2, and transient if n > 3. This fact will follow as a
very special case of the results of this chapter.
1. Transient solutions
For simplicity we shall consider only temporally homogeneous processes, i.e.,
solutions of systems of n stochastic differential equations with time-
independent coefficients,
di{t) - a(€(t)) dw(t) + b(€@) dt. A.1)
1И

1. "TRANSIENT SOLUTIONS 197
Set
n
n n
)t=i
We shall assume:
(Aj) For all x G Я n,
2 |b,(x)| + 2 l^(*)l < C(l + |x|) (C const);
for any Я > 0 there is a positive constant CR such that
2 H) 2 () Сл|х - y|
if |x| < Я, | y| < Я.
(A2) The matrix (aj;(x)) is positive definite for each
Let
A(x,|)=-^ 2 <h+
в(х) = 2
«.
and set
S(x, ?) - Ц '* {X' , S(x) = S(x, x). A.3)
We shall need the assumption:
(A3) There is a positive constant Яо such that
S(x) > 1 + e(|*|) if |x| > Яо, A.4)
where e(r) is a continuous function satisfying:
(h\dt< 00. A.5)
* J
Notice that A.5) holds for any of the functions
«(*) - d, c(«) - A/a. c(*)-fl/Iog*

198 9 RECURRENT AND TRANSIENT SOLUTIONS
where d, А, В are positive constants and В > 1.
Theorem 1.1. Let (A^Ag) hold. Then the solution of A.1) is transient,
i.e., for any xGR",
«>} = l. A.6)
Um
Proof. Let a > 0. By (A3), there is a continuous function в(г) defined for
r > a such that
S(x) > 9(\x\) if |x| > a, A.7)
0(r) = 1 + e(r) if r > Ло. A.8)
We shall construct a function f(x) = F(r), where r = |x|, such that
L/(x) < 0 if \x\ > a. As easily verified,
2L/(x) = A(x, x)F'(r) + ^ [B(x) - A(«, x) + C(x, x)]. A.9)
Hence, if
F'(r) < 0 for r > a, A.10)
F"(r) + — F'(r) =0 for r > a, (l.ll)
r
then, by A.7)-A.9),
Lf(x) < 0 if |x| > a. A.12)
Set
r9-^ds. A.13)
Then a solution of A.11) is given by
F{r) = (X e~IM ds; A.14)
•'r
the integral is convergent, by A.5), A.8). Notice that A.10) is also satisfied.
Hence A.12) holds when F is given by A.14).
By Ito's formula and A.12), if \x\ > a, then
E,F(|?(t)|) - E,F(|*|) = Ex Г L№*)) ds < 0 A.15)
Л)
where т is any bounded stopping time such that |?(s)| >aifO<s<r. Let
P > a, and let та/8 denote the exit time from the shell {y; a <\y\ < C).
Denote by Pa(a) the probability that |?(та/8)| ™ а (given |@) - x) and by

1. TRANSIENT SOLUTIONS 199
PX(P) the probability that |?(та/8)| = 0 (given ?@) = x). Setting т = Т л raf>
in A.15) and taking Г—» oo, we get (since та/9 < oo a.s.; see Theorem 8.1.1),
Taking /?—»oo and using the fact that F( /J)—»0 if /?—»oo, we get
limjx(a) <-^jj-. A.16)
Introduce the balls
Яр - { У? I У1 < P} @ < p < oo)
and the event
fi(a) = {?(*) hits the ball Ba for some * > 0}. A.17)
Then we can write A.16) in the form:
P,(G(a)) < -щ- . A.18)
Denote by tR the hitting time of the boundary of the ball BR by i-(t). By
Theorem 8.1.1, if |x| < Я, then Px{tR < oo} = 1. Introduce the event
Й*(а) = (|(t) hits the ball Ba at a sequence of times increasing to oo }.
A.19)
Thus Й*(а) is a subset of Й(а).
Let а < |x| < Я. Since Рж{*л < oo} = 1,
Px(?l*(a)) = Px{?(t + <д) hits Ba at a sequence of times increasing to oo }.
Using the strong Markov property (see Problem 3) we get
P,(O*(«)) = E,WH«)) < iyW0(a)) < 4 |^ = || ,
where A.18) has been used. Taking Я->оо, we get PI(fi*(a)) = 0. This
means that
im|
t—»oo
Since a is arbitrary, we get
i.e., A.6) holds.

200 9 RECURRENT AND TRANSIENT SOLUTIONS
We shall now replace the condition (A3) by:
(A3) As |x|—»oo,
ф)^, A.20)
? *b,(*)->0, A.21)
where the matrix (o^) has at least three positive eigenvalues.
Theorem lfc. Let (АД (A2), (A3) hold. Then the solution of A.1) is
transient.
Proof. We can perform an orthogonal transformation x->x' in Л" which
takes (o^) into (afi), where a$ = 0 if i ф /', a? — 1 if i = 1, 2, 3 and a? = 0
or 1 if 4 < i < n. In the new coordinates the condition (A3) holds with
e(s) = d where d is any positive constant < 1. Now apply Theorem 1.1.
2. Recurrent solutions
We shall replace the condition (A3) by:
(A4) For any zGR" there is a positive constant Rz such that
S(x, x - z) < 1 + e(|x - *|) if \x- z\> Rz, B.1)
where e(r) is a continuous function satisfying
/•oo i Г rt e(s) 1
- exp - I -^ ds dt= oo for some Й* > 0. B.2)
•'Я* * [ JR* S J
For simplicity we have taken e(r) to be independent of z; but the
subsequent results are unaffected if e(r) is allowed to depend on z.
Notice that the function e(r) = l/(log r) satisfies B.2).
Theorem 2.1. Let (Ax), (A2), (A4) hold. Then the solution of A.1) is
recurrent, i.e., for any xElR" and for any ball Ba(z) = { y;
\y - z\ < a}, a > 0
Px{U(t) hits Ba(z) at a sequence of times increasing to 00} = 1. B.3)
Proof. We take, for simplicity, z = 0 and write Ba — Ba@). We shall first
construct a function/(x) = F(r) for r = \x\ > a such that
Lf[x) > 0 if |x| > a. B.4)

2. RECURRENT SOLUTIONS 201
Let 9(r) be a continuous function such that
S(x) < 9(\x\) if |x| > a, B.5)
9(r) = 1 + e(r) if r > Ro. B.6)
In view of A.9), if F{r) satisfies A.10), A.11), then B.4) holds. With the
definition A.13), the function
F(r) = - С e~m ds B.7)
satisfies both A.11) and A.10). In view of B.6), B.2),
F(r)^> - oo if r->oo. B.8)
We shall now apply the equality in A.15) to the present function F(r).
Making use of B.4) and taking т = тар л T, we get, after letting T -> oo,
F(a)Px(a) + F( 0)PX( /?) - F(\x\) > 0. B.9)
Taking /?->oo in B.9) and using B.8), we conclude that Px(/?)->0 if /?^>oo.
Hence, Px{a) = 1 - Px(/8)-»l if ^^oo. This means that
Px(B(a)) - 1 B.10)
where fi(a) is defined in A.17).
For any p > 0, let 3Bp = { у; | J/| = p}. Let
a < Л,! < Я2 < • • • <«„,<••-, йт^оо if m^oo.
Introduce Markov times:
Tx = first time ?(t) hits Ba;
Oj = first time > т1 such that ^(f) hits ЗВЯ ;
in general,
тт = first time > am_1 such that ?(t) hits Ba;
am = first time > rm such that ?(t) hits ЗВд^.
By Theorem 8.1.1, on the set where rm < oo also am < oo. By B.10),
Px(t1 < oo) = 1. Hence РД^ < oo) = 1, and by the strong Markov prop-
property (see Problem 4),
Px(t2<oo) = ?x??WXti<00 = 1. B.11)
where B.10) has been used in the last equality.
We now proceed by induction. Assuming that rm < oo a.s., we get
^(Tm + l < 00) = ?r?e@X,1<oo = L
Now, at each time t — rm, ?(t) hits Ba. Further, since |?(O| = ^m ~* °°
as m -* oo, Umm_ocam — oo; hence also lirajn.,,,,^ >¦ oo. This completes the
proof of B.3) (in case z - 0).

202 9 RECURRENT AND TRANSIENT SOLUTIONS
We shall now replace the condition (A4) by:
(A4) As |x|-»oo,
and the matrix (o?) has precisely two positive eigenvalues.
Theorem 2.2. Let (Ax), (A2), (A4) hold. Then, the solution of A.1) is
recurrent.
Proof. We perform an orthogonal transformation x—»x' that takes (a?) into
a new matrix (a?) with afi = 0 if i ф / or if i = j > 3, and a? = 1 if
i = 1, 2. In the new coordinates, the condition (A4) is satisfied with
e(r) = l/(log r). Now apply Theorem 2.1.
Remark. Suppose (A4) is replaced by
ац(х) ~* a?i ^ \x\ ~* °°' 2 l^i(x)l= °( T~T I ^ \x\ —* °°
where the matrix (a^) has precisely one positive eigenvalue. Then the
assertion of Theorem 2.1 remains valid, with the same proof; here e(r) =
— d where d is any positive constant < 1.
Example. Consider the case where n > 2, b{ = 0, and a is such that
OO* = (Ощ), Ощ = Sy + — SjX, (f = |x|);
g(r) is a Lipschitz continuous function vanishing near r = 0, and
ц < g(r) < M where jx > — 1, M < 00 (/ч, М const).
The eigenvalues of (<fy(x)) are 1 (with multiplicity n — 1) and 1 + g. Hence
(a^(x)) is positive definite for all iER". Clearly,
S(ac) - 1 = — M = e(r).
Hence, if g(r) is such that e(r) = A/(log r) for some A > 1 (and all large r),
then the assertion of Theorem 1.1 holds. If g(r) is such that e(r) = l/(log r)
for all large r, then (A4) holds with e(s) = l/(log s) + C/s for some positive
constant C; consequently the assertion of Theorem 2.1 holds. This example
shows that conditions (A3), (A4) made in Theorems 1.1, 2.1 are rather sharp.

3. RATE OF WANDERING OUT TO INFINITY 203
This example also shows that the behavior asserted in Theorems 1.1 and
2.1 does not depend exclusively on the dimension n. In fact, given any e(r)
which converges to 0 as r—»oo, take
n — 2 — e /? „ , ч
g = x + ? (br all large r)
in the above example. Then the behavior of ?(t) does not depend on n; if
e(r) > A /log r for some A > 1, then ?(t) is transient, whereas if
e(r) < I/log r then ?(?) is recurrent.
3. Rate of wandering out to Infinity
In this section we return to the situation of Theorem 1.1. We shall
assume:
(A5) aJx), Ъг(х) are bounded functions in Rn, the а^(х) are uniformly
Holder continuous in Й", and, for all хёй", ?e«n,
n
> ao\U\2 (a0 positive constant).
We shall also assume that the function e(r) occurring in the condition (A3)
satisfies, for some r0 sufficiently large,
e(r) = d if r > r0 (d positive constant). C.1)
Theorem 3.1. Let (A^, (Ag) hold and let (A3) hold with e(r) satisfying C.1).
Then, for any 0 < в < ?, х?й",
^l C.2)
We first prove two lemmas.
Lemma 3.2. Let (Aj), (A5), (A3), and C.1) hold. Then there exists a positive
constant С such that for any a > r0, хёй",
РХ{Ш\ < a for some t > 0} < c( ^y ) . C.3)
Proof. Let R > a. Consider the Dirichlet problem:
LuR(x) - 0 If о < |x| < Д,
ы„ - 1 if |x| - а, ыя - 0 if |x| - Д.

204 9 RECURRENT AND TRANSIENT SOLUTIONS
Let f(x) = F(r) (r = \x\) be the function constructed in the proof of
Theorem 1.1. Then
satisfies:
Lv < 0 if a < |x| < Й,
v = 1 if |x| = a, v > 0 if |x| = R.
Hence, by the maximum principle,
0 < uR(x) < o(x). C.4)
By Ito's formula, if a < |x| < Й,
uR(x) = Px{U(t) hits ЬВа before it hits ЭВЯ). C.5)
Hence uR(x) f u(x) as R f oo, where
u(x) = Р»{|@ hits 9Ba for some t > 0}. C.6)
From C.4) we get
/r"exp[-!(*)]&
0 < u(x) < v(x) = -j- f—т/ \i j ¦
Using C.1) to estimate the right-hand side, we obtain
0<u(x)<C^ C.7)
where С is a constant independent of a, x.
Now, if \x\ < a, then the assertion C.3) is trivially true (with С = 1). If,
on the other hand, |x| > a, then the assertion C.3) follows from C.6), C.7).
Lemma 3.3. Let (A^, (A5), and (A3), C.1) hold, with d < n. Then there is a
positive constant С such that, if a > r0, \x\ < a/4, T > 0,
Px{Ut)\ < a for some t > T) < C'( ^ ) . C.8)
Proof. By the Markov property and Lemma 3.2,
P»{|?(t)| < a for some t>T) = ?*р?(яШ*I < a ior some
sl + J. C.9)
Denote by T(x, t, y) the fundamental solution of the parabolic operator

3. RATE OF WANDERING OUT TO INFINITY 205
L - 9/3*. By Theorem 6.4.5 and Problem 7, Chapter 6,
0 < Г(х, *,«/)< -Д exp| - Ml* ~ У| ] C.10)
where M, ц are positive constants.
We can write
I - f T(x, T, y) dy,
Ay\«*
J=Cctd[ -^T(x,T,y)dy.
J\y\>« \y\
We shall subsequently denote various positive constants by the same symbol
C. Substituting | у — x\ — pVT in the integral I and noting that
pVT = I у — x\ < 2a (since | y\ < a, \x\ < a),
we get
/ \n
Z^ /"• I л — 1 — до J ^ f~A 1
< С I p e ™ dp < Cl I .
JpVf <2a \ VY '
Substituting \y — x\ — pVT in the integral / and noting that
pVT = I у - x\ > a/2 (since \y\> a, \x\ < a/2),
| f/| > | у - x\ - \x\ > pVT /2 (since |x| < a/4 < | у - x\/2),
we get
KCa-f
since n — 1 — d>— 1. Substituting the estimates for I, / into C.9), the
assertion C.8) follows.
Proof of Theorem 3.1. Without loss of generality we may assume that
d < n. We apply Lemma 3.3 with T = 2m, a = 2<m+1)e where m is а
positive integer such that |x| < 2(m+1)e/4. We get
< te for some t,2m < t < 2m + 1}
< Px{\i{t)\ < 2(m+1" for some t > 2m) < J 2<>^" ] < C2m{"-l'2)d.

206 9 RECURRENT AND TRANSIENT SOLUTIONS
Since 2 2m('/2)d < oo, the Borel-Cantelli lemma implies that, with prob-
probability 1, the sequence of events
{|€(t)| < t" for some t, 2m < t < 2m + 1)
(where ?@) = x) occurs only finitely often. Hence
FxiW)\ > t" for a11 * sufficiently large) = 1.
Since this is true also for 9 replaced by any 9',9 < 9' <\, the assertion of
the theorem follows.
Corollary 3.4. Let (АД (A5), and (A3) hold. Then, for any 0 < 9 <\,
", the assertion C.2) holds.
Indeed, perform an orthogonal transformation as in the proof of Theorem
1.2. In the new coordinates the conditions (A3), C.1) hold.
Remark 1. By Theorem 3.6.1,
f — N*)l \
Px lim W =1=1.
Г0 V* log log t J
More generally, under the conditions of Theorem 8.4.2,
^U C.U)
J
where a is the inverse of шт1|;с|_+ооа(х). From C.11) it follows that
Thus the assertion C.2) (for any 9 <\) is rather sharp. However, this
assertion can still be strengthened as follows.
Let g(t) be a positive and monotone decreasing function for t > 0,
satisfying
00
(gBm)r<oo C.12)
m = l
where d is any constant < n such that C.1) holds. If in the proof of
Theorem 3.1 we replace t9 by t1//2g(t), then we conclude that the events
{\U(t)\ < *1/2g(*) for some t, 2m < t < 2m+1)
occur only finitely often. Consequently
Ш > ,) . X.

4. OBSTACLES 207
It can be shown (see Dvoretsky and Erdos [1]) that if the function g(t)
satisfies
2 (gBm))d=oo (d = n-2,n>3), C.14)
m = l
instead of C.12), then
Remark 2. Consider the example at the end of Section 2. If n = 2 and
g = -1/A + l/d) (d > 0), then the assertion of Theorem 3.1 holds. Thus,
even when n = 2, a diffusion process ?(?) may wander out to oo at a rate
> te, for any 0 < в < \.
4. Obstacles
We shall maintain the condition (Ax) but relax the condition j)
Let G be a closed bounded domain with C3 connected boundary 9G, and
let G = Rn\G.
Suppose the diffusion matrix (а^(х)) degenerates on 9G in such a way that
2 V*)'*'; if *g3G' DЛ)
where v = (vv . . . , vn) is the outward normal at x to 9G. This condition is
equivalent to
fc? (^ »*(*)*)= о,
i.e.,
2 ол(х)*4«0 if xG9G, 1 < к < п. D.2)
The expression 2 а-^(х)р{р, is called the normal diffusion to 9G at x.
Let p(x) = dist (x, 9G) for xG G и 9G, and let
G« - (lEGipW < e}.
Since 8G is in C3, p(x) is in C2(G,oU 9G) for some e0 > 0 sufficiently small.
Noting that
v{{x) -др(х)/дх, xGdG, 1 < t < n,

206 9 RECURRENT AND TRANSIENT SOLUTIONS
we deduce from D.2) that
" Эр(зс)
as
Taking the squares, we get
2 ^.)f f<«(.))- if ,E4, D.3)
where C is a positive constant.
Let D.1) hold and suppose a^EC1(G€o). By D.2), the vector
Tk = (alk anh) is tangent to 9G. Since the function 2 o-k dp/dxf
vanishes on 9G, its derivative with respect to the tangent vector of Tk also
vanishes on 9G. Consequently
^i ^> ^ V 9P
This gives, after using D.2) once more,
" Э2р п 9a«
2 аЦ "a—д~ = ~ 2 "д~ vi on 3G. D.4)
The vector with components
-i 1 ^ Ц
is called the Fichera drift. We shall impose the condition:
2 V, + I 2 «« ^4r > 0 on 9G. D.5)
If UjEC^GJ then, by D.4), this condition is equivalent to
2 (b*-| 2 ^)"«>o on9G-
i.e., the Fichera drift at 9G points into the exterior of G.
Notice that D.5) implies that
^ , Эр i » 92p
? h< "9*; + 2 4 2 x «. ^^ > " CP in Geo D.7)
where с is a positive constant.
Theorem 4.1. Let (Ax) ond D.1), D.5) hold. Then
Рх{?(*)EGforaut>0} -1 if xEG. D.8)

4. OBSTACLES 209
Proof. Let Я (x) be a C\6) function such that
X) tf * G G«, (fOT SOme
[ ' \\\ if |x| > M,
and ex < Л (ж) < M elsewhere; M is chosen so large that Ge с {x; \x\ < M}.
(To construct Л (ас), let
[рЫ if xec
[l [M if xeRn\G4
and take Л (ж) to be a moUifier of Л (х); see Problem 4, Chapter 4.)
Let V(ac) = 1/(Л (x)I for some e > 0. We have
LV--eR—X2 b4 |5-
i Щ
I V я I ,/ , i\R -«-2 ЭЛ ЭЛ R -?-i Э Д
1 V я I ,/, R
2 ^J '' [ v ' Эх,
„/ e v , ЭЛ , 1 v aH Г , , .. ЭЛ ЭЛ R Э2Д
1 Л T ' Э* 2 ^- я2 [ Эх Эл: Эх dx
2j i 57Г^
T ' Э*, 2 ^- я [ * / *
D.9)
In view of D.3) and D.7), the right-hand side of D.9) is bounded by ^V if
xEG(i, where ц is a positive constant. In view of the condition (A^, the
same estimate is valid also if |x| > M. It follows that
LV< pV for all x<=G. D.10)
Further,
V(x)^>oo if p(ac)->0, x G G. D.11)
We shall now use D.10), D.11) in order to prove D.8).
Denote by тр the hitting time of the set G1/p (p = 1, 2, . . .). By Ito's
formula,
ТрЛГ
e-"(W)v(?(Tp л Г)) - V(x) + Г e-'"Vie(€(*))o(€(*))dc(*)
•'о
+ f (LV-fiV)(€(*))e"'litfe. D.12)
Taking the expectation and using D.10), we get
and, as Г | oo,

210 9 RECURRENT AND TRANSIENT SOLUTIONS
Now, if D.8) is not true then the exit time т from G is finite on a set В of
positive probability. Since тр | т on B, the monotone convergence theorem
eives
gives
Since т < oo on B, we must then have PX(B) — 0, but this contradicts the
definition of B.
Theorem 4.1 motivates the following definition.
Definition. If the conditions D.1), D.5) are satisfied, then we say that 3G is
an obstacle from the outside.
Suppose D.5) is replaced by
on3G D.14)
with the same function p(x) as before. Define p(x) = dist (x, 9G) for x E. G,
and denote by v = (vx,. . . ,vn) the inward normal to 3G. Then D.14) holds
if and only if
2 Vi+ \ ? <>ц 4Az > 0 опЭС D.15)
where the left-hand side is evaluated by taking the limit as x tends to the
boundary from inside G. Using D.1), D.15), we can now duplicate the proof
of Theorem 4.1, taking Л (x) as a C2(G) function which coincides with p(x)
near 9G, and taking V(x) = 1/(й(х))'. We thus conclude:
Corollary 4.2. If (AY) and D.1), D.14) hold, then
Px{S(t) Eint G for all t > 0} = 1 if x eint G. D.16)
Definitions. If the conditions D.1), D.14) are satisfied, then we say that 3G
is an obstacle from the inside. If D.1) holds and if
? v, + !?«</ -Pb= ° on эс' DЛ7)
i = l л i,/ = l 0Xi 0Xi
then we say that 3G is a two-sided obstacle. In case D.17) is replaced by
either D.5) or D.14), we speak of a one-sided obstacle.
We shall now consider the case of degeneracy at one point z, and assume:
a4(z) = 0, bt(z) = 0 for 1 < i, / < n. D.18)
Notice that ац(г) = 0 for 1 < i, / < n if and only if aif(z) = 0 for
1 < i, j < n.

4. OBSTACLES 211
Using D.18) one can show that
Px{${t)^z for all t >0} = 1 if хфг. D.19)
Indeed, the proof is similar to the proof of Theorem 4.1. Here we take
V(x) = l/(K(x))e where R(x) = |x - z\.
A point z for which D.18) holds is called a point obstacle or a point of
total degeneracy.
We shall extend the previous considerations to the case where there are
several obstacles.
Let Gl5 . . . , Gk be mutually disjoint sets in Л"; if 1 < / < k0, G, consists
of one point z,, and, if fc0 + 1 < j < k, G(- is a closed bounded domain with
C3 connected boundary 3G^. Let
к
G = U Ц. G = Rn\G
p((x) = dist (x, №) if xgint G(..
We shall assume:
= 0, b^) = 0 if Ki,j < n, KhKko, D.20)
о^^^= G on ЭС^,, for k0 + 1 < h < k, D.21)
X V<+ i 2 °f/ a > ° on 9Gh' for k,y+ 1 < h < k.
D.22)
Theorem 4.3. 7/ (At) and D.20)-D.22) Ш, Леп
Px{?(*)eG/oraM * > 0} = 1 if x?G. D.23)
The proof is similar to the proof of Theorem 4.1. Here one takes
V = 1/Л* where Д(х) is a C2 function in G, satisfying:
Ph(x) if Ph(x) < €i (I < h < k),
|x| if |x| > M
and €x < i?(x) < M elsewhere, where tj is sufficiently small and M is
sufficiently large.
In the following two sections we shall replace the condition (A2) by the
weaker condition:
The matrix (ац(х)) is positive definite for all x E G.
We shall also assume that D.20)-D.22) hold and
dGh is C3 diffeomoфhic to a sphere, for *<, + 1 < h < k. D.24)

212 9 RECURRENT AND TRANSIENT SOLUTIONS
We shall then extend the results of Sections 1, 2. The following lemma
will be needed.
Lemma 4.4. Let (Aj), (A?), and D.24) hold. Then there exists a continuous
function R(x) in R" having the following properties:
(i) R (x) is in C2(G);
(ii) Л (x) > 0 in G;
(iii) R (x) = ph(x) if ph(x) < e0, Л (x) > ^ if minh ph(x) > e0, for
some e0 sufficiently small;
(iv) R (x)-> oo i/|x| -» oo;
» а2д(х)
(v) 2 ац(х) я < 0 i/ x?G, ОхЛ(х) = 0; there exist pre-
cisely к — 1 points x in G where DxЛ (х) = 0.
The number e0 will be smaller than
min dist (G{, G,).
Notice that the conditions D.20)-D.22) are not assumed in this lemma.
Proof. By the proof of the Schoenflies theorem (see Morse [1]) there is a
diffeomorphism у = f(x) of the exterior of U „^ onto the exterior of
U =1G{ in R", where G{, . . . , G^ are points situated on the y1 axis and
Gfco+i> • • • > Qt are balls with centers on the y1 axis; the center of G! lies to
the left of the center of Gf'+1. Furthermore, this diffeomorphism preserves
the distance functions (to (J Gf and to U -G-) as long as the distance is
sufficiently small. (The condition D.24) is needed in order to apply the
Schoenflies theorem.) Suppose for simplicity that k0 = 0, к = 2. Denote by
(Cj, 0, . . . , 0) the midpoint of the segment connecting the center
(<*!, 0, . . . , 0) of G{ to the center (a2, 0, . . ., 0) of G^, it is assumed that
a1 < a2. Construct a positive C2 function <j>( y') (where y' = (y2, . . . , yn))
on the plane yx = cv which increases radially, with grad ф( у') ф 0 if
у' ф 0, such that Эф/Эу( = 0, З2ф/Эу( byi = 0 B < i, / < n) at у' = 0.
Denote by ц{ the radius of G/. Construct a C2 function ^( j/j), positive for
ax + jx1 < yY < az — /*2> such that
i ~ ai ~ ^ for ° < !/i - ai - ^i < 5i>
a, - /la - ft for 0 < a2 - ft2 - ft < 8X
where 5j is sufficiently small, and such that ^'( ft) ф 0 if ft Ф c1 and
4- (Cl) = ф@, .... 0), ^.'(Cl) = 0, ^»(Cl) < 0.
We now construct a C2 positive function X( y) for у §?. (G[ U G2), which
extends the functions «f>, ^ and the distance function from G[ U Gj (as long

5. TRANSIENT SOLUTIONS FOR DEGENERATE DIFFUSION 213
as the distance is sufficiently small). This function is to satisfy:
gradX(!/)^0 if у Ф (clf 0, . . . , 0);
grad\(y) = O, -gp?, =0 if (i, /) Ф A, 1),
< 0 at у = (cl 0, . . . , 0);
M!/) = I У\ ^ I !/l is sufficiently large.
The construction of such a function \( y) can be accomplished by intro-
introducing a family of curves у , connecting (ab 0, . . . , 0) to (a2, 0, . . . , 0) and
intersecting the plane y1 = c1 orthogonally at (cl8 у'). Х( у) is defined along
у . such that its tangential derivative vanishes only at yx = cy
Define R (x) = \{f(x)). Clearly DXR (x) Ф 0 if x ф х* where f(x*) is the
point (cj, 0, . . . , 0). Furthermore, as easily seen,
This completes the proof if fc0 = 0, к = 2. The proof for any fc0, fc is similar.
Lemma 4.5. Let (Aj), (A?) ЬоИ and suppose that
Gh is C3 diffeomorphic to a closed ball, for k0 + 1 < h < fc. D.25)
T7ien Лете exists a function R(x) satisfying (i)-(v) of Lemma 4.4 and, in
addition,
(iv') R (x) = x if \x\ is sufficiently large.
For proof see Problem 8. By Milnor [1], D.24) implies D.25) if n > 5; the
same is true, by conformal mappings, if n = 2.
5. Transient solutions for degenerate diffusion
We shall extend the results of Section 1 to the case the diffusion matrix
(a{Ax)) degenerates in such a way that (A?) holds, and G-, G are as in Lemma
4.4.
For later references we state the following condition:
(Bj) (i) Gh = {zh} if h = 1, . . . , fc0, Gh is a bounded closed domain
with C3 connected boundary if h — k0 + 1, . . . , k.
(ii) The conditions D.20)-D.22) and D.24) hold.
(iii) The matrix (a^(x)) is nondegenerate for x6=G, where
G - Rn\G,G - Uj_,Cv

214 9 RECURRENT AND TRANSIENT SOLUTIONS
Let R (x) be the function constructed in Lemma 4.4 or 4.5, and set
/D 1 v / \ 9Д oR
« = 2 ь((х) f- + \ ±
Then, ifg(x) =
Lg(x) = 6В[Ф"(Л) + ? Ф'(Я)] + RQV(R). E.1)
If R (x) is as in Lemma 4.5, then the function S (x) defined in A.3) satisfies
Я2(х)О(х)
S(x) = H —— if |x| is sufficiently large.
As will be proved in Chapter 11, if lim Q(x) < 0 as i?(x)^0 and as
R(x)-»oo, then
minJph(^))]^Oif^«)}=l if xGG.
Thus, in this case, ?(t) neither wanders out to oo nor visits any open
neighborhood in G at a sequence of times increasing to oo. Hence, in order
to obtain the type of behavior asserted in Sections 1, 2, we shall have to
impose different conditions on Q(x).
As will be shown, in order to generalize the results of Sections 1, 2 to the
present case where (Bj) holds, we do not need to change the conditions (A3),
(A4) near infinity. We only need to impose a condition on Q(x) near
R(x) = 0. This condition is:
(B2) For some 0 < 50 < e0 there is a continuous function e(r), defined
for 0 < r < 50, such that
Q(x) >R4^<RW if
and
fS0 ^ Г -So e(t)
I - exp I —— dt
J. s r Л t
ds^oo if r^o. E.3)
We can take, for example, e(s) = - l/[log A/s)].
Remark. The condition: lim Q(x) < 0 as i?(x)-»O is a "stability condi-

5. TRANSIENT SOLUTIONS FOR DEGENERATE DIFFUSION 215
tion," meaning that G attracts ?(t) near the boundary. The condition (B2) can
be interpreted as a weak "repelling condition."
Theorem 5.1. Let (A^, (B^, (B2), and (A3) hold, Then the solution of A.1)
is transient in G, i.e.,
*ЛЙЗ,Ш\ -оо}-1 if xed E.4)
Let G be contained in a ball BH> = { y; \ y\ < iL).
We shall first prove a lemma.
Lemma 5.2. Let (Aj), (B^, (B2) /ioZd and let /3 > R.. Then
Px {?(*) hits the set ЭВ„ for some t > 0} = 1 if x E G П В„. E.5)
Here Bp is the ball { y; 11/| < /?} and ЭВд is its boundary.
Pro©/. We first construct a function g(x) = Ф(Л(х)) for xGGflBg, such
that
Lg(x) < 0 if xG G nB^g. E.6)
Denote by f l5 . . . , ffc_j the points in G where Dxi?(x) = 0. By slightly
modifying the proof of Lemma 4.4, we obtain a modified function R (x) for
which the points fj, . .. , ?k_1 lie outside the ball Bg. We shall work, in the
present proof, with this modified function Д(х); it coincides with the original
R(x) in the e0-neighborhood of G.
We claim that there is a continuous function в(г) satisfying:
л *(*)(?(*)
1 + > 6(RW if G G ПВ
6>(r) = i + e(r) if о < r < So. E.8)
Indeed, since & (x) Ф 0 if xG G, DxR(x) ^ 0, the left-hand side of E.7) is a
bounded function if xeGnBg, minhph(x) > 50. Using the assumption E.2),
the existence of 9(r) (satisfying E.7), E.8)) follows.
Let Ф(г) be a solution of
Ф"(г) + — Ф'(г) =0 if 0 < r < г* E.9)
Ф'(г) < 0 if 0 < r < r0 E.10)
where r0 = max^^^ Д(х). Then, upon using E.1), E.7) we conclude that
g(x) - Ф(Я(х)) satisfies E.6).

216 9 RECURRENT AND TRANSIENT SOLUTIONS
A solution of E.9), E.10) is given by
©W-j^eJjT*^ a|*. E.11)
In view of E.3),
Ф(г)^>оо if r->0. E.12)
By Ito's formula and E.6),
ЕХФ(№\) - ?*Ф(|*1) - Exf Lg(€(e)) ds < 0, E.13)
where т is any bounded stopping time such that ?(s) E G П Bg if 0 < s
< т. Denote by Gc (e > 0) the closed «-neighborhood of G, and denote by
аф the hitting time of the set Ge U ЭВ„. By Theorem 8.1.1, Px(ac0 < oo) = 1
if x ? Ge, x E B^,. Denote by Px(e) the probability that |(aeJ8) ? Ge (given
|@) = x), and by Px(/3) the probability that ?(aei8) E ЭВ„ (given ?@) = x).
Substituting т = o€/3 /\ T in E.13), and taking Г -» oo, we get
Taking e-»0 and using E.12), we deduce that Px(e)-»0 if e->0. Hence
Рх(Р)->1 if e^O. But this implies the assertion of the lemma.
Let R* < a < R and denote by^ tR the hitting time of the ball BH. By
Lemma 5.2, Px{tR < oo) = 1 if x E G. Hence, by the strong Markov property
(cf. the proof of Theorem 1.1),
Р,(Иа)) = ExPiitR)(U*(a)) < ЕхР,((й)(Я(а)),
where the notation A.17), A.19) is used.
Now, in the domain { y, | y\ > R*} the matrix (ajf( y)) is nondegenerate.
Since the condition (A3) holds, the estimate A.16) remains valid for
x = i(tR). Hence
^ (F(r)^Oifr^oo).
We can now complete the proof, as in the case of Theorem 1.1, by taking
R-»oo and noting that a can be arbitrarily large.
Theorem 5.3. Let (AY), (B^, (B2), and (A3) hold. Then for any x G G the
assertion E.4) holds.
Proof. Proceeding as in the proof of Theorem 5.1, it remains to establish
the estimate A.16). We now perform an orthogonal transformation as in the
proof of Theorem 1.2.

6. RECURRENT SOLUTIONS FOR DEGENERATE DIFFUSION 217
6. Recurrent solutions for degenerate diffusion
Theorem 6.1. Let (Ад), (В^, (B2), D.25) and (A4) hold. Then the solution of
A.1) is recurrent in G, i.e., for any ball Ba(z) = { у; \ у — z\ < a], a > 0,
lying entirely in G, the assertion B.3) holds.
Proof. For simplicity we take z = 0. Let R be any positive number such
that R > a and such that G is contained in the interior of the ball BH. We
shall prove: if x G 3BH, then
Px {?(*) hits the ball Ba for some t > 0} = 1. F.1)
Let 5 > 0 be sufficiently small so that 5 < e0, the closed 6-neighborhood
Ge of G lies in the interior of GR, and Gs n Ba = 0.
We shall construct a function f(x) = F(R (x)) (R (x) as in Lemma 4.5) in
Rn\Gs such that
Lf(x) > 0 if x ? Я" \ Gs, F.2)
F(r)-> -oo if r^oo. F.3)
Notice that at the points fm (m = 1, . . . , к - 1) where DXR (x) = 0, & =0
and, therefore, by the property (v) of R(x), Q < 0. Hence, Q(x) < 0 in a
neighborhood of each point fm. It follows that there is a continuous function
9(r),S<r< oo, such that
1 + ^? < в (R) if x G Я" \ G5. F.4)
In view of (A4), we can choose 0{r) so that, for all r sufficiently large,
9(r) = 1 + e(r) where e(r) satisfies B.2). If we now define F(r), for r > 5, by
F(f)- - fe-^ds, I(s)= f'^- dt
Js Js t
then F'(r) < 0, and F.2), F.3) hold. Arguing as in the proof of Theorem 2.1
(fallowing B.8)) with the present function/(x) = F(R(x)) and with the set
J^y; \y\ < a) replaced by Gs, we conclude that for any 16Rn\Gs,
Px {?(*) hits the set Gs for some t > 0} = 1. F.5)
Let j8 > R and denote by Ga/8S the domain bounded by 3G5, 3Ba, ЭВ^.
Denote by rap the exit time from this domain. By Theorem 8.1.1,
Рх(тар < oo) = 1 if xGGaj8S. Using the strong Markov property we get, for
any xG3Bj,,
Px {?(*) hits Ba for some t > 0} = Px {?(t + та0) hits Ba for some t > 0}
hits Ba for some t > 0}
inf Pu ii(t) hits Ba for some t > 0).

218 9 RECURRENT AND TRANSIENT SOLUTIONS
Denote by oaR the hitting time of BaU ЭВЯ. By Lemma 5.2, if у ? 3G5,
then P (oaR < oo) = 1. Hence, by the strong Markov property,
Py {?(*) hits Ba for some t > 0}
= Py{?(* + a^) hits Ba for some t > 0}
= ExP4(<w{?(f) hits Ba for some t > 0}
= РуШа«н)еэва} + (i - РуШааЯ)еэва}) inf p,{€(t)hitsBe
for some t>0]. "
Combining this with the previous inequality, and setting
Px(a) = Px{?{t) hits Ba for some t > 0},
we arrive at the inequality
+[1 - ^(y)] mf P,(a)}. F.6)
Note that yap(x) is the solution u(x) of the Dirichlet problem:
Lu = 0 in Gai85,
м = 1 on 3Ba
« = 0 on 3G8U ЭВ„.
Hence, by the strong maximum principle, yaj8(x) is positive on 3GH. Further,
yap(x) t if j8 f. Similar assertions are true for yi8(x). Since yai8(x)
+ у^(х) < 1, we conclude that
yfi(x)<9<l (хеЭСн) F.7)
where 9 is a constant independent of /?.
Notice that F.5) implies that
?,*(*) +Y/j(*M if ^oo (xG9GH). F.8)
Let
Let tj be a positive number, and choose Xq in 3BH so that
Pa > -Pxo(«) - V-
Let t/0 be a point in 3Ge such that

7. THE ONE-DIMENSIONAL CASE 219
Applying F.6) with x = x0, we get
К > Уа0Ы + УРЫ{ p(y0) +[1 " f*(«/o)R } - 2т,.
Hence
Taking j8 sufficiently large and using F.8) with x = x0, we get
Denote the expression in braces by Xg. From F.7) it follows that
Xo > 1 - в > 0. Hence
Since tj is arbitrary, Pa = 1. This implies that Pa(x) = 1 for all x G3GH, i.e.,
F.1) holds.
Having proved F.1), we can now easily complete the proof of Theorem
6.1 by the argument given in the proof of Theorem 2.1 (from B.10) on).
Instead of B.10) we use F.1), and instead of Theorem 8.1.1 we use Lemma
5.2.
Remark. Theorem 6.1 remains true if the condition (A4) is replaced by (A'4),
or by the conditions in the remark following Theorem 2.2.
7. The one-dimensionai case
Consider the case of one stochastic differential equation
d?(t) = o(i(t)) dw(t) + b(?(t)) dt. G.1)
We shall assume that the condition (At) of Section 1 holds in the present
case of n = 1, and that a(x) > 0 for all x. Let
2b{u)
|
exp
du
dz. G.2)
Л) а («)
This is a particular solution of
|aV + bv' = 0. G.3)
Notice that ф(х) is strictly monotone increasing. Set ф(оо) = Итх_+00ф(х),
ф(- oo) = Итх_>_00ф(х). These numbers may be finite or infinite.
Theorem 7.1. (а) 7/ф(-оо) - - oo, then
- 1. G.4)

220 9 RECURRENT AND TRANSIENT SOLUTIONS
1/ф(оо) = oo, then
Px{inf€(t)«-oo}-l. G.5)
(b) If ф(оо) = oo, ф(— oo) > — oo, then
Px{sup€(t)<oo}-1,
t>0
(c) If Ф(— oo) = -oo, ф(оо) < oo, then
Px{ inf €(t) > -oo} = 1,
(d) If ф(оо) < oo, ф(- oo) > - oo, then
-{.of «о > —} - '.{&«.)
ф(оо) — Ф\%)
ф( 00 ) — ф( — 00)
ф(х) — ф(— оо)
ф(оо) — ф( — оо)
1.
G.6)
G.7)
G.8)
G.9)
G.10)
G11)
•
G.12)
It follows that if ф(— oo) = — oo and ф(оо) = oo, then the solution of G.1)
is recurrent; in all other cases the solution of G.1) is transient.
Proof. Let x1 < x < x2 and denote by rx[xv x2] the first time ?(f) leaves
(xi> xz)> given |@) = x. By Ito's formula (cf. Problem 12(c), Chapter 8),
Noting that
Px { sup€(t) > x2) > Px {«(tJ*!, x2]) - x2), G,14)
and taking a^-» — oo in G.13) we obtain
Pxfsup€(t) > x2} = 1,
provided ф(— oo) = — oo. Since x2 can be arbitrarily large,
p
This proves G.4). The proof of G.5), in case ф(оо) = oo is similar.

7. THE ONE-DIMENSIONAL CASE 221
To prove (b), take xx-^> - oo in G.13) to obtain
ф(х)-ф(-оо)
Px{sup€(t) > x2) =
This implies that
p/ t/,w \ ф(ха) ф(х)
P{ sup€(t) < x2) = —-— . G.15)
loo ' ф(х2) - ф(-оо)
Taking sc2-»oo the assertion G.6) follows.
To prove G.7), let у < x and denote by ту the first time |(t) = y. By (a),
Рх(т < oo) = 1. Hence, by the strong Markov property, for any x2 > y,
Px { supK* + ry) > x2] = ExPy[ p l{ll\\
But
т„) > x2] = Px{ sup|(t) > x2] > Px{Timt(t) > x2].
Therefore,
Taking y^ — oo we conclude that
Р,[Шф) > x2}=0.
Here x2 is any real number. Taking x2—» — oo, G.7) follows.
The proof of (c) is similar to the proof of (b), and will be omitted.
We proceed to prove (d). The relation
ф(оо) — ф{— оо)
follows by taking x2—»oo in G.15). Similarly one proves that
- oo} -
ф(оо)-ф(-оо)
If we prove G.12), then the second equalities in G.10), G.11) follow from the
equalities in G.17), G.16) respectively. Thus it remains to prove G.12).
For any small € > 0 and large M > 0, choose xx < — M and x2 > M so
that Xj < x < x2,

222 9 RECURRENT AND TRANSIENT SOLUTIONS
and
, , ф(х2) - ф(М)
Px { inf Ш) > M ) = -j-^ -jr^- > 1 - «. G.19)
Let т = tJ*!, x2]. Then, by the strong Markov property,
Px {€(*) g [ - M, M] for t > r) = ?,P€(T){€(t) 2 [ - M, M] for f > 0}
= p,[€(t) = xb]pJ tof€(t) > м] + рх[€(т) = ^[шрад < -м]
> 1 - ?
by G.18), G.19). Consequently
Taking first с -» 0 and then M -» 00, the assertion G.12) follows.
PROBLEMS
1. The solution of @.1) is recurrent in G if and only if for every x E G and
for any open set V in G, @.2) holds with "finite random times tm" replaced
by "finite Markov times tm." [Hint: Let W be a closed domain contained in
V. Define tf = first time > ^_x + 1 such that |(t) hits W.]
2. Let ?(*) be a continuous process for * > 0, with values in R". Let
? = (?(f) hits a closed set Г at a sequence of random times tm increasing to
infinity}. Prove that there exists a sequence of bounded, Borel measurable
functions /к(х1} . . . , xk) such that
A(€(*i).-•¦.€&))->x* a-s-
and the sequence fk does not depend on the process ?(*). [Hint: Let { «/,} be
a dense set in Г. Show that E = {Bm i.o.} where
вт=П U U№)-y,l
i=l m<»y<m + l »=1
where {n} is the sequence of positive rational numbers; then cf. Problem 9,
Chapter 2.]
3. Let Г be a closed set, т the hitting time of Г by the solution |(t) of A.1).
Suppose Px(t < 00) = 1. Prove:
Px [i{t + t) hits Г at a sequence of times increasing to 00 j
= EIP^(T){|(f) hits Г at a sequence of times increasing to 00 j.
[Hint: Use the preceding problem and the method of solution of Problem 10,
Chapter 2.]

PROBLEMS 223
4. Prove the first relation in B.11). [Hint: Use the method of proof in
Problem 10, Chapter 2.]
5. Let G be a bounded domain with C3 boundary dG. Suppose that (A!)
holds and that the condition (B2) holds with R (x) replaced by
p(x) = dist (x, dG), xGG. Suppose further that (aJx)) is nonsingular for all
xG int G. Prove that t-(t) is recurrent in int G.
6. Consider G.1) and set
i \ Г00 i fz 2b(u) , 1 ,
(x) = | exp -J —— du\ dz.
•>% [ Jo a (m) J
Prove that if I^x) < oo, I2(x) < oo, then
where ?(*) is any solution of G.1).
7. Let x = 0 be a two-sided obstacle for G.1), i.e., ст@) = Ь@) = 0. As-
Assume that a(x) > 0 if x > 0 and set
a(x) = a(e*)e-*, b(x) = b(ex)e~x + \o\ex)e~^.
Let i//(jc) be the solution of
?afy" + fei//' = 0, <//@) = 0, <// @) = 1.
Prove that if ip(oo) = oo, \j/(- oo) > - oo, then
Px{|(«) > 0, (lim ф) = 0} = 1 if x > 0.
8. Prove Lemma 4.5. [Hinf: Let k0 = 0, Jfc = 1. By Palais [1] there is a
diffeomorphism у = f(x) of Я" \ (int Gx) onto | j/| > 1 such that f(x) = x
for x outside a neighborhood of Gv Take
with suitable ^8, /8 = 0 outside a neighborhood of Gl5 /8 = 1 in a smaller
neighborhood. For &Q = 0, ifc = 2 there exists (by applying Palais [1] twice) a
diffeomorphism у = f(x) of K" \ int(G! U G2) onto R" \ int(G^ U Gg) and
G{, G2 are as in Lemma 4.4. Replace in (*) |/(x)| by A(/(x)) where \(y) is
constructed as in the proof of Lemma 4.4.]

Bibliographical Remarks
Chapters 1-3. The material of these chapters is standard. We have used,
as sources, Breiman [1], Dynkin [1, 2], Doob [1], Varadhan [1], and McKean
Chapters 4-5. The stochastic integral, Ito's formula, and the basic theory
of stochastic differential equations are due to K. Ito [1, 2] (and the re-
references given there). These results are presented in a book form in Doob [1]
and, in more detail, in McKean [1] and Gikhman and Skorokhod [1, 2]. The
present treatment is based on Gikhman and Skorokhod [2], but it in-
incorporates some results from McKean [1], such as the exponential martingale
inequality. Polynomial mollifiers occur in Courant and Hilbert [1]. For more
details on mollifiers, see Friedman [2].
Stroock and Varadhan [1] have developed existence and uniqueness
theory, for stochastic differential equations, in which the drift coefficient is
any bounded measurable function and the diffusion matrix is continuous,
bounded, and uniformly positive definite. In [3] they allow the diffusion
matrix to degenerate. They have also extended, in [2], their results from [1]
to disfusion processes restricted to a given domain B, i.e., when the path hits
3 0 it is returned into 0 in a prescribed direction. (Some earlier results in this
area are described in McKean [1], Gikhman and Skorokhod [1, 2], and in the
references given there.)
Stochastic differential equations with dw(t) replaced by the differential of
a Poisson process are studied in Gikhman and Skorokhod [2], Komatsu [1],
and Stroock [1]. Some applications in stochastic control are given in Won-
ham [1].
Chapter 6. Theorem 1.2 is due to Freidlin [1] and to Phillips and Sarason
[1]. The material of Sections 2, 3, 4 is based on Friedman [1]. The connection
between solutions of partial differential equations and stochastic differential
224

BIBLIOGRAPHICAL REMARKS 225
equations is discussed in detail in Gikhman and Skorokhod [2], and in an
expository article by Freidlin [2].
Chapter 7. The results of this chapter are based on Girsanov [1]. Earlier
work in this direction was done by Cameron and Martin [1].
Chapter 8. The results of this chapter are due to Friedman [3]. Theorem
4.1 for n = 1 is due to Kulinic [2] (see also Gikhman and Skorokhod [2]) who
also proved, for n = 1, other related results [1, 3, 4].
Chapter 9. Sections 1-3, 5-6 are due to Friedman [4]. The concept of
obstacle and the results of Section 4 are due to Friedman and Pinsky [1]. The
results of Section 3, in the special case of Brownian motion, were first proved
by Dvoretzky and Erdos [1]. Theorem 7.1 is given in Gikhman and Skorok-
Skorokhod [2].

References
L. Breiman
[1] Probability. Addison-Wesley, Reading, Massachusetts, 1968.
R. H. Cameron and W. T. Martin
[1] The transformation of Wiener integrals by nonlinear transformations, Trans. Amer. Math.
Soc. 66 A949), 253-283.
R. Courant and D. Hilbebt
[1] Methods of Mathematical Physics, Vol. 1. Wiley (Interscience), New York, 1953.
J. L. Doob
[1] Stochastic Processes. Wiley, New York, 1967.
A. Dvohetsky and F. Erdos
[1] Some problems on random walk in space, Proc. 2nd Berkeley. Symp. Math. Statist.
Probability, 1950. Univ. of California Press, Berkeley, California, pp. 353-367.
E. B. Dynkin
[1] Foundation of die Theory of Markov Processes. Pergamon Press, New York, 1960.
[2] Markov Processes, Vols. 1, 2. Springer-Verlag, Berlin, 1965.
W. Feller
[1] An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York, 1971.
M. I. Fbeidlin
[1] On the factorization of nonnegative definite matrices, Theor. Probability Appl. 13 A968),
354-356 [Tear. Verojatnost. i Primenen 13 A968), 375-378].
[2] Markov processes and differential equations, in Progress in Mathematics, Vol. 3. Plenum
Press, New York, 1969.
A. Friedman
[1] Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, New
Jersey, 1964.
[2] Partial Differential Equations. Holt, New York, 1969.
[3] Limit behavior of solutions of stochastic differential equations, Trans. Amer. Math. Soc. 170
A972), 359-384.
[4] Wandering out to infinity of diffusion processes, Tram. Amer. Math. Soc. 184 A973),
185-203.
226

REFERENCES 227
A. Friedman and M. A. Pinsky
[1] Asymptotic stability and spiraling properties of solutions of stochastic equations, Trans.
Amer. Math. Soc. 186 A973), 331-358.
I. I. Gikhman and A. V. Skorokhod
[1] Introduction to the Theory of Random Processes. Saunders, Philadelphia, 1965.
[2] Stochastic Differential Equations. Naukova Dumka, Kiev, 1968. (Springer-Verlag, Vol. 72,
New York, 1973).
I. V. Gihsanov
[1] On transforming a certain class of stochastic processes by absolutely continuous substitu-
substitution of measures, Them. Probability Appl 5 A960), 285-301 [Tear. Verojatnost. i Primenen.
5 (I960), 314-330].
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[2] Lectures on Stochastic Processes. Tata Inst. Fundamental Res., Bombay, 1961.
К. 1тб and H. P. McKean
[1] Diffusion Processes and Their Sample Paths. Springer-Verlag, Berlin, 1965.
T. Komatsu
[1] Markov processes associated with certain integro-differential operators, Osaka J. Math. 10
A973), 271-303.
G. L. KuLiNii
[1] On the limit behavior of the distribution of the solution of a stochastic diffusion equation,
Theor. Probability Appl. 12 A967), 497-199 [Tear. Verojatnost. i Primenen 12 A967),
548-551].
[2] Limiting behavior of distributions of a solution of the stochastic diffusion equation, Ukrain.
Mat. Z. 19 A967), 119-125.
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Ukrain. Mat. Z. 20 A968), 396-400.
[4] Limit distributions of a solution of the stochastic diffusion equation, Theor. Probability
Appl 13 A968), 478-482 [Tear. Verojatnost. i Primenen. 13 A968), 502-506].
P. Levy
[1] Theorie de I'AddiHon de Variables Aleatoires. Gauthier-Villars, Paris, 1937.
H. P. McKean
[1] Stochastic Integrals. Academic Press, New York, 1971.
J. Milnoh
[1] Lectures on the h-Cobordism Theorem. Princeton Math. Notes, Princeton, New Jersey,
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113-115.
P. S. Palais
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228 REFERENCES
D. W. Stroock
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D. W. Stroock and S. R. S. Varadhan
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345-400; Part II, ibid., 479-530.
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Comm. Pure Appl. Math. 24 A972), 651-713.
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W. M. Wonham
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Mathematics, Vol. 2, pp. 131-212. Academic Press, New York, 1970.

Index
Absorbing point, 34
Adapted to, 58,165
Exponential martingale inequality,
93
88,
В
Backward parabolic equation,
149
Barrier, 134, 138
Blumenthal's zero-one law, 29
Brownian motion, 36
n-dimensional, 50, 156
normalized, 36
Feller property, 23
124, Fichera drift, 20,
First boundary value problem, 134
First initial-boundary value problem,
138
Formal adjoint, 142
Forward parabolic equation, 149
Fundamental solution, 141,143
Cameron-Martin-Girsanov's theorem,
157
Cauchy's problem, 139
Chapman-Kolmogorov's equation, 18
Countable stochastic process, 2, 4
Diffusion process, 114
Dirichlet problem, 134
Discrete time stochastic process,
Distribution functions, 1,3,5
consistent, 3, 5
Drift, 36
Dynkln's formula, 31
Elliptic operator,
Exit tlm», 26
132
G
Girsanov's formula, 169
Girsanov's theorem, 169
Green's identity, 143
H
Hitting time, 25
Indefinite integral, 67
Infinitesimal generator, 30
Intrinsic clock, 76
Intrinsic time, 76
Iterated logarithm, law of, 41, 50,186
Ito'e formula. 81. 84, 90
Ito'e Integral. 59, 62
Ito'e process, 166

INDEX
Jensen's inequality, 16
Joint normal distribution, 53
from the outside,
one-sided, 210
two-sided, 210
210
Kolmogorov's construction, 1, 4, 6
Kolmogorov's equation, 124
Kolmogorov's extension theorem, 2
Kolmogorov's model, 21
Parabolic boundary, 138
Parabolic operator, 134
Penetration time, 26
Point of total degeneracy, 211
Point obstacle, 211
Polynomial modifier, 94
Laws of the iterated logarithm, 41, 50,
186
Linear stochastic differential equation,
126
Markov process, 18, 21
time-homogeneous, 30
temporally homogeneous, 30
Markov property, 21
strong, 25
Markov time, 24, 30
Markov transition function, 18
Martingale, 9, 10, 15
Martingele inequality, 9
exponential, 88, 93
Maximum principle for elliptic equa-
equations,
strong, 133
weak, 132
Msximum principle for parabolic equa-
equations,
strong, 135
weak, 135
Modifier, 94
N
Nonanticipative function,
Normal boundary, 138
Normal diffusion, 207
Normal distribution, 53
55
Obstacle, 210
from the Inside,
210
Reccurrent process, 196
S
Sample paths, 6
Separable stochastic process, 6
Separant, 6
Set of separability, 6
Stationary transition probability, 30
Step function, 56
Stochastic differential, 78, 90
Stochastic differential equations, 98
solution of, 98, 111
uniqueness of solutions of, 113
Stochastic functional differential equa-
equations, 170
Stochastic integral, 59, 62, 89, 165
Stochastic process, 3, 4
continuous, 7
continuous in probability, 7
left continuous, 7
right continuoua, 7
separable, 6
version of, 6
Stochastic sequence, 2, 4
Stopping time, 11, 24, 30
extended, 11, 30
Strong Markov property, 25
Submartingale, 9, 10
Temporally homogeneous Markov pro-
process, 30
Time-homogeneous Markov process,
30

INDEX
Transient process, 196 V
Transition density function, 149 .
Transition probability, 18 Variance, 36
и w
Uniform ellipticity, 133 Wandering out to infinity, 203
Uniform parabolicity, 134 Wiener process, 36
Uniformly integrable sequence, 11 n-dimensional, 50

Probability and Mathematical Statistics
A Series of Monographs and Textbooks
Editors Z. W. Birnbaum E. Lukacs
University of Washington Bowling Green State University
Seattle, Washington Bowling Green, Ohio
1. Thomas Ferguson. Mathematical Statistics: A Decision Theoretic Approach. 1967
2. Howard Tucker. A Graduate Course in Probability. 1967
3. K. R. Parthasarathy. Probability Measures on Metric Spaces. 1967
4. P. Revesz. The Laws of Large Numbers. 1968
5. H. P. McKean, Jr. Stochastic Integrals. 1969
6. B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev. Mathematical Methods of
Reliability Theory. 1969
7. Demetrios A. Kappos. Probability Algebras and Stochastic Spaces. 1969
8. Ivan N. Pesin. Classical and Modern Integration Theories. 1970
9. S. Vajda. Probabilistic Programming. 1972
10. Sheldon M. Ross. Introduction to Probability Models. 1972
11. Robert B. Ash. Real Analysis and Probability. 1972
12. V. V. Fedorov. Theory of Optimal Experiments. 1972
13. K. V. Mardia. Statistics of Directional Data. 1972
14. H. Dym and H. P. McKean. Fourier Series and Integrals. 1972
15. Tatsuo Kawata. Fourier Analysis in Probability Theory. 1972
16. Fritz Oberhettinger. Fourier Transforms of Distributions and Their Inverses: A
Collection of Tables. 1973
17. Paul Erdos and Joel Spencer. Probabilistic Methods in Combinatorics. 1973
18. K. Sarkadi and I. Vincze. Mathematical Methods of Statistical Quality Control. 1973
19. Michael R. Anderberg. Cluster Analysis for Applications. 1973
20. W. Hengartner and R. Theodorescu. Concentration Functions. 1973
21. Kai Lai Chung. A Course in Probability Theory, Second Edition. 1974
22. L. H. Koopmans. The Spectral Analysis of Time Series. 1974
23. L. E. Maistrov. Probability Theory: A Historical Sketch. 1974
24. William F. Stout. Almost Sure Convergence. 1974

25. E. J. McShane. Stochastic Calculus and Stochastic Models. 1974
26. Z. Govindarajulu. Sequential Statistical Procedures. 1975
27. Robert B. Ash and Melvin F. Gardner. Topics in Stochastic Processes. 1975
28. Avner Friedman. Stochastic Differential Equations and Applications, Volumes 1
and 2.1975
29. Roger Cuppens. Decomposition of Multivariate Probabilities. 1975
In Preparation
Eugene Lukacs. Stochastic Convergence, Second Edition
Harry Dym and Henry P. McKean. Gaussian Processes: Complex Function Theory
and the Inverse Spectral Method
A
В 5
С 6
D 7
E 8
F 9
G 0
H 1
I 2
J 3