Lecture Notes in Mathematics 1702
Editors:
A. Dold, Heidelberg
F. Takens, Groningen
B. Teissier, Paris

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Jin Ma Jiongmin Yong
Forward-Backward
Stochastic
Differential Equations
and Their Applications
^i Springer

Authors
Jin Ma JiongminYong
Department of Mathematics Department of Mathematics
Purdue University Fudan University
West Lafayette, IN 47906-1395 Shanghai, 200433, China
USA e-mail: jyong@fudan.edu.cn
e-mail: majin@math.purdue.edu
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Ma, Jin:
Foreward backward stochastic differential equations and their
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Tokyo : Springer, 1999
(Lecture notes in mathematics ; 1702)
ISBN 3-540-65960-9
Mathematics Subject Classification A991): Primary: 60H10, 15, 20, 30; 93E03;
Secondary: 35K15, 20, 45, 65; 65M06, 12, 15, 25; 65U05; 90A09, 10, 12, 16
ISSN 0075-8434
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To
Yun and Meifen

Preface
This book is intended to give an introduction to the theory of forward-
backward stochastic differential equations (FBSDEs, for short) which has
received strong attention in recent years because of its interesting structure
and its usefulness in various applied fields.
The motivation for studying FBSDEs comes originally from stochastic
optimal control theory, that is, the adjoint equation in the Pontryagin-type
maximum principle. The earliest version of such an FBSDE was introduced
by Bismut [1] in 1973, with a decoupled form, namely, a system of a usual
(forward) stochastic differential equation and a (linear) backward stochastic
differential equation (BSDE, for short). In 1983, Bensoussan [1] proved the
well-posedness of general linear BSDEs by using martingale representation
theorem. The first well-posedness result for nonlinear BSDEs was proved
in 1990 by Pardoux-Peng [1], while studying the general Pontryagin-type
maximum principle for stochastic optimal controls. A little later, Peng [4]
discovered that the adapted solution of a BSDE could be used as a
probabilistic interpretation of the solutions to some semilinear or quasilinear
parabolic partial differential equations (PDE, for short), in the spirit of the
well-known Feynman-Kac formula. After this, extensive study of BSDEs
was initiated, and potential for its application was found in applied and
theoretical areas such as stochastic control, mathematical finance, differential
geometry, to mention a few.
The study of (strongly) coupled FBSDEs started in early 90s. In his
Ph.D thesis, Antonelli [1] obtained the first result on the solvability of an
FBSDE over a "small" time duration. He also constructed a
counterexample showing that for coupled FBSDEs, large time duration might lead to
non-solvability. In 1993, the present authors started a systematic
investigation on the well-posedness of FBSDEs over arbitrary time durations, which
has developed into the main body of this book. Today, several methods have
been established for solving a (coupled) FBSDE. Among them two are
considered effective: the Four Step Scheme by Ma-Protter-Yong [1] and the
Method of Continuation by Hu-Peng [2], and Yong [1]. The former provides
the explicit relations among the forward and backward components of the
adapted solution via a quasilinear partial differential equation, but requires
the non-degeneracy of the forward diffusion and the non-randomness of the
coefficients; while the latter relaxed these conditions, but requires
essentially the "monotonicity" condition on the coefficients, which is restrictive
in a different way.
The theory of FBSDEs have given rise to some other problems that are
interesting in their own rights. For example, in order to extend the Four
Step Scheme to general random coefficient case, it is not hard to see that
one has to replace the quasilinear parabolic PDE there by a quasilinear
backward stochastic partial differential equation (BSPDE for short), with a

viii Prefeice
strong degeneracy in the sense of stochastic partial differential equations.
Such BSPDEs can be used to generalize the Feynman-Kac formula and even
the Black-Scholes option pricing formula to the case when the coefficients of
the diffusion are allowed to be random. Other interesting subjects generated
by FBSDEs but with independent flavors include FBSDEs with reflecting
boundary conditions as well as the numerical methods for FBSDEs. It is
worth pointing out that the FBSDEs have also been successfully applied to
model and to resolve some interesting problems in mathematical finance,
such as problems involving term structure of interest rates (consol rate
problem) and hedging contingent claims for large investors, etc.
The book is organized as follows. As an introduction, we present several
interesting examples in Chapter 1. After giving the definition of
solvability, we study some special FBSDEs that are either non-solvable or easily
solvable (e.g., those on small durations). Some comparison results for both
BSDE and FBSDE are established at the end of this chapter. In Chapter
2 we content ourselves with the linear FBSDEs. The special structure of
the linear equations enables us to treat the problem in a special way, and
the solvabiUty is studied thoroughly. The study of general FBSDEs over
arbitrary duration starts from Chapter 3. We present virtually the first
result regarding the solvability of FBSDE in this generality, by relating the
solvability of an FBSDE to the solvability of an optimal stochastic control
problem. The notion of approximate solvability is also introduced and
developed. The idea of this chapter is carried on to the next one, in which
the Four Step Scheme is established. Two other different methods leading
to the existence and uniqueness of the adapted solution of general FBSDEs
are presented in Chapters 6 and 7, while in the latter even reflections are
allowed for both forward and backward equations. Chapter 5 deals with a
class of linear backward SPDEs, which are closely related to the FBSDEs
with random coefficients; Chapter 8 collects some applications of FBSDEs,
mainly in mathematical finance, which in a sense is the inspiration for much
of our theoretical research. Those readers needing stronger motivation to
dig deeply into the subject might actually want to go to this chapter first
and then decide which chapter would be the immediate goal to attack.
Finally, Chapter 9 provides a numerical method for FBSDEs.
In this book all "headings" (theorem, lemma, definition, corollary,
example, etc.) will follow a single sequence of numbers within one chapter
(e.g.. Theorem 2.1 means the first "heading" in Section 2, possibly followed
immediately by Definition 2.2, etc.). When a heading is cited in a different
chapter, the chapter number will be indicated. Likewise, the numbering
for the equations in the book is of the form, say, E.4), where 5 is the
section number and 4 is the equation number. When an equation in different
chapter is cited, the chapter number will precede the section number.
We would like to express our deepest gratitude to many people who
have inspired us throughout the past few years during which the main
body of this book was developed. Special thanks are due to R. Buck-
dahn, J. Cvitanic, J. Douglas Jr., D. Duffie, P. Protter, with whom we

Prefeice ix
enjoyed wonderful collaboration on this subject; to N. El Karoui, J. Jacod,
I. Karatzas, N. V. Krylov, S. M. Lenhart, E. Pardoux, S. Shreve, M. Soner,
from whom we have received valuable advice and constant support. We
particularly appreciate a special group of researchers with whom we were
students, classmates and colleagues in Fudan University, Shanghai, China,
among them: S. Chen, Y. Hu, X. Li, S. Peng, S. Tang, X. Y. Zhou. We also
would like to thank our respective Ph.D. advisors Professors Naresh Jain
(University of Minnesota) and Leonard D. Berkovitz (Purdue University)
for their constant encouragement.
JM would like to acknowledge partial support from the United States
National Science Fundation grant #DMS-9301516 and the United States
Office of Naval Research grant #N00014-96-l-0262; and JY would Uke to
acknowledge partial support from Natural Science Foundation of China, the
Chinese Education Ministry Science Foundation, the National Outstanding
Youth Foundation of China, and Li Foundation at San Francisco, USA.
Finally, of course, both authors would like to take this opportunity to
thank their families for their support, understanding and love.
Jin Ma, West Lafayette
Jiongmin Yong, Shanghai
January, 1999

Contents
Preface vii
Chapter 1. Introduction 1
§1. Some Examples 1
§1.1. A first glance 1
§1.2. A stochastic optimal control problem 3
§1.3. Stochastic differential utility 4
§1.4. Option pricing and contingent claim valuation 7
§2. Definitions and Notations 8
§3. Some Nonsolvable FBSDEs 10
§4. Well-posedness of BSDEs 14
§5. SolvabiUty of FBSDEs in Small Time Durations 19
§6. Comparison Theorems for BSDEs and FBSDEs 22
Chapter 2. Linear Equations 25
§1. Compatible Conditions for Solvability 25
§2. Some Reductions 30
§3. Solvability of Linear FBSDEs 33
§3.1. Necessary conditions 34
§3.2. Criteria for solvability 39
§4. A Riccati Type Equation 45
§5. Some Extensions 49
Chapter 3. Method of Optimal Control 51
§1. Solvability and the Associated Optimal Control Problem 51
§1.1. An optimal control problem 51
§1.2. Approximate Solvability 54
§2. Dynamic Programming Method and the HJB Equation 57
§3. The Value Function 60
§3.1. Continuity and semi-concavity 60
§3.2. Approximation of the value function 64
§4. A Class of Approximately Solvable FBSDEs 69
§5. Construction of Approximate Adapted Solutions 75
Chapter 4. Four Step Scheme 80
§1. A Heuristic Derivation of Four Step Scheme 80
§2. Non-Degenerate Case—Several Solvable Classes 84
§2.1. A general case 84
§2.2. The case when h has linear growth in z 86
§2.3. The case when m = 1 88
§3. Infinite Horizon Case 89

xii Contents
§3.1. The nodal solution , 89
§3.2. Uniqueness of nodal solutions 92
§3.3. The Umit of finite duration problems 98
Chapter 5. Linear, Degenerate Backward Stochastic
Partial Differential Equations 103
§1. Formulation of the Problem 103
§2. Well-posedness of Linear BSPDEs 106
§3. Uniqueness of Adapted Solutions Ill
§3.1. Uniqueness of adapted weak solutions Ill
§3.2. An Ito formula 113
§4. Existence of Adapted Solutions 118
§5. A Proof of the Fundamental Lemma 126
§6. Comparison Theorems 130
Chapter 6. The Method of Continuation 137
§1. The Bridge 137
§2. Method of Continuation 140
§2.1. The solvability of FBSDEs Unked by bridges 140
§2.2. A priori estimate 143
§3. Some Solvable FBSDEs 148
§3.1. A trivial FBSDE 148
§3.2. Decoupled FBSDEs 149
§3.3. FBSDEs with monotonicity conditions 151
§4. Properties of Bridges 154
§5. Construction of Bridges 158
§5.1. A general consideration 158
§5.2. A one dimensional case 161
Chapter 7. FBSDEs with Reflections 169
§1. Forward SDEs with Reflections 169
§2. Backward SDEs with Reflections 171
§3. Reflected Forward-Backward SDEs 181
§3.1 A priori estimates 182
§3.2 Existence and uniqueness of the adapted solutions 186
§3.3 A continuous dependence result 190
Chapter 8. AppUcations of FBSDEs 193
§1. An Integral Representation Formula 193
§2. A Nonlinear Feynman-Kac Formula 197
§3. Black's Consol Rate Conjecture 201
§4. Hedging Options for a Large Investor 207
§4.1. Hedging without constraint 210
§4.2. Hedging with constraint 219
§5. A Stochastic Black-Scholes Formula 226

Contents xiii
§5.1. Stochastic Black-Scholes formula 227
§5.2. The convexity of the European contingent claims 229
§5.3. The robustness of Black-Scholes formula 231
§6. An American Game Option 232
Chapter 9. Numerical Methods for FBSDEs 235
§1. Formulation of the Problem 235
§2. Numerical Approximation of the Quasilinear PDEs 237
§2.1 A special case 237
§2.1.1. Numerical scheme 238
§2.1.2. Error analysis 240
§2.1.3. The approximating solutions {u(")}~=i 244
§2.2 General case 245
§2.2.1. Numerical scheme 247
§2.2.2. Error analysis 248
§3. Numerical Approximation of the Forward SDE 250
Comments and Remarks 257
References 259
Index 269

Chapter 1
Introduction
§1. Some Examples
To introduce the forward-backward stochastic differential equations (FBS-
DEs, for short), let us begin with some examples. Unless otherwise
specified, throughout the book, we let {fl,J-, {.7^i}t>o,P) be a complete filtered
probability space on which is defined a d-dimensional standard Brownian
motion W{t), such that {J-t}t>o is the natural filtration oiW{t), augmented
by all the P-null sets. In other words, we consider only the Brownian
filtration throughout this book.
§1.1. A first glEince
One of the main differences between a stochastic differential equation (SDE,
for short) and a (deterministic) ordinary differential equation (ODE, for
short) is that one cannot reverse the "time". The following is a simple
but typical example. Suppose that d = 1 (i.e., the Brownian motion is
one-dimensional), and consider the following (trivial) differential equation:
A.1) dY{t) = 0, te[0,T],
where T > 0 is a given terminal time. For any ^ £ R we can require
either y@) = ^ or Y{T) = ^ so that A.1) has a unique solution Y{t) = ^.
However, if we consider A.1) as a stochastic differential equation (with
null drift and diffusion coefficients) in Ito's sense, things will become a
little more complicated. First note that a solution of an Ito SDE has to
be {J"t}t>o-adapted. Thus specifying y@) and Y{T) will have essential
difference. Consider again A.1), but as a terminal value problem:
,T„^ jdYit)=0, te[0,T],
where ^ £ L^^{Q,;K), the set of all .T^r-measurable square integrable
random variables. Since the only solution to A.2) is Y{t) = ^, Vt £ [0,7],
which is not necessarily {.7^t}t>o-adapted unless ^ is a constant, the
equation A.2), viewed as an Ito SDE, does not have a solution in general!
Intuitively, there are two ways to get around with this difficulty: A)
modify (or even remove) the adaptedness of the solution in its definition;
B) reformulate the terminal value problem of an SDE so that it may
allow a solution which is {.7^t}t>o-adapted. We note here that method A)
requires techniques such as new definitions of a backward Ito integral, or
more generally, the so-called anticipating stochastic calculus. For more on
the discussion in that direction, one is referred to the books of, say, Kunita

2 Chapter 1. Introduction
[1] and Nualart [1]. In this book, however, we will content ourselves with
method B), because of its usefulness in various applications as we shall see
in the following sections.
To reformulate A.2), we first note that a reasonable way of modifying
the solution Y{t) = ^ so that it is {J"i}i>o-adapted and satisfies Y{T) = ^
is to define
A.3) Yit)^E{^\Tt}, te[0,T].
Let us now try to derive, if possible, an (Ito) SDE that the process Y{-)
might enjoy. An important ingredient in this derivation is the Martingale
Representation Theorem (cf. e.g., Karatzas-Shreve [1]), which tells us that
if the Rltration {Tt}t>o is Brownian, then every square integrable
martingale M with zero expectation can be written as a stochastic integral with
a unique integrand that is {.7"t}t>o-progressjVei7 measurable and square
integrable. Since the process Y{-) defined by A.3) is clearly a square
integrable {.7^i}i>o-martingale, an application of the Martingale Representation
Theorem leads to the following representation:
A.4) Y{t) = y(o) + / z{s)dwis), vt e [o,t],
Jo
a.s.,
where Z(-) £ L%{0,T;WI), the set of all {J"t}t>o-adapted square integrable
processes. Writing A.4) in a differential form and combining it with A.3)
(note that ^ is .T^x-measurable), we have
jdY{t) = Z{t)dW{t), te[0,T],
^'■'^ \YiT)=^.
In other words, if we reformulate A.2) as A.5); and more importantly,
instead of looking for a single {.7^t}t>o-adapted process Y{-) as a solution
to the SDE, we look for a pair {Y{-), Z{-)) (although it looks a little strange
at this moment), then finding a solution which is {.7^t}t>o-adapted becomes
possible! It turns out, as we shall develop in the rest of the book, that
A.5) is the appropriate reformulation of a terminal value problem A.2)
that possesses an adapted solution {Y, Z). Adding the extra component
Z(-) to the solution is the key factor that makes finding an adapted solution
possible.
As was traditionally done in the SDE literature, A.5) can be written
in an integral form, which can be deduced as follows. Note from A.4) that
A.6) y@) = Y{T) - / Z{s)dW{s) ^i- f Z{s)dW{s).
Jo Jo
Plugging A.6) into A.4) we obtain
A.7) Y{t) = y@) + / Zis)dW{s) =i~ j Z{s)dW{s), \/t e [0,T].

§1. Some examples 3
In the sequel, we shall not distinguish A.5) and A.7); each of them is called
a backward stochastic differential equation (BSDE, for short). We would like
to emphasize that the stochastic integral in A.7) is the usual (forward) Ito
integral.
Finally, if we apply Ito's formula to |l^(i)P (here | • | denotes the usual
Euclidean norm, see §2), then
A.8) E\e = E\Y{t)\'+1 E\Z{s)\'ds, Vte[0,T].
Thus ^ = 0 implies that Y = 0 and Z = 0. Note that equation A.7) is
linear, relation A.8) leads to the uniqueness of the {.7^t}t>o-adapted solution
(y(-), Z{-)) to A.7). Consequently, if ^ is a non-random constant, then by
uniqueness we see that Y{t) = ^ and Z{t) = 0 is the only solution of
A.7), as we expect. In the following subsections we give some examples
in stochastic control theory and mathematical finance that have motivated
the study of the backward and forward-backward SDEs.
§1.2. A stochastic optimsil control problem
Consider the following controlled stochastic differential equation:
j dX{t)=[aX{t)+bu{t)]dt + dW{t), te[0,T],
\x@) = :r,
where X{-) is called the state process, u(-) is called the control process.
Both of them are required to be {.7^i}t>o-adapted and square integrable.
For simplicity, we assume X, u and W are all one-dimensional, and a and
h are constants. We introduce the so-called cost functional as follows:
A.10) J{u) = \e[J^ [\X{t)\^ + \u{t)\^]dt+\X{T)\^].
An optimal control problem is then to minimize the cost functional A-10)
subject to the state equation A.9). In the present case, it can be shown
that there exists a unique solution to this optimal control problem (in fact,
the mapping u i-> J{u) is convex and coercive). Our goal is to determine
this optimal control.
Suppose u(-) is an optimal control and X{-) is the corresponding
(optimal) state process. Then, for any admissible control v{-) (i.e., an {J-t}t>0'
adapted square integrable process), we have
Q^i^ + ^'")- Ji^)
A.11)
e
/■T
-^ e{ / [X{t)i{t) + u{t)v{t)\dt + X(T)^(T)}, e -^ 0,
where ^(■) satisfies the following variational system:.
[ dm = Hit) + bv{t)]dt, t e [o,T],
1 ^@) = 0.

4 Chapter 1. Introduction
In order to get more information from A.11), we introduce the following
adjoint equation:
(dYit) = -[aYit)+Xit)]dt + Zit)dWit), te[0,T],
^ ' ' \y{T)=X{T).
and we require that the processes Y{-) and Z() both be {.7^t}i>o-adapted.
It is clear that A-13) is a BSDE with a more general form than the one we
saw in §1.1, since Y{-) is specified at t = T, and X{T) is .T^r-measurable in
general.
Now let us assume that A.13) admits an adapted solution {Y{-),Z{-)).
Then, applying Ito's formula to Y{t)^{t), one has
E[XiT)aT)]=E[YiT)aT)]
A.14) =eI {[-aYit)-Xit)]m+Yit)[aat)+bvit)]}dt
= E [ [- X{t)^{t) + bY{t)v{t)]dt.
Jo
Hence, A.11) becomes
A.15) 0<E f [bY{t) + u{t)]v{t)dt.
Jo
Since v{-) is arbitrary, we obtain that
A.16) u{t) = -bY{t), a.e.t £ [0,T], a.s.
We note that since Y{-) is required to be {.7^t}t>o-adapted, the process
u(-) is an admissible control (this is why we need the adapted solution for
A.13)!). Substituting A.16) into the state equation A.9), we finally obtain
the following optimality system:
A.17)
dX{t) = \aX{t) - b'^Y{t)\dt + dW{t),
te[0,T],
dY{t) = -[aY{t)+X{t)\dt + Z (t) dW (t),
X{0)=x, Y{T) = X{T).
We see that the equation for X{-) is forward (since it is given the initial
datum) and the equation for Y(-) is backward (since it is given the final
datum). Thus, A.17) is a coupled forward-backward stochastic differential
equation (FBSDE, for short). It is clear that if we can prove that A.17)
admits an adapted solution {X{-),Y{-), Z{-)), then A.16) gives an optimal
control, solving the original stochastic optimal control problem. Further, if
the adapted solution {X{-),Y{-), Z{-)) of A.17) is unique, so is the optimal
control u(-).
§1.3. Stochastic differential utility
Two of the most remarkable applications of the theory of BSDEs (a
special case of FBSDEs) in finance theory have been the stochastic differential

§1. Some examples 5
utility and the contingent claim valuation. In this and the following
subsections, we describe these problems from the perspective of FBSDEs.
Stochastic differential utility is an extension of the notion of recursive
utility to a continuous-time, stochastic setting. In the simplest discrete,
deterministic model (see, e.g., Koopmans [1]), the problem of recursive utility
is to find certain utility functions that satisfy a recursive relation. For
example, assume that the consumption plans are denoted by c = {co,ci, ■ • •},
where ct represents the consumption in period t, and the current utility is
denoted by Vt, then we say that V = {Vt : t = 0,1, ■ ■ ■} defines a recursive
utility if the sequence Vq , Vi, • • • satisfies the recursive relation:
A.18) Vt = W{ct,Vt+i), t = 0,l,---,
where the function W is called the aggregator. We should note that in
A.18), the recursive relation is backwards. The problem can also be stated
as finding a utility function U defined on the space of consumption plans
such that, for any t = 0,1, • • •, it holds that Vt — V{{ct, ct+\ ,••}), where V
satisfies A.18). In particular, the utility function V can be simply defined
by f/({co,ci,-- •}) = Vo, once A.18) is solved.
In the continuous-time model one often describes the consumption plan
by its rate c = {c(t) : t > 0}, where c(t) > 0, Vt > 0 (hence the accumulate
consumption up to time t is L c(s)ds). The current utility is denoted by
Y{t) ^ U{{c{s) : s>t}), and the recursive relation A.18) is replaced by a
differential equation:
A.19) ^ = -f{c{t),Y{t)),
where the function / is the aggregator. We note that the negative sign in
front of / reflects the time-reverse feature seen in A.18). Again, once a
solution of A.19) can be determined, then U{c) = Y{0) defines a unitiliy
function.
An interesting variation of A.18) and A.19) is their finite horizon
version, that is, there is a terminal time T > 0, such that the problem is
restricted to 0 < t < T. Suppose that the utility of the terminal consumption
is given by u{c{T)) for some prescribed utility function u, then the
(backward) difference equation A.18) with terminal condition Vt = u{c{T)) can
be solved uniquely. Likewise, we may pose A.19), the continuous
counterpart of A.18), as a terminal value problem with given Y{T) = u{c{T)), or
equivalently,
A.20) Y{t)=u{c{T)) +j f{c{s),Y{s))ds, t e [0,T].
In a stochastic model (model with uncertainty) one assumes that both
consumption c and utility Y are stochastic processes, defined on some
(filtered) probability space {fl,T,{Tt}t>o,P)- A standard setting is that at
any time t >0 the consumption rate c{t) and the current utility Y{t) can
only be determined by the information up to time t. Mathematically, this

6 Chapter 1. Introduction
axiomatic assumption amounts to saying that the processes c and Y are
both adapted to the filtration {J-t}t>o- Let us now consider A.20) again,
but bearing in mind that c and Y are {.7^t}i>o-adapted processes. Taking
conditional expectation on both sides of A.20), we obtain
A.21) Yit) = E{Yit)\Tt} = E{uiciT)) + J ficis),Yis))ds\Tt},
for all t G [0,T]. In the special case when the filtration is generated by a
given Brownian motion W, just as we have assumed in this book, we can
apply the Martingale Representation Theorem as before to derive that
A.22) Y{t) = u{c{T)) + f f{c{s),Y{s))ds-I Z{s)dWs, t e [0,T].
That is, {Y,Z) satisfies the BSDE A.22). A more general BSDE that
models the recursive utility is one in which the aggregator / depends also
on Z. The following situation more or less justifies this point. Let U be
another utility function such that U = ip o U ioi some C^ function ip with
f'ix) > 0, Va; (in this case we say that U and U are ordinally equivalent).
Let us define u = ipou, Y{t) = ipiY{t)), 'Z{t) = (p'{Y{t))Z{t), and
f{c, y,z) = (p {if {y))f{c, if (y)) - . ,. .. ^-
Then an application of Ito's formula shows that (Y, Z) satisfies the BSDE
A.22) with a new terminal condition u{c{T)) and a new aggregator /, which
now depends on z.
The BSDE A.22) can be turned into an FBSDE, if the consumption
plan depends on other random sources which can be described by some
other (stochastic) differential equations. The following scenario, studied by
DufRe-Ceoffard-Skiadas [1], should be illustrative. Consider m agents
sharing a total endowment in an economy. Assume that the total endowment,
denoted by e, is a continuous, non-negative, {.7^i}i>o-adapted process; and
that each agent has his own consumption process c* and utility process Y^
satisfying
A.23) Y\t) = Uiic\T)) + I ric\s),Y'is))ds + I Z'{s)dW{s),
for t G [0,T]. For a given weight vector a G R!^, we say that an allocation
Ca = (c^, • ■ ■,c^) is a-efficient if
m m m
A.24) ^aif/i(ci) =snp{j2<^iUi{c') \ ^c\t) < e(t), t G [0,T],a.s.},
i=l i=l i=l
where Uiic') = Y'{0).
It is conceivable that the a-efRcient allocation Ca is no longer an
independent process. In fact, using techniques of non-linear programming

§1. Some examples 7
it can be shown that, under certain technical conditions on the
aggregators /*'s and the terminal utility functions Uj's, the process Ca takes the
form: Ca(t) = K{X{t),e{t),Y{t)), for some R"*-valued function K, and
A = (A^, • • •, A"*), derived from a first-order necessary condition of the
optimization problem A.24), satisfies the differential equation:
A.25) dX'it) = X'{t)b'{t,X{t),Y{t))dt; t £ [0,T],
with b%X,y,cu) = ^qifll^^^^^^^^^^^^^^. Thus A.23) and A.25) form
an FBSDE.
§1.4. Option pricing Eind contingent claim valuation
In this subsection we discuss option pricing problems in finance and their
relationship with FBSDEs. Consider a security market that contains, say,
one bond and one stock. Suppose that their prices are subject to the
following system of stochastic differential equations:
a 26) (dPoit) = rit)Poit)dt, (bond);
^' ' \dP{t)=P{t)b{t)dt + P{t)a{t)dW{t), (stock),
where r(-) is the interest rate of the bond, 6(-) and a{-) are the appreciation
rate and volatility of the stock, respectively.
An option is by definition a contract which gives its holder the right to
sell or buy the stock. The contract should contain the following elements:
1) a specified price q (called the exercise price, or striking price);
2) a terminal time T (called the maturity date or expiration date);
3) an exercise time.
In this book we are particularly interested European options, which
specify the exercise time to be exactly equal to T, the maturity date. Let
us take the European call option (which gives its holder the right to buy)
as an example. The decision of the holder will depend, conceivably, on
P{T), the stock price at time T. For instance, if P{T) < q, then the
holder would simply discard the option, and buy the stock directly from
the market; whereas if P{T) > q, then the holder should opt to exercise the
option to make profit. Therefore the total payoff of the writer (or seller) of
the option at time t = T will be {P{T) — g)+, an .T^x-measurable random
variable. The (option pricing) problem to the seller (and buyer alike) is
then how to determine a premium for this contract at present time t = 0.
In general, we call such a contract an option if the payoff at time t = T can
be written explicitly as a function of P{T) (e.g., {P{T) — g)+). In all the
other cases where the payoff at time t = T is just an .T^x-measurable random
variable, such a contract is called a contingent claim, and the corresponding
pricing problem is then called contingent claim valuation problem.
Now suppose that the agent sells the option at price y and then invests
it in the market, and we denote his total wealth at each time t by Y{t).
Obviously, 1^@) = y. Assume that at each time t the agent invests a
portion of his wealth, say 7r(t), called portfolio, into the stock, and puts

8 Chapter 1. Introduction
the rest {Y{t) — 7r(t)) into the bond. Also we assume that the agent can
choose to consume so that the cumulative consumption up to time t is
C{t), an {^t}t>o-adapted, nondecreasing process. It can be shown that the
dynamics of Y{-) and the portfolio/consumption process pair Gr(-),C(-))
should follow an SDE as well:
J dVit) = {rit)Yit) + Zit)9it)}dt + Z{t)dW{t) - dC{t),
where Z{t) — Tr{t)a{t), and 6{t) = a~^{t)[b{t) — r{t)] (called risk premium
process). For any contingent claim H £ I/jr^(n,]R), the purpose of the
agent is to choose such a pair (tt, C) as to come up with enough money
to "hedge" the payoff H at time t = T, that is, Y{T) > H. Such a
consumption/investment pair, if exist, is called a hedging strategy against
H. The fair price of the contingent claim is the smallest initial endowment
for which the hedging strategy exists. In other words, it is defined by
A.28) y* = mi{y = Y{0); 3Gr,C), such that y"'^(T) > H}.
Now suppose H = g{P{T)), and consider an agent who is so prudent that
he does not consume at all (i.e., C = 0), and is able to choose tt so that
Y{T) = H = g{P{T)). Namely, he chooses Z (whence tt) by solving the
following combination of A.26) and A.27):
A.29)
' dP{t) = P{t)h{t)dt + P{t)a{t)dW{t),
dY{t) = {r{t)Y{t) + Z{tN{t)]dt + Z{t)dW{t),
,PiO)=p, Y{T) = g{P{T)),
which is again an FBSDE (an decouped FBSDE, to be more precise). An
interesting result is that if A.29) has an adapted solution {Y, Z), then the
pair Gr,0), where tt = Za~^, is the optimal hedging strategy and y = Y{0)
is the fair price! A more complicated case in which we allow the interaction
between the agent's wealth/strategy and the stock price will be studied
in details in Chapter 8. In that case A.29) will become a truly coupled
FBSDE.
§2. Definitions and Notations
In this sections we list all the notations that will be frequently used
throughout the book, and give some definitions related to FBSDEs.
Let ]R" be the n-dimensional Euclidean space with the usual Euclidean
norm I • I and the usual Euclidean inner product (•,•). Let R"*'"* be the
Hilbert space consisting of all (m x d)-matrices with the inner product
B.1) {A,B)^tT{AB^}, \/A,BeWr'"'.
Thus, the norm \A\ of A induced by inner product B.1) is given by \A\ =
^ytT {AA'^}. Another natural norm for A £ JR"*^'* could be taken as

§2. Definitions and notations 9
ll^ll =: y/iaaxa{AA'^) if we regard ^4 as a linear operator from JR"* to IR'*,
where a{AA'^) is the set of all eigenvalues of AA^. It is clear that the
norms | • | and || • || are equivalent since JR"*^** is a finite dimensional space.
In fact, the following relations hold:
mxd
B.2) Mil < ^JtT{AAT} = \A\ < ^/;^^^^||yl||, Vyl £ ]R'
where m A d = min{m,d}. We will see that in our later discussions, the
norm | • | in R"*^'* induced by B.1) is more convenient.
Next, we let T > 0 be fixed and {fl,J-, {.7"t}t>o,P) be as assumed at
the beginning of §1. We denote
• for any sub-u-field Q of T, L^{fl; IR"*) to be the set of all ^-measurable
IR'^-valued square integrable random variables;
• Ljr(n;L^@,T;]R")) to be the set of all {J"i}i>o-progressively
measurable processes X{-) valued in ]R" such that /^ E\X{t)\'^dt < oo. The
notation Ljr@,T; ]R") is often used for simplicity, when there is no
danger of confusion.
• Ljr(n;C([0,T];]R")) to be the set of all {.7^t}t>o-progressively
measurable continuous processes X{-) taking values in ]R", such that
-£^SUPtg[o,T]l^(*)P < 00.
Also, for any Euclidean spaces M and N, we let
• L%{0,T;W^'°°{M;N)) be the set of all functions / : [0,T] x M x
Cl -^ N, such that for any fixed 6 e M, {t,oj) i-> f{t,6;uj) is {Tt}t>o-
progressively measurable with f{t, 0; w) £ L^@,T; N), and there exists
a constant L > 0, such that
\fit,e;u!) - fit,e;ij)\ < L\e-e\, ye,eeM, a.e.t e [o,t], a.s.;
• L%^{n; W^'°°{WC-,Wr)) be the set of all functions ^ : ]R" x n -> jR"*,
such that u! i-> g{x; w) is .T^r-measurable for all x £ ]R" and x i-> g{x; w)
is uniformly Lipschitz in a; £ ]R" and g{0;uj) £ -Ljr(n;]R'").
Further, we define
B.3) M[0,T]^L%{n;C{[0,T];nn) ^ L'A^;C{[0,T];BJ^))
xL^@,T;]R^).
The norm of this space is defined by
\\iXi.),Yi-),Zi-))\\ = {E sup \Xit)\'+E sup \Yit)\^
^ te[o,T] te[o,T]
+ EJ \Z{t)\^dt] ,
for all {X{-),Y{-),Z{-)) £ M[Q,T]. It is clear that M[Q,T] is a Banach
space under norm B.4).

10
Chapter 1. Introduction
We are now ready to give the formal description of an FBSDE. Let us
consider an FBSDE in its most general form:
{dX{t) = b{t,X{t), Y{t), Z{t))dt + (T{t,X{t), Y{t), Z{t))dW{t),
dY{t) = h{t, X{t), Y{t),Z{i))dt + a{t,X{t), Y{t), Z{t))dW{t),
X@) = X, Y{T) = g{X{T)).
Here, the initial value x oi X{-) is in R"; and 6, a, h, a and g are some
suitable functions which satisfy the following Standing Assumptions: denoting
M = R" X jR"* X ]R^ one has
( b e L%-{0,T;W^'°°{M;WC)), a e L%-{0,T;W^'°°{M;WC'"')),
B.6) {
h e L^@,T;H^i'°°(M;]R'")), a e L%{0,T;W^''^iM;Wr'"^))
B.7)
Uei^^(n;W^^'°°(]R";]R'")).
Definition 2.1. A process {X{-),Y{-),Z{-)) e M[0,T] is called an adapted
solution of B.5) if the following holds for any t £ [0,T], almost surely:
Xit)=x+ f b{s,X{s),Y{s),Z{s))ds
Jo
+ f a{s,X{s),Y{s),Z{s))dW{s),
Jo
Yit) = g{X{T)) - I his, Xis), Yis), Z{s))ds
- j d{s,X{s),Y{s),Z{s))dW{s).
Furthermore, we say that FBSDE B.5) is solvable if it has an adapted
solution. An FBSDE is said to be nonsolvable if it is not solvable.
In what follows we shall try to answer the the following natural
question: for given b,a,h,a and g satisfying B.6) and for given x £ R", is
B.5) always solvable? In fact, what makes this type of SDE interesting is
that the answer to this question is not affirmative, although the standing
assumption B.6) is already quite strong from the standard SDE point of
view.
§3. Some Nonsolvable FBSDEs
In this section we shall first present some nonsolvability results, and then
give some necessary conditions for the solvability.
It is well-known that two-point boundary value problems for ordinary
differential equations do not necessarily admit solutions. On the other
hand, an FBSDE can be viewed as a two-point boundary value problem for
stochastic differential equations, with extra requirement that its solution is
adapted solely to the forward filtration. Therefore, we do not expect the
general existence and uniqueness result, even under the conditions that are

§3. Some nonsolvable FBSDEs 11
usually considered strong in the SDE literature; for instance, the uniform
Lipschitz conditions.
The following result is closely related to the solvability of two-point
boundary value problem for ordinary differential equations.
Proposition 3.1. Suppose that the following two-point boundary value
problem for a system of linear ordinary differential equations does not admit
any solution:
C.1)
X{0) = X, Y{T) = GXiT),
where A{-) : [0,T] -> ]r("+'") x ("+"») jg a deterministic integrabie function
and G £ R"*^". Tiien, for ajiyproperiy defined a{t,x, y, z) and d{t,x, y, z),
the following FBSDE:
C.2) <j "^ [ Y{t) ) - ■^^^) [ Y{t) ) "^^ + [a{t, X{t),Y{t), Z{t)) ) "^^^^'^
X{0) = X, y(T) = GX{T),
does not admit any adapted solution.
Here, by properly defined a, we mean that for any {X, Y, Z) £ M[0, T]
the process a{t,X{t),Y{t),Z{t)) is in L%{0,T;WC'^). The similar holds
for a.
Proof. Suppose C.2) admits an adapted solution {X,Y,Z) £ A^[0,T].
Then, {EX{-), EY{-)) is a solution of C.1), a contradiction. This proves
the assertion. D
There are many examples of systems like C.1) which do not admit
solutions. Here is a very simple one: (n = m = 1)
C.3)
We can easily show that for T = kn + ^ {k, nonnegative integer), the
above two-point boundary value problem does not admit a solution for any
a; e R \ {0} and it admits infinitely many solutions for a; = 0.
Using C.3) and time scaling, we can construct a nonsolvable two-point
boundary value problem for a system of linear ordinary differential
equations of C.1) type over any given finite time duration [0,T] with the
unknowns X, Y taking values in R" and R"*, respectively. Then, by
Proposition 3.1, we see that for any duration T > 0 and any dimensions n,m,, £ and
d for the processes X,Y,Z and the Brownian motion W{t), nonsolvable
FBSDEs exist.
X = Y,
Y=-X,
X{0) = X,
Y{T) =
-X{T)

12 Chapter 1. Introduction
The case that we have discussed in the above is a little special since
the drift of the FBSDE is linear. Let us now look at some more general
case. The following result gives a necessary condition for the solvability of
FBSDE B.1).
Proposition 3.2. Assume that b, a, h and a satisfy B.6). Assume further
that a and a are continuous in (t, x, y) uniformly in x, for each u! £ H;
and that g e C"^ <! Cj (]R";]R'") and is deterministic. Suppose for some
X e ]R", there exists a T > 0, such that B.5) admits an adapted solution
iX,Y,Z) eM[0,T] with
C.4) tv{gl^{X){aa''){-,X,Y,Z)}eL^AO,T;B.), l<i<m.
Then,
■mi \aiT,X{T),giX{T)),z)
C.5) ^eR'
- g,iX{T))aiT, X{T), g{X{T)),z)\ = 0, a.s.
Fbrtiermore, suppose there exists a To > 0, such that for all T £ @,To],
B.5) admits an adapted solution (X, Y, Z) (depending onT > 0) satisfying
the following:
C.6) / E{\h{s,X{s\ Y{s),Z{s))\^ + \a{s, X{s), Y{s),Z{s))f]ds < C,
Jo
for some constants C > 0 and /? > 2, independent ofTe @,To]. Tien,
C.7) E mi \d{0,x,g{x),z)-ga;{x)a{0,x,g{x),z)\ = 0, a.s.
Proof. Let {X,Y,Z) £ M[0,T] be an adapted solution of B.5). We
denote
his) = ih\s),---,h"^is)f,
h' = h'-{gi,b)-^tT{gi,aa''), l<i<m.
Here, we have suppressed X, Y, Z and we will do so below for the notational
simplicity. Clearly, h £ L%{0,T;'R"'). Next, for any i = 1,2, •••,m, by
Ito's formula
0=E\Y'{T)-g'{X{T))\'^
C.8)
^E\Y\t)-g\X{t))\^+E j^ \d'-gia\
+ E j^ 2[Y\s)-g\X{s))\[h'-{gi,h)-\tr{gl,aa'^)\ds
= E\Y\t) - g'{X{t))\'' +E j \d'- gia\''ds
+ eJ 2\Y'{s)-g\X{s))]h}{s)ds.

§3. Some nonsolvable FBSDEs 13
On the other hand, by B.5) and Ito's formula, we have
Y\s) - g'iXis)) = Y^s) - Y\T) + g\X{T)) - g\X{s))
C.9)
= - / h\r)dr - I (a'- gia)dW{r).
Combining C.8) and C.9), we obtain that
' ds
E\Y{t)-g{X{t))\^+E j \d~g,a\
= -'^eJ {Y{s)-g{X{s)),h{s))ds
C.10) =2E I {[ h{r)dr+ f [a - g^a]dWir),his))ds
= 2E I { f h{r)dr,h{s)) ds
<{T-t) f E\hir)\'^dr = o{T-t).
It
In the above, we have used the fact that
eUI [d-g,a]dW{r)Ms))\^Q.
J 8
Consequently, we have that
E inf \d{s,X{s),Y{s),z)-g^{X{s))a{s,X{s),Y{s),z)\^ds
C.11) ■'' '^^
<E \d- gx(j\^ds = o(T - t).
Since a and a are continuous in {t,x,y), uniformly in z, the process
F{s) ^ inf.gR. \a{s, X{s),Y{s),z) - g,{X{s))a{s, X{s), Y{s),z)\^ is
continuous, and an easy application of Lebesgue's Dominated Convergence
Theorem and Differentiation Theorem leads to that
EFiT) = ^^u^^E{^j"Fis)ds} = 0,
proving C.5) since F{T) is nonnegative. Finally, if C.6) holds, then by the
forward equation in B.5) one has
C.12) lim E|X(T) - a;p = 0,
uniformly (note that {X{-),Y{-), Z{-)) depends on the time duration [0,T]
on which B.5) is solved). Hence, C.7) follows. D
We note that C.4) holds if both g].^ and a are bounded, and C.6) holds
if both b and a are bounded.

14 Chapter 1. Introduction
An interesting corollary of Proposition 3.2 is the following nonsolvable
result for FBSDEs.
Corollary 3.3. Suppose a is continuous in (t, x, y, z) and uniformly Lips-
chitz continuous in (x, y, z). Suppose there exists an e > 0, such that
C.13) {a{0,x,y,z)\zen^}cn"''"'\B,{ao), a.s.
for some {x,y) £ ]R" x JR"* and some do £ R"^'', wiere Be{ao) is the
closed ball in JR"*^'* centered at ctq with radius e. Then there exist smooth
functions b, a, h and g, such that the corresponding FBSDE B.1) does not
have adapted solutions over all small enough time durations [0,T].
Proof. In the present case, we may choose b, a, h and g such that C.6)
holds but C.7) does not hold. Then our claim follows. D
Since we are mainly interested in the case that FBSDEs do have
adapted solutions, we should avoid the situation C.13) happening. A
natural way of doing that is to assume that
C.14) {ct@, x,y,z)\ze R^} = WT""^, V(a;, y) £ R" x BT, a.s.
This implies that £ > md. Further, C.14) suggests us to simply take
C.15) a{t,x,y,z) = z, V(t,a;,2/) G [0,T] x ]R" x JR"*,
with z e R'"'"*. From now on, we will restrict ourselves to such a situation.
Hence, B.5) becomes
{dX{t) = b{t,X{t),Y{t), Z{t))dt + G{t,X{t), Y{t), Z{t))dW{t),
dYit) = h{t, X{t), Y{t), Z{t))dt + Z{t)dW{t),
X{0)=x, Y{T) = g{X{T)).
Also, B.3) now should be changed to the following:
C 17) M[0,T]^L%in;C{[0,T];nn)^LU^;C{[0,T];nn)
xL2^@,T;R'"^'*).
We keep B.4) as the norm of M[0,T], but now |Z(t)|2 = tr {Z{t)Z{t)'^}.
§4. Well-posedness of BSDEs
We now briefly look at the well-posedness of BSDEs. The purpose of this
section is to recall a natural technique used in proving the well-posedness
of BSDEs, namely, the method of contraction mapping.
We consider the following BSDE (compare with C.16)):
(dY{t) = h{t,Y{t),Z{t))dt + Z{t)dW{t), te[0,T],
where ^ £ L%^{n;Wl"') and h e L^@,T;H^i'°°(R'" x R'"^'*;R'")) i.e.,
(recall from §2), h : [0,T] x R"* x R"*""* x fi -> R"*, such that {t,oj) i->

§4. Well-posedness of BSDEs 15
h{t, y, z; w) is {Ji}i>o-progressively measurable for all {y, z) e R"* x jR"*^'*
with h{t,0,0;oj) £ L|-@,T;]R'") and for some constant L > 0,
.4 2) \h{t,y,z)-h{t,y,z)\ <L{\y-y\ + \z- z\},
\/y,yeWr, z,zen"''"^, a.e.te[0,T], a.s.
Denote
D.3) Af[0,T] = L''^{n-C{[0,T\;Wi"')) X L%i0,T;Wr'"'),
and
D.4) \\{Y{-),Z{-))\\mio,t] = {e sup \Y{t)\'+E [ |Z(t)pdA'^'.
"■ 0<t<T Jo '
Then, A/'[0,T] is a Banach space under norm D.4). We can similarly define
M[t,T],ioTte [0,T).
Let us introduce the following definition (compare with Definition 2.1).
Definition 4.1. A processes {¥{■), Z{-)) e A/'[0,T] is called an adapted
solution of D.1) if the following holds:
D 5) ^^*^^^~l M*'^(*)'^(*))'^*-/ Zis)dWis),
\/te [0,T], a.s.
The following result gives the existence and uniqueness of adapted
solutions to BSDE D.1).
Theorem 4.2. Let h e L%{0,T;W^^°°{B."' x ]R'"^'*;]R'")). Tien, for any
^ e L%^iO,; JR"*), BSDE D.1) admits a unique adapted solution {¥{■), Z{-)).
Proof. For any {y{-),z{-)) £ A/'[0,T], we know that
D.6) h{-) = h{-,y{-),<-))eL'AO,T;nn.
Now, we define
M{t) = E{i- / h{s)ds\Tt],
D.7) I ^\ te[0,T].
Y{t) = E{i- [ h{s)ds\Tt},
\ Jt
Then M{t) is an {.7^t}i>o-martingale (square integrable), and
D.8) M{Q) = E{i- f h{s)ds]^Y{Q).
Jo
Therefore, by the Martingale Representation Theorem, we can find a Z{-) £
L^@,T;]R'"^'*), such that
D.9) M{t) = M{0)+ [ Z{s)dW{s), \/te[0,T].
Jo

16 Chapter 1. Introduction
Since ^ is .T^r-measurable, we see that (note D.7)-D.8))
D.10) ^- f h{s)ds = M{T)=Y{0)+ f Z{s)dW{s).
Jo Jo
Consequently, by D.7)-D.10), we obtain
Y{t) = M{t) + f h{s)d.s
Jo
= y@) + / Z{s)dW{s) + f his)ds
Jo Jo
D.11) =^- [ his)ds- [ Z{s)dW{s)
Jo Jo
+ / his)ds+ [ Z{s)dW{s)
Jo Jo
= i- f h{s)ds - I Z{s)dW{s).
It is not very hard to show that actually {¥{■),Z{-)) £ U[Q,T] (See below
for a similar proof). Thus, we obtain an adapted solution {¥{■), Z{-)) to
the following equation:
(A 1o^ {dY{t) = h{t,y{t),z{t))dt + Z{t)dW{t),
^ ^ ^ 1 Y{T) = e
Now, let {y{-),z{-)) e Af[0,T] and (F(-),^(-)) £ ^f[0,T] be the
corresponding solution of D.12). Then, by Ito's formula and D.2), we have
D.13)
E\Y{t)-Y{t)\^+E I \Z{s)-Z{s)\''ds
< 2LE j \Y{s)-Y{s)\{\y{s)-y{s)\ + \z{s)-z{s)\]ds.
Next, we set
D 14) \v{t) = {E\Y{t)-Y{t)\^y^\
\ V>(t) = {E\y{t) -y{t)?y'' + {E\z{t) - z{t)?V'^.
Then, D.13) implies
D.15) ip{tf +E f \Z{s) - Z(s)|2ds <2L f ip{s)ip{s)ds, t e [0,T].
We have the following lemma.
Lemma 4.3. Let D.15) hold. Then,
\Z{s)-'Z{s)\^ds<L'^{j V(s)d4 , Vte[0,T].

§4. Well-posedness of BSDEs 17
Proof. We call the right hand side of D.15) 2Le{t). Then, by D.15),
D.17) e'{t) = ~ipm{t) > -ip{t)^2Le{t),
which yields
D.18) {v^}'>-V^V(i)-
Noting e{T) = 0, we have
D.19) -y/6{f)>-s/Lj2 I ■il){s)ds.
Consequently,
D.20) e{t)<^[j ^|>{s)ds]\ Vte[0,T].
Hence, D.16) follows from D.15) and D.20). D
Now, applying the above result to D.13), we obtain
rT
E\Y{t)-Y{t)\^+E I \Z{s)-Z{s)\''ds
{ f {{E\y{s) - y{s)\^)''' + {E\z{s) - z{s)\'Y'']ds\
D.21) ^^ ^ -
*■ Jt
<C{T ~t)\\{y{-),z{-)) - {y{-),z{-))\\lf[t,TV
Then, by Doob's inequality, we further have
ii(n-),^(-))-(F(-),^(-))ikt,T]
<C(T-t)||B/(.),^(-))-(y(-),^(-))llW]' Vte[0,T].
Here C > 0 is a constant depending only on L. By taking 5 = jfe, we
see that the map {y{-)-,z{-)) i-> (y(-),Z(-)) is a contraction on the Banach
space M\T — 6,T]. Thus, it admits a unique fixed point, which is the
adapted solution of D.1) with [0,T] replaced by [T - E,T]. By continuing
this procedure, we obtain existence and uniqueness of the adapted solutions
to D.1). D
We now prove the continuous dependence of the solutions on the final
data ^ and the function h.
Theorem 4.4. Let h,h £ L^@,T;_W^i'°°(]R'" x ]R'"^'*;]R'")) and i,l £
L^^(n;]R'"). Let {¥{■), Z{-)),{Y{-),Z{-)) e M[0,T] be the adapted
solutions of D.1) corresponding to (h,^) and (h,^), respectively. Then
||(y(.)-F(.),z(-)-z(-))|IVT]
D-23) r - f^ - -.
< c{e\^ -e+EJ \h{s,Y{s),Z{s)) - h{s,Y{s),Z{s))\'ds],

18 Chapter 1. Introduction
with C > 0 being a constant oniy depending on T > 0 and the Lipschitz
constants of h and h.
Proof. We denote
D 24) p(-) = n-)-F@, Zi-) = Zi.)-Zi.),
Applying Ito's formula to |y(-)P, we obtain
\Yit)\'+l \Z{s)\'ds
= \i\^-2 j {Y{s), h{s, Y{s), Zis)) - h{s,Y{s),Z{s))) ds
-21 {Y{s),Z{s)dWis))
D.25) <|^p+2^ {\Y{s)\\h{s)\+L\Y{s)\{\Y{s)\ + \Z{s)\)}ds
-21 {Y{s),Z{s)dW{s))
< ifP + I\il + 2L + 2L^)\Y{s)\^ + i|ZE)p + \his)\^}ds
-21 {Y{s),Z{s)dWis)).
Taking expectation in the above, we have
D.26)
£|y(t)|2 + Ie f \Z{s)\''ds < E|^|2 + E f \h{s)\''ds
^ h Jo
+ A+2L + 2L2)e / \Y{s)\''ds, te[0,T].
Thus, it follows from Gronwall's inequality that
E\Y{t)\^+Ef \Z{s)\''ds
D.27) ^'
<
c{e\^\' + eI \h{s)\^ds], Vte[0,T].
On the other hand, by Burkholder-Davis-Gundy's inequality (see Karatzas-

D.28)
§5. Solvability of FBSDEs in small time durations 19
Shreve [1]), we have from D.25) that (note D.27))
E{ sup \Y{t)\^]<c{E\i\^+E j \h{s)\^ds\
te[o,T] •■ Jo '
+ 2E sup I / {Y{s),Z{s)dW{s))\
te[o,T] I Jt '
<c[e\1\^+eJ \h{s)\^ds]
+ cAe sup \Y{t)\''\'\E [ \Z{s)\^ds\'\
^ te[o,T] ' ^ Jo '
Now D.23) follows easily from D.28) and D.27). D
We see that Theorems 4.2 and 4.4 give the well-posedness of BSDE
D.1). These results are satisfactory since the conditions that we have
imposed are nothing more than uniform Lipschitz conditions as well as certain
measurabihty conditions. These conditions seem to be indispensable, unless
some other special structure conditions are assumed.
§5. Solvability of FBSDEs in Small Time Durations
In this section we try to adopt the method of contraction mapping used in
the previous section to prove the solvability of FBSDE C.16) in small time
durations. The main result is the following.
Theorem 5.1. Let 6, a, h and g satisfy B.6). Moreover, we assume that
{\a{t, X, y, z; w) - a{t, x, y,z; u))] < Lo\z-z\,
^{x,y)eWCxWr, z,zeWr''\ a.e.t>0, a.s.
\g{x;uj) - g{x;uj)\ < Li\x -x\, Wx,x eWi"', a.s.
with
E.2) LoLi < 1.
Tien tiere exists a Tq > 0, such that for any T £ @, To] and any x £ R",
C.16) admits a unique adapted solution {X,Y, Z) £ jM[0,T].
Note that condition E.2) is almost necessary. Here is a simple example
for which E.2) does not hold and the corresponding FBSDE does not have
adapted solutions over any small time durations.
Example 5.2. Let n — m — d— I. Consider the following FBSDEs;
{dX{t) = Z{t)dW{t),
dY{t) = Z{t)dWit),
X@) = 0, YiT) = X{T)+^,
where ^ is .Tr-measurable only (say, ^ = W{T)). Clearly, in the present
case, Lo = Li — 1. Thus, E.2) fails. If E.3) admitted an adapted solution

20 Chapter 1. Introduction
{X,Y,Z), the:
the following:
{X, Y, Z), then the process t] = Y — X would be {.7^t}t>o-adapted and satisfy
,,,, jdv{t) = o, te[o,T],
^'■'^ b(T) = e
We know from §1 that E.4) does not admit an adapted solution unless ^ is
deterministic.
Proof of Theorem 5.1. Let 0 < Tq < 1 be undetermined and T £ @, Tq].
Let X e ]R" be fixed. We introduce the following norm:
E.5) ||(y,Z)|U^ sup {E\Y{t)\^+E r\Z{s)\^dsY'\
tG[0,T] Jt
for all {Y, Z) £ ATfO, T]. It is clear that norm E.5) is weaker than D.4). We
let ATfO, T] be the completion of M[Q, T] in L%{Q, T; WT) x L%{0, T; WT'"')
under norm E.5). Take any {Yi,Zi) £ 7V'[0,T], i = 1,2. We solve the
following FSDE for Xi:
(dXi = bit,Xi,Y,Zi)dt + ait,X,,Yi,Zi)dWit), te[0,T],
^'■'^ [ XM = -
It is standard that under our conditions, E.6) admits a unique (strong)
solution Xi e L%{n;C{[0,T];WC)). By Ito's formula and the Lipschitz
continuity of b and a (note E.1)), we obtain
E\Xiit) - X2it)f
<E I {2L|Xi - X2I (|Xi - X2I + 1^1 - Y2\ + |Zi - Z2I)
E.7) ° f X2-,
+ (l{\X, - X2I + lYi - ^2!) + Lo\Z, - Z2IJ ]ds
<E j' [C,(\X, - X2P + 1^1 - y2p) + {Ll + e)\Z, - Z2p}d5,
where Cg > 0 only depends on L, Lq and e > 0. Then, by Gronwall's
inequality, we obtain
E.8) E|Xi.(t)-X2(t)P<e^^^Ey" [Ce\Y,-Y2\''HLl+e)\Z,-Z2\^]ds.
Next, we solve the following BSDEs: (i = 1,2)
{dYi = h{t,Xi,Yi,Zi)dt + 'ZidW{t), te[0,T],
\Y,{T) = g{X,{T))-
We see from Theorem 4.2 that (for i= 1,2) E.9) admits a unique adapted
solution (Fi,Zi) e A/'[0,T] C J[[Q,T]. Thus, we have defined a map

§5. Solvability of FBSDEs in small time durations 21
T rJV'fO,T]_-> Jf[Q,T] by {Yi,Zi) i-> (Fi,Zi). Applying Ito's formula
to 1^1 (t) - Y-i{t)\^, we have (note E.1) and E.8))
T
E\Y,{t)-Y2{t)\''+E j \Z,-Z2\''ds
< LlE\XiiT) - X2iT)\
E.10)
+ 2LEf \Y, - Y2\{\X, - X2I + 1^1 - ^2! + \Z, - Z2\)ds
<L\E\X,{T)-X2{T)\^+CeE j \Y,-Y2\^ds
rp rp
+ eE f \Zi-Z2\''ds + E f {\Xi - X2\''+ \Yi-Y2\'')ds
< {L\ + T)e^=^E / [Ce\Y, - ^2!' + {Ll+e)\Z, - Z2\'']ds
Jo
rp rp
+ eE [ \Z^-Z2\^ds + E f \Y^-Y2\''ds
Jo Jo
+ CeE j \Y^-Y2\^ds.
In the above, Ce could be different from that appeared in E.7)-E.8). But
Ce is still independent of T > 0. Using Gronwall's inequality, we have
E\Y,{t)-Y2{t)\^+E j |Zi-Z2pds
E.11)
<e'^''^[CeE j^ \Y,-Y2\^ds
+ [e+{L\+T){Ll+e)e^''^]E j \Z, - Z2\^ds^
< e'^'^lCeT + e+{L\+ T){Ll + e)e^=^]
.\\{Y,,Z,)-{Y2,Z2)\\^^^,^y
where Cg > 0 is again independent of T > 0. In the above, the last
inequality follows from the fact that for any {Y, Z) £ ATfO, T],
rE|y(t)|2<||(y,z)|||^j„_yj, vte[o,T],
E.12) \ rT
[| E\Z{t)\'dt<\\{Y,Z)\\^^^,^,
Since E.2) holds, by choosing e > 0 small enough then choosing T > 0
small enough, we obtain
E.13) ||(Fi,Zi)-(F2,Z2)||j^[o,y, <a||(yi,Zi)-A^2,^2I1

22 Chapter 1. Introduction
for some 0 < a < 1. This means that the map T : 7J[Q,T] -> 7J[Q,T] is
contractive. By the Contraction Mapping Theorem, there exists a unique
fixed point (y, Z) for T- Then, similar to the proof of Theorem 4.2 we can
show that actually (y, Z) £ A/'[0, T]. Finally, we let X be the corresponding
solution of E.6). Then {X,Y,Z) £ A^[0,T] is a unique adapted solution
of C.16). The above argument applies for all small enough T > 0. Thus,
we obtain a Tq > 0, such that for all T £ @,To] and all a; £ R", C.16) is
uniquely solvable. D
In the above proof, it is crucial that the time duration is small enough,
besides condition E.2). This is the main disadvantage of applying the
Contraction Mapping Theorem to two-point boundary value problems. Starting
from the next chapter, we are going to use different methods to approach
the solvability problem for the FBSDE C.16).
§6. Comparison Theorems for BSDEs and FBSDEs
In this section we study an important tool in the theory of the BSDEs—
Comparison Theorems. The main ingredients in the proof of the desired
comparison results are "linearization of the equation" plus a change of
probability measure. We should also note that in the coupled FBSDE case
the situation becomes quite different. We shall give an example in the end
of this section to show that the simple-minded generalization from BSDEs
to FBSDEs fails in general.
To begin with, we consider two BSDEs: for i = 1,2,
F.1) Y\t)=e+j h\s,Y\s),Z\s))ds-1 {Z\s)dWis),
where TV is a d-dimensional Brownian motion, and naturally the dimension
of y's and Z's are assumed to be 1 and d, respectively. Assume that
F.2) CeL%{n;n); h^ e L''^{0,T;W''°°{B.'^+\n)), i=l,2,
where L%-{0,T;W^'°°{Wi'''^\'R)) is defined in §2. Since under these
conditions both BSDEs are well-posed, we denote by {Y^,Z^), i = 1,2 the two
adapted solutions respectively. We have
Theorem 6.1. Suppose that assumption F.2) holds, and suppose that
^1 > ^^ and h^{t,y,z) > h'^{t,y,z), for all {y,z) £ ]R'*+S P-almost surely.
Then it holds that Y^{t) > Y'^{t), for all t £ [0,T], P-a.s.
Proof. Denote Y{t) = Y^{t) - Y'^{t), Z{t) = Z\t) - Z'^{t), Vt £ [0,T];
h{t) = h\t,Y\t),Z\t))-h\t,Y\t),Z\t)), t£[0,T].
Clearly, h is an {.7^t}t>o-adapted, non-negative process; and Y satisfies the

§6. Comparison theorems for BSDEs and FBSDEs 23
following (linear!) BSDE:
Yit) =UI {[h\s,Y\s),Z\s)) -h\s,Y^{s),Z^{s))]+his)}ds
F.3) - f Z{s)dW{s)
= i+ I {ais)Y{s) + Pis)Zis)+his)}ds - f Z{s)dWis),
where
a{s) = [ hlis, Y^{s) + XY{s), Z\s) + XZ{s))dX;
Jo
P{s)= [ hl{s,Y^{s) + XY{s),Z^{s) + XZ{s))dX.
Jo
Clearly, a and P are {.7"t}t>o-adapted processes, and are both uniformly
bounded, thanks to F.2). In particular, f3 satisfies the so-called Novikov
condition, and therefore the process
M{t)=exp[l li{s)dW{s)-]^j \li{s)\^ds], te[0,T]
is an P-martingale. We now define a new probability measure P by
d'P
dP ^ '
Then by Girsanov's theorem, W{t) = W{t) — f^ I3{s)ds is a P-Brownian
motion, and under P, Y satisfies
F.4) Y{t)=^+ f [a{s)Y{s) + his)]ds- f Z{s)dW{s).
Now define r(t) = explj^ a{s)ds}, then Ito's formula shows that
T{T)^-r{t)Y{t) = - f T{sNhis)ds+ f Z{s)dWit).
Taking conditional expectation E^{-\Tt} on both sides above, and noticing
the adaptedness of r(-)y(-) we obtain that
r{t)Y{t) = E^ I r{T)'i+ f r{s)h{s)ds\j^t > > o, vt e [o,t],
P-almost surely, whence P-almost surely, proving the theorem. D

24
Chapter 1. Introduction
An interesting as well as important observation is that the comparison
theorem fails when the BSDE is coupled with a forward SDE. To be more
precise, let us consider the following FBSDEs: for i = 1,2,
X\t) = x'+ [ b\s,X\s),Y'{s),Z'{s))ds
Jo
+ f a\s,X'{s)X{s),Z\s))dW{s)
Jo
Y\t) = g'{X'{T))+ [ h^{s,X\s),Y\s),Z'{s))ds
Jo
F.5)
/
Jo
Z'{s)dW{s).
We would like to know whether gi{x) > gi{x), Va; would imply Yi{t) >
Y-i{t), for all t? The following example shows that it is not true in general.
Example 6.2. Assume that d= I. Consider the FBSDE:
X{t)
F.6)
dX{t) =
dY{t) =
{Z{t)-Y{t)r + l
Z{t)
{Z{t) - Y{t)r + 1
[ X{0) = x; Y{T) = g{X{T))
dt + X{t)dW{t),
dt + Z{t)dW{t),
We first assume that ^(a;) = ^^(a;) = x. Then, one checks directly that
X^{t) = Y^{t) = Z^{t) = xexp{W{t) + t/2}, t e [0,T] is an adapted
solution to F.6). (In fact, it can be shown by using Four Step Scheme of
Chapter 6 that this is the unique adapted solution to F.6)!)
Now let g'^{x) = x + I. Then one checks that X'^{t) = Z'^{t) and
y2(t) = X2(t) + 1 = Z2(t) + 1, Vt e [0,T] is the (unique) adapted solution
to F.6) with g'^{x) = X -\- 1. Moreover, solving F.6) explicitly again we
have y2(i) = 1 +pexp{H^(t)}.
Consequently, we see that Y^{t) - Y'^{t) = pe^(*)[e*/2 - 1] - 1, which
can be both positive or negative with positive probability, for any t > 0,
that is, the comparison theorem of the Theorem 6.1 type does not hold!
D
Finally we should note that despite the discouraging counterexample
above, the comparison theorem for FBSDEs in a certain form can still be
proved under appropriate conditionis on the coefficients. A special case
will be presented in Chapter 8 (§8.3), when we study the applications of
FBSDE in Finance.

Chapter 2
Linear Equations
In this chapter, we are going to study linear FBSDEs in any finite time
duration. We will start with the most general case. By deriving a necessary
condition of solvability, we obtain a reduction to a simple form of linear
FBSDEs. Then we will concentrate on that to obtain some necessary and
sufRcient conditions for solvability. For simplicity, we will restrict ourselves
to the case of one-dimensional Brownian motion in §§1-4. Some extensions
to the case with multi-dimensional Brownian motion will be given in §5.
§1. Compatible Conditions for Solvability
Let (n, T, {.7-t}t>o, P) be a complete filtered probability space on which
defined a one-dimensional standard Brownian motion Wif), such that {.7-t}t>o
is the natural filtration generated by W{t), augmented by all the P-nuU sets
in T. We consider the following system of coupled linear FBSDEs:
' dX{t) = {AX{t) + BY{t) + CZ{t) + Dh{t)]dt
+ {A^X{t) + B^Y{t) + CiZ{t) + DMt)]dW{t),
dY{t) = {AX{t) + BY{t) + CZ{t) + Db{t)}dt
+ {A,X{t) + B,Y{t) + CiZ{t) + Diait)}dWis),
te[o,T],
{ X{0) = X, Y{T) = GX{T) + Fg.
A.1)
In the above, A,B,C etc. are (deterministic) matrices of suitable sizes, b,
a, b and a are stochastic processes and ^ is a random variable. We are
looking for {.7^t}t>o-adapted processes X{-), Y{-) and Z{-), valued in R",
R"* and R^, respectively, satisfying the above. More precisely, we recall
the following definition (see Definition 2.1 of Chapter 1):
Definition 1.1. A triple {X,Y,Z) £ A^[0,T] is called an adapted solution
of A.1) if the following holds for all t £ [0,T], almost surely:
X{t) = x+ f {AX{s) + BY{s) + CZ{s) + Db{s)}ds
Jo
+ f {A,X{s) + B,Y{s) +CiZ{s) + Diais)}dWis),
Jo
Y{t) = GX{T) +Fg-f {AXis) + BY{s) + CZ{s) + Db{s)}ds
- f {AiX{s) + BiY{s) + CiZ{s) + Dia{s)}dW{s).
A.2) <^

A.3)
26 Chapter 2. Linear Equations
When A.1) admits an adapted solution, we say that A.1) is solvable.
In what follows, we will let
'A,Ai GR"^"; 5,5i GR"^'"; C,Ci eWC''^;
£)eR"^"; £)ieR"^"^; 5eR'"^'^;
5i eR'">^'^\- FeR"*^*;
b e L%{0,T; R"); ae L^@,T; R"^);
6eL|-@,T;R'^); 5 £ L|.@,r;R'^^);
Uei|-^(fi;iR''); a^eR"-
Following result gives a compatibility condition among the coefficients
of A.1) for its solvability.
Theorem 1.2. Suppose there exists a T > 0, such that for all b, a, b, a, g
and X satisfying A.3), A.1) admits an adapted solution {X, Y, Z) e M[0, T].
Then
A.4) n{Ci - GCi) D 7^(F) + 7^El) + niGDi),
wiere TZiS) is the range of operator S. In particular, if
A.5) 7^(F)+7^El)+7^(G£)l) = R'",
tien Ci - GCi £ R"*^^ is onto and thus e>m.
To prove the above result, we need the following lemma, which is
interesting by itself.
Lem.m.a 1.3. Suppose that for any a e L^@,T;R*) and any g £
L^^(n;R*), there exist h e L^@,T;R'") and / £ L|.(n;C([0,T];R'")),
such that the following BSDE admits an adapted solution (Y, Z) £
L2,(n;C([0,T];R'")) X L%{0,T-n^):
idYit) = hit)dt + [f{t)+C,Zit)+Da{t)]dW{t), t e [0,T],
\ Y{T) = Fg.
where Ci £ R'
A.7)
mxi
and D em:
mxk
Then,
■R{Ci) Dn{F) + n{D).
Proof. We prove our lemma by contradiction. Suppose A.7) does not
hold. Then we can find an ry £ R"* such that
A.8)
T]'Ci=0, butr?''F^O, or rj^Dj^O.

§1. Compatible conditions for solvability
Let C{t) = T]^Y{t). Then C(-) satisfies
f dC{t) = h{t)dt + [f{t) + ri^^a{t)]dW{t),
27
A.9)
U{T)=r^'
Fg,
where h{t) = T]^h{t), f{t) = rj^ f{t). We claim that for some choice of g and
&{■), A.9) does not admit an adapted solution ({■) for any h £ L^{0,T;WC)
and / e Ljr(n;C([0,T];]R)). To show this, we construct a deterministic
Lebesgue measurable function C satisfying the following:
A.10)
(P{s) = ±l, Vse[0,T],
\{se[Ti,T]\P{s) = l}\ =
\{se[Ti,T]\C{s) = -l}\ =
T-Tj
2 '
T-Ti
i > 1,
for a sequence Ti'fT, where |{---}| stands for the Lebesgue measure of
{■■■}. Such a function exists by some elementary construction. Now, we
separate two cases.
Case 1. rj^F ^ 0. We may assume that |F^ry| = 1.
Let us choose
A.11) g=i^j li{s)dW{s))F'^r], a{t) = Q.
Then, by defining
A.12) at) = [J l3{s)dW{s)), t e [0,T],
we have
1 CiT) - C(T) = 0.
Applying Ito's formula to \({t) - C(*)Pi we obtain
E\at)-m' + El^'^\f{s)-f3{s '
(s)\'ds
A.14)
= -2E^ {Cis)-Cis)Ms))ds
= 2E I {[ h{r)dr + j [f{r)-li{r)]dW{r)Ms))ds
= 2E f { f h{r)dr, h{s)) ds
I h{s)ds ^ <{T-i) I E\h{s)\^ds.
= E

28 Chapter 2. Linear Equations
Consequently, (note h £ L%{0,T;WI) and f e L%{n;C{[0,T];WC)))
EJ \f{T)-P{s)fds
(-^-^5) <2E^ \f{s)^P{s)\''ds + 2El^ \fiT)-fis)fds
<2{T-t) f E\h{s)\^ds + 2E j \f{T)-f{s)\''ds
= o(T - t).
On the other hand, by the definition of /?(■), we have
E [ \fiT) - (i{s)\^ds
A.16) Jt,
= T-lIi[E\J{T) - Ip +E|/(T) + Ip), Vz > 1.
Clearly, A.16) contradicts A.15), which means rj^F ^ 0 is not possible.
Case 2. ry^F = 0 and tj^D ^ 0. We may assume that \D t]\ = I.
—T
In this case, we choose a{i} = l3{t)D t] with /?(•) satisfying A.10).
Thus, A.9) becomes
. . / dm = h{t)dt + [Rt] + /3(t)]dw^(t), t e [0,T],
^ • '^ \C(T) = 0.
Then the argument used in Case 1 applies. Hence, ry^D ^ 0 is impossible
either, proving A.7). D
Proof of Theorem 1.2. Let {X,Y,Z) e_X[0,T] be an adapted solution
of A.1). Set Y{t) = Y{i) - GX{t). Then Y{-) satisfies the following BSDE:
' dY= {{A-GA)X + {B-GB)Y
+ {C - GC)Z + Db-GDb}dt
A-18) < + {{A, - GA,)X + {B, - GB,)Y
+ (Ci - GCi)Z + Did - GDiG}dW{t),
. Y{T) = Fg.
Denote
j h = {A- GA)X + {B- GB)Y + {C - GC)Z-^Dh - GDb,
[ f = {Ai - GAi)X + 01 - GBi)Y.
We see that h £ L%{0,T-Wr) and f e L|.(n;C([0,T];]R'")). One can
rewrite A.18) as follows:
A 20) i"^^ ^'^^ ^^^^ ^^' ~ ^^'^^ ^ ^'^ ~ *^^i^>'^^(*)'
I Y{T) = Fg.

§1. Compatible conditions for solvability 29
Then, by Lemma 1.3, we obtain A.4). The final conclusion is obvious.
D
To conclude this section, let us present the following further result,
which might be less useful than Theorem 1.2, but still interesting.
Proposition 1.4. Suppose that the assumption of Theorem 1.2 holds.
For any b, a, b, a, g and x satisfying A.3), let {X,Y, Z) £ jM[0, T] be an
adapted solution of A.1). Then it holds
^^ [A,-GA, + {B,-GB,)G]X{T)
+ {Bi-GBi)Fge'll{Ci~GCi), a.s.
If, in addition, the following holds:
A.22)
then
A.23)
' TliA + BG) + n{BF) C n{D),
TliAi +BiG)+ 7^ElF) C n{Di),
■Jl{A + BG) + TliBF) C n{D),
[ Kill + BiG)+ 7^ElF) C 7^El),
7^(ll - GAi + (Bi - GBi)G^ + 7^((Bl - G5i)f)
C7^(Cl-GCl).
Proof. Suppose ry £ R"* such that
A.24) r?^(Ci-GCi)=0.
Then, by A.4), one has
A.25) T]^F = 0, r?^5i=0, T]^GDi=0.
Hence, from A.20), we obtain
A 26) I '^[^''^(*)] = v''hit)dt + ri''fit)dWit), t e [0,T],
\ ri^Y{T) = 0.
Applying Ito's formula to \r}'^Y{t)\'^, we have (similar to A.14))
F|r?^F(t)p +E I \'ri^f{s)\^ds = -2E f r?^F(s)r?^/i(s)ds
A.27) =2f/ [/ T]'^h{r)dr+ f ri^f{r)dW{r)\Ti^h{s)ds
= e\ f r}'^his)ds ^ <{T~t) I E\Ti^h{s)\^ds.

30 Chapter 2. Linear Equations
Dropping the first term on the left side of A.27), then dividing both sides
by T — t and sending t -> T, we obtain
A.28) E\rj^fiT)\' = 0.
By A.19), and the relation Y{T) = GX{T) + Fg, we obtain
A.29) r?^[li - GAi + {Bi - GBi)G\X{T) + r?^(Bi - GBi)Fg = 0, a.s.
Thus, A.21) follows. In the case A.22) holds, for any a; £ R" and ff £ R"*
(deterministic), by some choice oi b, a, b and a, A.1) admits an adapted
solution {X,Y,Z) = {x,Gx + Fg,0). Then, A.21) implies A.23). D
§2. Some Reductions
In this section, we are going to make some reductions under condition A.5).
We note that A.5) is very general. It is true if, for example, F = I e JR"*^"*,
which is the case in many applications. Now, we assume A.5). By Theorem
1.2, if we want A.1) to be solvable for all given data, we must have Ci—GCi
to be onto (and thus £ >m). Thus, it is reasonable to make the following
assumption:
Assumption A. Let ^ = m and Ci - GCi £ jR"*^"* be invertible.
Let us make some reductions under Assumption A. Set Y = Y — GX.
Then Y{T) = Fg and (see A.18))
dY = {AX + BY + CZ + Db)dt
+ (liX + B^Y + CiZ + Dia)dW
- G{AX + BY + CZ + Db)dt
- G{AiX + B^Y + CiZ + Dia)dW
B.1) =^[A-GA + {B-GB)G]X + iB-GB)Y
+ {C- GC)Z + Db-GDb^dt
+ {[Ai- GA, + 01 - GBi)G]X + {Bi - GB^)Y
+ (Ci - GCi)Z + Did - GDia^dW.
Define
B.2)
Z =[Ai- GAi + (Bi - GBi)G]X + (Bi - GBi)Y
+ ((?i - GCi)Z + Did - GDiG.
Since {Ci — GCi) is invertible, we have
Z= {Ci - GCi)-'{'Z- [Ai - GAi + {Bi - GBi)G]X
-{Bi-GBi)Y-{Dia~GDia)}.

§2. Some reductions
Then, it follows that
31
B.4)
where
' dX = (J.X + 'BY + C^ + b)dt
+ (a^X + BiY + Ci'Z + w] dW,
dV = (^oX + BoY + CoZ + h)dt + 'ZdW,
[ X{0) = X, Y{T) = Fg,
'A = A + BG-C{Ci -GCi)-M^i -GAi +{Bi-GBi)G],
B = B- C{Ci - GCi)-\Bi - GBi),
C = C{Ci -GCi)-\
b = Db-C{Ci -GCi)-\Did-GDiG),
- Ci(Ci - GC,)-'[A, - GA, + Ei - GB,)G],
^i = B,- C,(Ci - GC,)-\B, - GB,),
C, = C,{C,-GC,)-\
a = DiG-Ci{Ci -GCi)-^iDid-GDia),
Ao = A-GA + {B-GB)G
-{C- GC)iCi - GCi)-'[A, - GA, + {B, ~ GB,)G],
Bo = B ~ GB - {C - GC)iCi - GCi)-^(Bi - GBi),
Co = {C - GC){C, - GC,)-\
[h = Db-GDb -{C- GC)iCi - GCi)-\Dia - GD^a).
The above tells us that under Assumption A, A.1) and B.4) are equivalent.
Next, we want to make a further reduction. To this end, let us denote
B.5)
B.6)
A =
~A,=
A B
Ao Bo
0 0
C =
Ci =
C
Co
Ci
Let '$'(•) be the solution of the following:
B.7)
j d^{t) = A^{t)dt + Ai'^it)dWit), t>0,
[ *@) = /.

32 Chapter 2. Linear Equations
Then B.4) is equivalent to the following: For some y £ R"*,
B.8)
+
h{s)
-4i
ais)
+
Jo
C,Z{s) +
a{s)
0
dW{s),
ds
te[o,T],
with the property that
B.9)
Fg =@, /)*(T) ( y ) + @, mm ^ *(s)-^ [(C - ^iCi)ZE)
^[h{s))-^'[ 0
+ @,/)*(T)jr *E)-i[CiZE)+ (^^[j'))]dW^E).
ds
Clearly, B.9) is equivalent to the following: For some y £ R"* and Z(-) £
L^@,T;]R'"), it holds
r?^Fff-@,/)*(T)(J
B.10)
-@,/)*(T) / *(s)-i
-@,/)*(T)^ *(s)-i(^^[j'))dW^(s)
= @, /)*(T) (y ) + @, /)*(T) ^ ns)-\C - AC,)Z{s)ds
+ @,/)*(T)/ ^{s)-'Ci^{s)dW{s).
Jo
Thus, if we can solve the following:
B.11) Hf)={^(|)+Cz}dt+{:4.(f)+C.z}dH^,
[ X{0) = 0, Y{T) = T],
with T] being given by B.10), then for such a pair y = Y{0) and Z(-) = Z(-),
by setting {X,Y) as B.8), we obtain an adapted solution {X,Y,Z) £
A^[0,T] of B.4). The above procedure is reversible. Thus, by the
equivalence between B.4) and A.1), we actually have the equivalence between
the solvability of A.1) and B.11). Let us state this result as follows.

§3. Solvability of linear FBSDEs
33
Theorem 2.1. Let F = I e ]R'"X'" and e = m. Then A.1) is solvable for
all b, a, b, a, x and g satisfying A.3) if and only if B.11) is solvable for all
We note that by Theorem 1.2, F = / and £ = m imply Assumption
A. Based on the above reduction, in what follows, we concentrate on the
following FBSDE:
B.12)
' dX = {AX + BY + CZ)dt
+ {AiX + BiY + CiZ)dW{t),
dY= {AX + BY + CZ)dt + ZdW{t),
tX@)=0, YiT) = g.
te[o,T],
By denoting
B.13)
A =
Ai =
C =
A,
0
5i
0
Ci =
I
we can write B.12) as follows:
'X\ r . /X
B.14)
X@) = 0, Y{T) = rj.
+ CiZ^dW,
In what follows, we will not distinguish B.12) and B.14), and we will let
f d$(t) = A^t)dt + AiHt)dW{t), te [0,T],
\ $@) = /.
B.15)
If we call {X,Y) the state and Z the control, B.12) is called a
(linear) stochastic control system. Then, the solvability of B.12) becomes the
following controllability problem: For give g £ L'jr^{Cl;]EC^), find a control
Z e L%{0,T;Wr), such that some initial state {X{0),Y{0)) £ {0} x R"*
can be steered to the final state {X{T),Y{T)) e L%-^{Q,;'R"') x {g} at the
moment t = T, almost surely. This is referred to as the controllability of
the system B.12) from {0} x WC to L|.^(n;]R") x {g}. We note that g is
an ^T-measurable square integrable random vector, and we need exactly
control Y{T) to g.
§3. Solvability of Linear FBSDEs
In this section, we are going to present some solvability results for linear
FBSDE B.12). The basic idea is adopted from the study of controllability
in control theory. For convenience, we denote hereafter in this chapter that

34 Chapter 2. Linear Equations
H = L|.^(n;]R'") and n = L%.{0,T;Wr) (which are Hilbert spaces to
which the final datum g and the process Z{-) belong, respectively).
§3.1. Necessary conditions
First of all, we recall that if $ is the solution of B.15), then, $~^ exists
and it satisfies the following linear SDE:
r d$-i = _$-! [A - Al]dt - ^-^AidW{t), t > 0,
^^'^^ I $-1@)=/.
Moreover, {X,Y, Z) e M[0,T] is an adapted solution of B.12) if and only
if the following variation of constant formula holds:
(^J|JJ)=$(t)Q)+$(t)^*$(.)-nC-.4iCi)ZE)d5
+ $(t)/ $(s)-iCiZ(s)dW^(s), te[0,T],
Jo
for some y £ R"* with the property:
g = @,/){$(T) (^°) + $(T) ^ Hs)-\C - A,C,)Z{s)ds
C-3) °
+ $(T)/ ^{s)-'CiZis)dW{s)y
Let us introduce an operator /C : "H -> i? as follows:
rT
C.4)
/CZ=@,/){$(T) /" $(s)-'(C-.4iCi)Z(s)ds
+ $(T) / $(s)-iCiZ(s)dW^(s)}.
Then, for given g E H, finding an adapted solution to B.12) is equivalent
to the following: Find y E WC" and Z e Ti, such that
C.5) g=iO,IMT)(j^y + ICZ,
and define {X, Y) by C.2). Then {X, Y, Z) e M[0, T] is an adapted solution
of B.12). Hence, the study of operators $(T) and /C is crucial to the
solvability of linear FBSDE B.12). We now make some investigations on
$(■) and /C. Let us first give the following lemma.
Lemma 3.1. For any f £ L>@,T;]R"+'") and h e L|.@,T;]R"+'"), it

§3. Solvability of linear FBSDEs 35
holds
C.6) E{ml^Hsrfis)ds}=l\^i^~^^Efis)ds, ^^jo^^j
£{$(t) f ^{s)-^h{s)dWis)^ =0,
Also, it holds that
C.7) E sup |$(t)p*^+E sup |$(t)-^|2* <oo, Vfc > 1.
0<t<T 0<t<T
Proof. Let us first prove the second equality in C.6). The other two in
C.6) can be proved similarly. Set
C.8) m = Ht) [ ^isr'f{s)ds, t e [0,T].
Jo
Then ^Q) satisfies the following SDE:
^'^o^ jdat) = [Aat)+f{t)]dt + A,mdW{t), te[0,T],
^^•^^ U(o) = o.
Taking expectation in C.9), we obtain
jd[Em] = [AEat)+Ef{t)]dt, te[0,T],
^'•^'^ \ Em = 0.
Thus,
C.11) Em= I e-^^'-'^E}{s)ds, te[0,T],
Jo
proving our claim.
Now, we prove C.7). For any ^o £ ©."+'", process i{t) = $(t)^o satisfies
the following SDE:
,o,oA {d^{t) = Amdt + A,mdW{t), te[0,T],
Then, by Ito's formula, Burkholder-Davis-Gundy's inequality and Gron-
wall's inequality, we can show that
C.13) E sup \m?'' < K\io?\ k > 1,
0<t<T
for some constant K > 0. Thus, the first term on the left hand side of
C.7) is finite. Similarly, one can prove that the second term is finite as
well. D

36 Chapter 2. Linear Equations
From C.7), we see that /C : "H -> i? is a bounded linear operator. Now,
applying C.6) to C.3), we obtain that B.12) admits an adapted solution,
then
C.14) Eg = iO,I){e^''(j)y + l e^^^'-'Xc - A,C,)EZis)ds],
for some y £ R"* and EZ{-) £ L'^{0,T;'R"'). This leads to the following
necessary condition for the solvability of B.12).
Theorem 3.2. Suppose B.12) is solvable for all g E H. Then
rank|@,/)(e^^f^Vc-XiCi,^(C-^iCi),
C.15) ^ ^ VV
•••,^"+'"-nC-ACi))}=m.
Proof. Set C =C — AiCi and define
Cu{-)= [ e^^^-'^Cu{s)ds, Vu(-)£L'@,T;]R'").
Jo
Then £ : -L^@,T; JR"*) -> ]R" is a linear bounded operator. We claim that
C.16) 7^(£) = 7^(C) + TliAC) + ■■■ + n{A"+"'-'C).
In fact, if a; £ 7^(£)-L, then, for any u(-) £ L'^{0,T;Wr), it holds
0 = x^jru{-)= / a;^e-^(^-^)Cu(s)ds,
which yields
x^e^'C = 0, Vs£[0,T].
Consequently,
This implies that
X £ {7^(C) + TliAC) + ■■■ + 7^(^"+'"-lC)}^,
which results in
7^(C) + -R-iAC) + ■■■ + 7^(^"+'"-lC) C 7^(£).
The above proof is reversible with the add of Calay-Hamilton's theorem.
Thus, we obtain the other inclusion, proving C.16). Then C.15) follows
easily. D

§3. Solvability of linear FBSDEs 37
We note that in the case C = AiCi, C.15) becomes
C.17) det{@,/)e^^E)}^0.
This amounts to say that the FBSDEs B.12) (with C - AiCi) is solvable
for all g £ H implies that the corresponding two-point boundary value
problem for ODEs:
(I(';,')--^(?S)' -I"'-''
C.18)
■ X@) = 0, y(T) = 9,
admits a solution for all § £ R"*.
Let us now present another necessary condition for the solvability of
B.12).
Theorem 3.3. Let C = 0. Suppose B.12) is solvable for all g e H. Then,
C.19) det{@,/)e-^*Ci} > 0, Vt £ [0,T].
Consequently, if
C.20) f = inf {T > 0 | det [@, /)e-^^Ci] = 0} < oo,
then, for any T >T, there exists a g E H, such tha.t B.12) is not solvable.
Remark 3.4. The above result reveals a significant difference between
the solvability of FBSDEs and that of two-point boundary value problems
for ODEs. We note that for C.18) to be solvable for all 5 £ BT, if and
only if C.16) holds. Since the function 11-> det {@, /)e-^* I ^ 1 } is analytic
(and it is equal to 1 at t = 0), except at most a discrete set of T's, C.16)
holds. That implies that for any To £ @,00), if it happens that C.18) is not
solvable for T = Tq with some g £ R"*, then, at some later time T > To,
C.18) will be solvable again for all g £ WC. But, in the above FBSDEs
case, if T < 00, then for any T > T, we can always find a. g E H, such
that B.12) (with C = 0) is not solvable. Thus, FBSDEs and the two-point
boundary value problem for ODEs are significantly different as far as the
solvable duration is concerned.
Proof of Theorem 3.3. Suppose there exists an so £ [0,T), such that
C.21) det {@,1)e^(-^-'°ki} = 0.
Note that so < T has to be true. Then there exists an ry £ R"*, \t]\ = 1,
such that
C.22) r?^@,/)e^(^-^'')Ci =0.

38
Chapter 2. Linear Equations
We are going to prove that for any e > 0 with sq + e < T, there exists a
g e -Ljr^ ^ (ri; R"*) C H, such that B.12) has no adapted solutions. To this
end, we let P : [0,T] -> R be a Lebesgue measurable function such that
C.23)
(P{s) = ±l, Vse[0,so + e]; P{s) = 0, Vse(so+e,T];
Sk - So
\{se[so,Sk]\f3{s)^l}\ =
\{se[so,Sk]\P{s) = -l}\--
2 '
Sk - So
fc> 1,
for some sequence Sk-lsq and Sk <T — e. Next, we define
rt
C.24)
m
= [ mdWis), te[0,T],
Jo
and take g = C{T)t] G L%^ ^jn;]R'") C H. Suppose B.12) admits an
adapted solution {X,Y,Z) e M[0,T] for this g. Then, for some y eWC",
we have (remember C = 0)
C(T)ry = @,/){e^^(°
C.25) ^ '
+ ^ e^(^-)[^, (^J[;j j +C,Z{s)]dW{s)}.
Applying rj^ from left to C.25) gives the following:
C.26)
where
C.27)
T
C(T)=a+ / {j{s) + {i^{s),Z{s))}dW{s),
Jo
=^^'^le]R,
a = T]^{0,I)e^
7(-)=r?^@,/)e^(^-Ui (J[.j) eL%in;Ci[0,T];m,
is analytic, V(*o) = 0.
>(-)=[r?^@,/)e^(^-)C,
Let us denote
C.28) e{t)=a+ [ [j{s) + {iP{s),Z{s))]dW{s), t e [0,T].
Jo
Then, it follows that
. . (d[e{t)-at)] = Mt) + {m,zit))-p{t)]dwit), te[o,T],
^' ' [ [e{T) - C(T)] = 0.

§3. Solvability of linear FBSDEs 39
By Ito's formula, we have
o=E\e{t)-c{t)\'
C.30) fi'
+ eJ \^{s)+{iP{s),Z{s))-P{s)\^ds, te[0,T].
Thus,
C.31) P{s)-j{s) = {tp{s),Z{s)), a.e.se [0,T], a.s.
which yields
C.32) [ " E\li{s)--f{s)\^ds= [ " E\{il){s),Z{s))\^ds, Vfc > 1.
•'So •'So
Now, we observe that (note 7 £ 1,3^@; C([0,T];]R)) and C.23))
r E\fi{s)~^{s)\''ds
J So
C.33) > 1 r E\P{s) - j{so)\'ds - r E\j{s) - j{so)\'ds
J So J So
>'-^^E[\l-j{so)\^ + \l+j{so)\^]-o{sk-so), k>h
On the other hand, since V(') is analytic with V(*o) = 0, we must have
C.34) ^P{s) = {s-So)ii{s), se[0,T],
for some V'(-) which is analytic and hence bounded on [0,T]. Consequently,
C.35) f " E\{i){s),Z{s)) pds < K{sk - so)' / ' E\Z{s)\''ds.
J So J So
Hence, C.32)-C.33) and C.35) imply
'" ~ '°e[|1 - 7Eo)r + |1 +7(so)r] - 0E, - so)
C.36) ^
<K{sk-sof f " E\Z{s)\'^ds, Vfc>l.
J so
This is impossible. Finally, noting the fact that det{@,/)e-^*Ci}| = 1,
we obtain C.19). The final assertion is clear. D
It is not clear if the above result holds for the case C ^ 0 since the
assumption C = 0 is crucial in the proof.
§3.2. Criteria for solvability
Let us now present some results on the operator /C (see C.4) for
definition) which will lead to some sufRcient conditions for solvability of linear
FBSDEs.

40 Chapter 2. Linear Equations
Lem.m.a 3.5. Tie range TZ{IC) ofK. is closed in H.
Proof. Let us denote Ho = L%^{n;WC) and H = Ho >^ H =
L2 (n;]R"+'"). Define
C.37)
K.Z=^T)[ ^s)-\C-AiCi)Z{s)ds
Jo
+ $(T) /" ^s)-^CiZis)dW{s), Zen.
Jo
Then, by C.7), /C is a bounded linear operator and /C = @,1)k.. We claim
that the range TZ{fC) of /C is closed in H. To show this, let us take any
convergence sequence
(^y'^^pA = JCZk-^ C, inH,
C.38)
where {Xk,Yk) is the solution of the following:
C.39) -
^ \ykio)
]dt+{A, (^^)+C,Zk}dW{t),
0.
Then, by Ito's formula, we have
C.40) =E{\Xk{T)\^ + \Yk{T)\^
-r<(ss)'-(ts)-^*'>4
We note that (recall Ci = ('^j))
A,{ y^]+C,Zk^
C.41) ={{I + Cl^C,)Zk,Zk) +
1
A,
Xk
Yk
+ 2(Cj^.4i(^M,Z,)
>^\Zk\^-C{\Xk\^ + \Yk\%

+ ^'^
§3. Solvability of linear FBSDEs 41
for some constant C > 0. Thus, C.40) implies
E{|x,(t)|2 + |y,(t)|2 + | \Zk{s)\''ds]
C.42) <CE{\Xk{T)\^ + \Yk{T)\^
I {\Xkis)\^ + \Ykis)\^)ds}, te[0,T].
Using Gronwall's inequality, we obtain
C.43) ^{l^-^WI' + l^-^WI' + l^ \Zkis)\'ds}
<CE{|X,(T)p + |n(T)|2}, te[0,T].
From the convergence C.38) and C.41), we see that Zk is bounded in "H.
Thus, we may assume that Z^ -> Z weakly in 'H- Then it is easy to see
that ICZ = C, proving the closeness of TZ{fC).
Now, TZ{IC) is a Hilbert space with the induced inner product from that
of H. In this space, we define an orthogonal projection Ph ■ H -^ H hy
the following:
Then the space
C.45) PH{n{K.)) = {0} X 7^(/C)
is closed in TZ{fC) and so is in H. Hence, TZ{IC) is closed in H. D
The following result gives some more information for the operator /C
when C = AiCi = 0, which is equivalent to the conditions: C = 0, C = 0
and AiCi + Bi = 0. Note that Ai, Bi and C\ are not necessarily zero.
Lemma 3.6. Let C = 0 and let C.19) hold. Then
C.46) 7^(/C) = {T]eH\ET] = 0} = Af{E),
C.47) Ar(/C) ={Z e ^ I /CZ = 0} = {0}.
Proof. First of all, by Lemma 3.5, we see that TZ{IC) is closed. Also,
by C.4) and Lemma 3,1, 7^(/C) C M{E) (since C = AiCi). Thus, to show
C.46), it suffices to show that
C.48) A^(E)fl7^(/C)^ = {0}.
We now prove C.48). Take tj G Af{E). Suppose
0 = E{t],JCZ)
C.49) fT
= E(r?,@,/)$(T) / ^s)-^CiZ{s)dW{s)), MZ eH.
Jo

42
Denote
Chapter 2. Linear Equations
C.50) (^y^^^ = ^{t) j ^{s)-'C,Z{s)dW{s), t e [0,T].
Then, hj C = AiCi = 0, we have
C.51)
'd(y) =A(^y^dt+{Ai(^y^+C,z}dWit),
{[¥{0)) ''■
By Ito's formula and Gronwall's inequality, we obtain
C.52) S{|X(t)|2 + |F(t)n <K [ E\Zis)\''ds, t e [0,T].
Jo
Also, we have
C.53) (^g) = ^*e^(*-){^i (^[^*J) +CiZE)}dW^E), t £ [0,T].
Since £ry = 0 and t] E H, by Martingale Representation Theorem, there
exists a C £ "H, such that
C.54) T]= [ C{s)dW{s).
Jo
Then, from C.49) and C.53), we have
0 = E(r?,/CZ)=E(r?,@,/)(^^[Jj))
C.55)
This yields
= I^E{Cis),{0,I)e^^''-'^{A, (^I^'j) +C,Zis)})ds.
C.56)
■Te^'(^-^U^j]as),Zis))ds
Jo
=-f^(^fe-<-.(;)<w.(fw),..
By C.49), the above holds for all Z £ ^. Now, let 0 < E < T and take
C.57) Z{s) = Cre^"(^-) (^ j) C(*)X[t-..t](*), s £ [0,T].

§3. Solvability of linear FBSDEs 43
Then X(s) = 0, Y{s) = 0 for all s e [0,T - 6]. Consequently, C.56) and
C.52) result in
/J^E|cfe^^(^-^)(;)c(^)fd.
C.58) kkJ (^E\as)\'y^\f E\Z{r)\'dry^'ds
r^ / \ 1/2 / /"* \ 1/2
<K (E\C{s)f] ( E\Cir)\'dr) ds.
By C.19), we obtain
/ E\Cis)\^ds<K r E|cJ^e^"(^-^)f°)c(s)|'d5
Jt-5 Jt-S ' V-'/ '
C.59) -^Ir^ {E\as)\'y^\f £|C(r)pdr)'^'d5
<l f E\as)\''ds + K [ f E\ar)\''drds.
^ JT-& Jt-SJT~S
Thus, it follows that
rp rp
C.60) / E\C{s)\^ds <K6 f E\C{s)\'^ds,
Jt-5 Jt~s
with K > 0 being an absolute constant (independent of 6). Therefore, for
6 > 0 small, we must have
C.61) C(s) = 0, a.e. s e [T - 6,T], a.s.
This together with C.56) implies that
C.62)
^''~'E(Cre^^(^-)E)c(^),ZE))d5
=-f-^(^re-^(-)(;)c(.),(fg))..
Then, thanks to C.19), we can continue the above procedure to conclude
that C.61) holds over [0,T] and hence it follows from C.54) that rj = 0.
This proves C.48). _ _
We now prove C.47). Suppose /CZ = 0. Again, we let {X{-),Y{-)) be
defined by C.50). Then, for any C e ?/, by C.53), we have
C.63)
0 = E{ f C{s)dW{s),K:Z)
Jo
= Ej\as),iO,I)e^^^-^^{A, (^Jg) +C,Zis)})ds.

44 Chapter 2. Linear Equations
This implies that
C.64) @,/)e^(^-^){^i (y^f}) +CiZ(s)} = 0, a.e.s e [0,T], a.s.
By C.19), we easily see that
s ^ Bis) ^ {@, /)e^(^-^)Ci }-\0, /)e^(^-»)^i
is analytic and hence bounded over [0,T]. From C.64), we obtain
C.65) Z{s) = -B{s) (y^'J^^ , a.e.5 e [0,T], a.s.
Then, {X,Y) is the solution of
d(y) =A(^yyt+[A,-Bit)] (^y^Wit),
C.66)
/^^@)\_o
Hence, we must have {X,Y) = 0, which yields Z = 0 due to C.65). This
proves C.47). D
A consequence of the above is the following.
Theorem 3.7. Let C = AiCi = 0. Tien, iinear FBSDE B.12) is solvable
for all g G H if and only if C.17) and C.19) hold. In this case, the adapted
solution to B.12) is unique (for any given g £ H).
Proof. Theorems 3.2 and 3.3 tell us that C.17) and C.19) are necessary.
We now prove the sufficiency. First of all, for any g £ H,hy C.17), we can
find y e R"*, such that C.14) holds (note C = 0). Then we have
C.67) g-iO,I)HT)(jyeMiE).
Next, by C.46), there exists a, Z &%, such that
C.68) g-{Q,imT)(^j]y = K,Z.
For this pair {y,Z) G BT x H, we define iX,Y) by C.2). Then one can
easily check that {X,Y,Z) e M[0,T] is an adapted solution of B.12). The
uniqueness follows easily from C.47) and C.17). D
The above result gives a complete solution to the solvability of linear
FBSDE B.12) with C = AiC,, = 0. By Theorems 1.2, 2.1 and 3.7, we can
obtain the solvability result for the original linear FBSDE A.1). We omit
the precise statement here.

§4. A Riccati type equation 45
§4. A Riccati Type Equation
In this section, we present another method. R will give a sufficient condition
for the unique solvability of B.12). We will obtain a Riccati type equation
and a BSDE associated with B.12). Let us now carry out a heuristic
derivation.
Suppose {X,Y,Z) e M[0,T] is an adapted solution of B.12). We
assume that X and Y are related by
D.1) Yit) = P{t)X{t)+pit), \/t e [0,T], a.s.
where P : [0,T] -> JR"*^" is a deterministic matrix-valued function and
p : [0,T] X n -> IR"* is an {.7^t}t>o-adapted process. We are going to derive
the equations for P(-) and p(-). First of all, from D.1) and the terminal
condition in B.12), we have
D.2) g = P{T)X{T)+p{T).
Let us impose
D.3) P{T) = 0, p(T) = g.
Since g e Ljr^(n;]R'") and p(-) is required to be {.7^t}t>o-adapted, we
should assume that p(-) satisfies a BSDE:
'"* U(T) =
I dp{t) = a{t)dt + q{t)dW{t), t e [0,T],
9,
with a(-), q{-) e -Ljr@,T;]R'") being undetermined. Next, by Ro's formula,
we have (for simplicity, we suppress t below):
dY = {PX + P[AX + BY + CZ] + a}dt
+ {P[AiX + B^Y + CiZ] + q}dW
D.5)
= {[P + PA + PBP]X + PCZ + PBp + a}dt
+ {[PAi + PBiP]X + PCiZ + PBip + q}dW,
Now, compare D.5) with the second equation in B.12) (note D.1)), we
obtain that
D.6) [P + PA + PBP]X + PCZ + PBp +a=[A + BP]X + CZ + Bp,
and
D.7) {PAi+PBiP)X + PCiZ + PBip + q= Z.
By assuming / — PCi to be invertible, we have from D.7) that
D.8) Z={I- PCi)-'{iPAi + PBiP)X + PBip + q}.

46
Chapter 2. Lineai Equations
Then, D.6) can be written as
0=[P + PA + PBP -A-BP
+ {PC - C){I - PCi)-^iPAi + PBiP)]X
+ [PB-B + {PC - C){I - PCi)-^PBi]p
+ {PC-C){I-PCi)-^q + a.
D.9)
Now, we introduce the following differential equation for R"*^"-valued
function P(-):
D.10)
' P + PA + PBP-A-BP
+ (PC - C){I - PCi)-'(PAi + P5iP) = 0, t e [0,T],
. P{T) = 0.
We refer to D.10) as a Riccati type equation. Suppose D.10) admits a
solution P(-) over [0,T] such that
D.11) [I-P{t)Ci]-^ is bounded for t e [0,T].
Then, D.9) gives
a = -[PB-B + {PC- C){I - PCi)-'PBi]p
-{PC-C){I-PC,)-'q.
Combining this with D.4), we see that one should introduce the following
BSDE:
D.12)
dp = -{[P5 - 5 + (PC - C)(/ - PCi)-^P5i]p
+ {PC - C){I - PCi)-ig}dt + qdW, t e [0,T],
When D.10) admits a solution P(-) such that D.11) holds, by Theorem 3.2
of Chapter 1, BSDE D.12) admits a unique adapted solution (p(-))9@) ^
A/'[0,T]. Then we can define the following:
D.13)
f 1 = A + 5P + C(/ - PC{)-^{PA^ + P5iP),
li = Ai + 5iP + Ci(/ - PCi)-i(PAi + P5iP),
b = Bp + C{I- PCi)-'{PBip + q),
[a = Bip+Ci{I- PCi)-' {PBip + q).
It is clear that A and Ai are time-dependent matrix-valued functions and b
and a are {.7"t}t>o-adapted processes. Further, under D.11), the following
SDE admits a unique strong solution:
D.14)
dX = {AX + b)dt + {AiX + a)dW,
X{0) = X.
te[0,T],

§4. A Riccati type equation 47
The following gives a representation of the adapted solution of FBSDE
B.12).
Theorem 4.1. Let D.10) admits a solution P(-) such that D.11) holds.
Then FBSDE B.12) admits a unique adapted solution {X,Y,Z) e M[Q,T]
which is determined by D.14), D.1) and D.8).
Proof. First of all, a direct computation shows that the process
{X,Y,Z) determined by D.14), D.1) and D.8) is an adapted solution of
B.12). We now prove the uniqueness. Let {X,Y,Z) e jV([0,T] be any
adapted solution of B.12). Set
(y = px+p,
[Z=il- PC,)-' [{PA, + PB,P)X + PBip + q],
where P and (p, q) are (adapted) solutions of D.10) and D.12), respectively.
Denote Y = Y — Y and Z = Z — Z. Then a direct computation shows that
' dY = [{PB - B)Y
D.16) \ +(PC-C)Z]dt+[P5iy-(/-PCi)Z]dW(t),
. y(T) = 0.
By D.11), we may set
D.17) Z = P5iy-(/-PCi)Z,
to get the following equivalent BSDE (of D.16)):
' ^ = {\PB-B-^[PC - C){I - PCi)-'PBi]Y
D.18) I -{PC-C)iI-PCi)-'Z}dt + ZdW{t),
JiT) = 0.
It is clear that such a BSDE admits a unique adapted solution (Y, Z) = 0
(see Chapter 1, §3). Consequently, Z = 0. Hence, by D.15), we obtain
(Y = PX+p,
D 19) <
\z={I-PCi)-'[{PAi+PBiP)X + PB,p + q],
This means that any adapted solution {X, Y, Z) of B.12) must satisfy D.19).
Then, similar to the heuristic derivation above, we have that X has to be
the solution of D.14). Hence, we obtain the uniqueness. D
The following result tells us something more.
Proposition 4.2. Let D.10) admits a solution P(-) such that D.11) holds
for t e [To,T] (with some Tq > Oj. Tien, for any f e [0,T - To], iinear
FBSDE B.12) is uniquely solvable on [0,f].

48 Chapter 2. Linear Equations
Proof. Let
D.20) P{t) = P{t + T-f), te[0,f].
Then P(-) satisfies D.10) with [0,T] replaced by [0,f] and
D.21) [I-P{t)Ci]-^ is bounded for te [0,T].
Thus, Theorem 4.1 applies. D
The above proposition tells that if D.10) admits a solution P(-)
satisfying D.11), FBSDE B.12) is uniquely solvable over any [0,T] (with T < T).
Then in the case C = A\Ci, by Theorem 3.2, the corresponding two-point
boundary value problem C.17) of ODE over [0,T] admits a solution for all
g e R"*, of which a necessary and sufficient condition is
D.22) det{@,/)e^* (/) } > °' ^* ^ t^'^l-
Therefore, by Theorem 3.7, compare D.22) and C.17), we see that the
solvability of Riccati type equation D.10) is only a sufficient condition for
the solvability of B.12) (at least for the case C = A\C\ =0).
In the rest of this section, we concentrate on the case C = 0. We do
not assume that A\Ci = 0. In this case, D.10) becomes
D 23) {P + PA + PBP-A-BP = Q, te[0,T],
\ P{T) = 0,
and the BSDE D.12) is reduced to
D 24) idp=[B- PB]pdt + qdW{t), t e [0, T],
\p{T) = g.
We have seen that D.22) is a necessary condition for D.23) having a solution
P(-) satisfying D.11). The following result gives the inverse of this.
Theorem 4.3. Let C = 0, C = 0. Let D.22) hold. Then D.23) admits a
unique solution P(-) which has the following representation:
D.25) P(t) = -[@,/)e^(^-*)E)]"\0,/)e^(^-*)(j), t G [0,T].
Moreover, it holds
D.26)
I-Pit)C, = [@,/)e^(^-*) E)]"[@,/)e^(^-*) (^^)
te[0,T].
Consequentiy, if in addition to D.22), C.19) holds, then D.11) holds and
the linear FBSDE B.12) (with C = 0) is uniquely solvable with the
representation given by D.14), D.1) and D.8).

§5. Some extensions
49
Proof. Let us first check that D.25) is a solution of D.23). To this end,
we denote
D.27) e(t) = @,/)e^(^-*)(^5), te[0,T].
Then we have (recall B.13) for the definition of A)
D.28) Q{t) = -@,/)e^(^-*) (J) 5 - e(t)B.
Hence,
= e-^{-(o,/)e^(^-*)(JM-eB}(-p)
D.29) +e-@,/)e^(^-)D)
= (P5-5)(-P)+e-'@,/)e^(^-*) (^Aa + A
= -PBP + BP-PA + A.
Thus, P(-) given by D.25) is a solution of D.23). Uniqueness is obvious
since D.23) is a terminal value problem with the right hand side of the
equation being locally Lipschitz. Finally, an easy calculation shows D.26)
holds. Then we complete the proof. D
§5. Some Extensions
In this section, we briefly look at the case with multi-dimensional
Brownian motion. Let W{t) = {W^ (t), • • •, W^^t)) be a d-dimensional Brownian
motion defined on (H, J", {Ji}t>o,P) with {Jt}t>o being the natural
filtration of W{-) augmented by all the P-null sets. Similar to the case of
one-dimensional Brownian motion, we may also start with the most general
case, by using some necessary conditions for solvability to obtain a reduced
FBSDE. For simplicity, we skip this step and directly consider the following
FBSDE:
' dX= {AX + BY)dt
d
E.1)
i=l
ie[0,T],
dY = {AX + BY)dt + Y^ Z'dW'it),
i=l
[ X{0) = 0, Y{T) = g,

50
Chapter 2, Linear Equations
where A,B, etc. are certain matrices of proper sizes. Note that we only
consider the case that Z does not appear in the drift here since we have
only completely solved such a case. We keep the notation A as in B.13)
and let
E.2)
A\ =
A\ B{
0 0
c\
c\
Ki <d.
If we assume X{-) and Y{-) are related by D.1), then, we can derive a
Riccati type equation, which is exactly the same as D.23). The associated
BSDE is now replaced by the following:
E.3)
dp=[B-PB]pdt + Y,Q'dW'it), te[0,T],
i=l
[piT) = 9.
Also, D.13), D,14) and D.8) are now replaced by the following:
'A = A + BP, b = Bp,
A\ = A\ + 5JP
+ Cj(/ - PCi)~^iPA\ + PB\P), l<i<d,
{ y = B{p + Cj(/ - PC{)-'{PB\p + q'),
E.4)
E.5)
dX = {AX + h)dt-irY^{A^iX + '5')dW\t), tG [0,T],
[X@) = 0,
i=l
E.6) Z' = {I-PCi)-'^{{PA\+PBiP)X + PBip + q'}, l<i<d.
Our main result is the following.
Theorem 5.1. Let D.22) hold and
E.7) det {@,/)e-^*Cj} > 0, Vt e [0,T], 1 < i < d
Tien D.23) admits a unique, solution P{-) given by D.25) such that
E.8) [/ - Pit)Ci]~^ is bounded for t e [0,T], 1 < i < d,
and the FBSDE E.1) admits a unique adapted solution (X, Y, Z) e M[0, T]
which can be represented by E.5), D.1) and E.6).
The proof can be carried out similar to the case of one-dimensional
Brownian motion. We leave the proof to the interested readers.

Chapter 3
Method of Optimal Control
In this chapter, we study the solvability of the following general nonlinear
FBSDE: (the same form as C.16) in Chapter 1)
@.1)
' dX{t) = b{t,X{t),Yit),Z{t))dt + a{t,X{t),Y{t),Z{t))dW{t),
dY{t) = h{t,X{t), Y{t), Z{t))dt + Z{t)dW{t), t e [0,T],
[ X@) = X, Y{T) = g{X{T)).
Here, we assume that functions 6, u, h and g are all deterministic, i.e., they
are not explicitly depending on w e fi; and T > 0 is any positive number.
Thus, we have an FBSDE in a (possibly large) finite time duration. As we
have seen in Chapter 1, §4, under certain Lipschitz conditions, @.1) admits
a unique adapted solution {X{-),Y{-),Z{-)) e M[0,T], provided T > 0 is
relatively small. But, for general T > 0, we see from Chapter 2 that even if
h, a, h and g are all affine in the variables X, Y and Z, system @.1) is not
necessarily solvable. In what follows, we are going to introduce a method
using optimal control theory to study the solvability of @.1) in any finite
time duration [0, T]. We refer to such an approach as the m,ethod of optim,al
control.
§1. Solvability and the Associated OptimEil Control Problem
§1.1. An optimal control problem
Let us make an observation on solvability of @.1) first. Suppose {X{-),
Yi-),Zi-)) e M[0,T] is an adapted solution of @.1). By letting y = Y{0) e
R"*, we see that {xl),Y{-)) satisfies the following FSDE:
A.1)
' dX{t) = b{t,X{t),Y{t),Z{t))dt + a{t,X{t),Y{t),Z{t))dW{t),
dY{t) = h{t,X{t),Y{t),Z{t))dt + Z{t)dW{t), t£ [0,T],
[ X@) = X, y@) = y,
with Z(-) e Z[0,T] = L%-{0,T;'R"''"') being a suitable process. We note
that y and Z(-) have to be chosen so that the solution {X{-),Y{-)) of A.1)
satisfies the following terminal constraint:
A.2) Y{T) = g{X{T)).
On the other hand, if we can find an y e R"* and a Z{-) G Z[0, T], such that
A.1) admits a strong solution {X{-),Y{-)) with the terminal condition A.2)
being satisfied, then {X{-),Y{-),Z{-)) G M[0,T] is an adapted solution of
@.1). Hence, @.1) is solvable if and only if one can find an y e R"*
and a Z(-) e Z[0,T], such that A.1) admits a strong solution {X{-),Y{-))
satisfying A.2).

52 Chapter 3. Method of Optimal Control
The above observation can be viewed in a different way using the
stochastic control theory. Let us call A.1) a stochastic control system with
{X{-),Y{-)) being the state process, Z{-) being the control process, and
{x,y) e R" X R"* being the initial state. Then the solvability of @.1) is
equivalent to the following controllability problem for A.1) with the target:
A.3) T={ix,gix))\x&m"}.
Problem (C). For any a; e R", find an y e R"* and a control Z(-) e
2:[0,T], such that
A.4) (X(T),y(T))er, a.s.
Problem (C) having a solution means that the state {X{t),Y{t)) of
system A.1) can be steered from {a;} x R"* (at time t = 0) to the target T,
given by A.3), at time t = T, almost surely, by choosing a suitable control
z(-) e Z[0,T].
In the previous chapter, we have presented some results related to this
aspect for linear FBSDEs. We point out that the above controllability
problem is very difficult for nonlinear case. However, the above formulation
leads us to considering a related optimal control problem, which essentially
decomposes the solvability problem of the original FBSDE into several
relatively easier ones; and we can treat them separately. Let us now introduce
the optimal control problem associated with @.1).
Again, we consider the stochastic control system A.1). Let us make
the following assumption:
(HI) Functions b{t,x,y,z), a{t,x,y,z), h{t,x,y,z) and ^(a;) are
continuous and there exists a constant L > 0, such that ioi (p = b, a, h,g, it holds
that
A.5)
' \'Pit,x,y,z) -(fi{t,x,y,z)\ < L{\x-x\ + \y -y\ + \z-z\),
|^(i,0,0,0I, \a{t,x,y,0)\ < L,
Vt e [0,T], x,x e R", y,y e R"*, z,z e R"*""^.
Under the above (HI), we see that for any (a;, y) e R" xR"*, and Z(-) e
2^[0,T], A.1) admits a unique strong solution, denoted by, {X(-),Y{-)) =
{X{- •,x,y,Z{-)),Y{- ■,x,y,Z{-))), indicating the dependence on {x,y,Z{-)).
Next, we introduce a functional (called cost functional). The purpose is
to impose certain kind of penalty on the difference Y{T) — g{X{T)) being
large. To this end, we define
A.6) f{x,y) = ^/l + \y-g{xW-l, V(a:,y) S R" x R"
Clearly, / is as smooth as g and satisfying the following:
U{x,y)>0, V(a:,2/)eR"xR'",
1 f{x, y) = 0, if and only if y = g{x).

§1. Solvability and the associated optimal control problem 53
In the case that (HI) holds, we have
d 8) l-^(^'y) - f('^^y)\ <L\x-x\ + \y-y\,
\/ix,y),ix,y)GB"xn"'.
Now, we define the cost functional as follows:
A.9) J{x,y;Z{.))^Ef{X{T;x,y,Zi-)),Y{T;x,y,Zi.))).
The following is the optimal control problem associated with @.1).
Problem (OC). For any given {x,y) e R" x JR"*, find a Z(-) e Z[0,T],
such that
A.10) V{x,y)^^^mi^^^^J{x,y;Zi-)) = J{x,y;^{-)).
Any Z{-) e -E[0,T] satisfying A.10) is call an optimal control, the
corresponding state process
(X{.),Y{.))^{X{.;x,y,M.)),Y{.;x,y,^{-)))
is called an optimal state process. Sometimes, {X{-),Y{-),Z{-)) is referred
to as an optimal triple of Problem{OC).
We have seen that the optimality in Problem{OC) depends on the
initial state {x,y). The number V{x,y) (which depends on {x,y)) in A.10)
is called the optimal cost function of Problem{OC). By definition, we have
A.11) V{x,y)>0, y{x,y)eWC xWT.
We point out that in the associated optimal control problem, it is possible
to choose some other function / having similar properties as A.7). For
definiteness and some later convenience, we choose / of form A.6).
Next, we introduce the following:
A.12) M(V) ={ix,y) e R" X R'" I Vix,y) = 0}.
This set is called the nodal set of function V. We have the following simple
result.
Proposition 1.1. For x e R", FBSDE @.1) admits an adapted solution
if and only if
A.13) M{V)f][{x}xn"']^cl>,
and for some (x, y) G J\f{V°), there ejcists an optimal control Z{-) € Z[0, T],
such that
A.14) V{x,y) = J{x,y;Z{-))=0.
Proof Let {X{-),Y{-),Z{-)) e M[0,T] be an adapted solution of @.1).
Let y = y@) e R"*. Then A.14) holds which gives {x,y) e J\f{V) and
A.13) follows.

54 Chapter 3. Method of Optimal Control
Conversely, if A.14) holds with some {x,y) e R" x R"* and Z(') e
Z[0,T], then {X{■),¥{■),Z{-)) e M[0,T] is an adapted solution of @.1).
D
In light of Proposition 1.1, we propose the following procedure to solve
the FBSDE @.1):
(i) Determine the function V{x,y).
(ii) Find the nodal set Af{V) oiV; and restrict a; e R" to satisfy A.13).
_(iii) For given a; e R" satisfying A.13), let y G R"* such that {x,y) e
M(V). Find an optimal control Z(-) e Z[0,T] of Problem{OC) with the
initial state ix,y). Then the optimal triple {X{-),Y{-),Z{-)) e M[0,T] is
an adapted solution of @.1).
It is clear that in the above, (i) is a PDE problem; (ii) is a minimizing
problem over R"*; and (iii) is an existence of optimal control problem.
Hence, the solvability of original FBSDE @.1) has been decomposed into
the above three major steps. We shall investigate these steps separately.
§1.2. Approximate solvability
We now introduce a notion which will be useful in practice and is related
to condition A.13).
Definition 1.2. For given x g R", @.1) is said to be approximately
solvable if for any e > 0, there exists a triple {Xe{-),Ye{-), Zs{-)) e A^[0,T],
such that @.1) is satisfied except the last (terminal) condition, which is
replaced by the following:
A.15) E\Y,{T) - g{X,{T))\ < e.
We call {Xg{-),Yg{-),Zg{-)) an approximate adapted solution of @.1) with
accuracy e.
It is clear that for given x e R", if @.1) is solvable, then it is
approximately solvable. We should note, however, even if all the coefficients of an
FBSDE are uniformly Lipschitz, one still cannot guarantee its approximate
solvability. Here is a simple example.
Example 1.3. Consider the following simple FBSDE:
A.16)
' dXit) = Yit)dt + dW{t),
dY{t) = -X{t)dt + Z{t)dW{t),
^X{0) = x, Y{T) = -X{T),
with T = ^ and a; ^ 0. It is obvious that the coefficients of this FBSDE are
all uniformly Lipschitz. However, we claim that A.16) is not approximately
solvable. To see this, note that by the variation' of constants formula with

§1. Solvability and the associated optimal control problem 55
y = Y{0), we have
/X{t)\ _ / cost sinA /a;
A17) UWy ~V-sint cos*; [y
fWcosit-s) sin(t-.)\/ 1 \
J^y-smit-s) cosit-s)J \Zis)J
Plugging t = T = ^ into A,17), we obtain that
X{T)+Y{T) = -V2x+ f T]{s)dW{s),
Jo
where t] is some process in L^{0,T;K). Consequently, by Jensen's
inequality we have
E\Y{T) - g{XiT))\ = E\X{T) + Y{T)\ > \E[X{T) + Y{T)]\ = V2\x\ > 0,
for all (y, Z) e R"* x Z[0,T]. Thus, by Definition 1.2, FBSDE A.16) is not
approximately solvable (whence not solvable). D
The following result establishes the relationship between the
approximate solvability of FBSDE @.1) and the optimal cost function of the
associated control problem.
Proposition 1.4. Let (HI) hold. For a given x G R", tie FBSDE @.1) is
approximately solvable if and only if the following holds:
A.18) inf V{x,y) = 0.
Proof. We first claim that the inequality A.15) in Definition 1.2 can
be replaced by
A.19) Ef{X,{T),Y,{T))<e.
Indeed, by the following elementary inequalities:
A.20) '^-^ < x/l + r2 - 1 < r, Vre[0,oo),
we see that if A.15) holds, so does A.19). Conversely, A.20) implies
E/(X,(T),n(T)) > ^-E{\Y,iT) - giX,iT))\^I^\Y.^T)-gix.iT))\<i))
+ iE(|n(T) -ff(X,(T))|/(|y,(T)-,(X.(T))|>l)).
Consequently, we have
A.21) E\Y,{T)-g{X,{T))\ < 3Ef{X,{T),Y,{T)) + ^3Ef{X,{T),Y,{T)).
Thus A.19) implies A.15) with e being replaced by e' = 3e + v3e. Namely,
A.18) is equivalent to the approximately solvability, by Definition 1.2 and
the definition oiV. D

56 Chapter 3. Method of Optimal Control
Using Proposition 1.4, we can now claim the non-approximate
solvability of the FBSDE A.16) in a different way. By a direct computation using
A.21), one shows that
J{x,y;Z{.))=Ef{X{T),Y{T))
>l[^V2\x\ + \-l]'>0, ^Zi-) G Z[0,T].
V{x,y)>\[JV2\x\ + \-h'>0,
Thus,
s'-y ' ' 4 2^
violating A.18), whence not approximately solvable.
Next, we shall relate the approximate solvability to condition A.13).
To this end, let us introduce the following supplementary assumption.
(H2) There exists a constant L > 0, such that for all {t,x,y,z) e
[0,T] X R" X R"* X WT'"', one of the following holds:
\b{t,x,y,z)\ + \a{t,x,y,z)\ < L(l + |a:|),
{hit,x,y,z),y)>-L{l + \x\\y\ + \y\''),
A-22) , ,,„ _ ,2
,, „„, j {h{t,x,y,z),y)>-L{l + \yf),
^'^^ \\9{:c)\<L.
Proposition 1.5. Let (HI) hold. Then A.13) implies A.18); conversely,
ifV{x,-) is continuous, and (H2) holds, then A.18) implies A.13).
Proof. That condition A.13) implies A.18) is obvious. We need only
prove the converse. Let us first assume that V is continuous and A.22)
holds.
Since A.18) implies the approximately solvability of @.1), for every
e e @,1], we may let {Xs,Ys,Zs) e A^[0,T] be the approximate adapted
solution of @.1) with accuracy e. Some standard arguments using Ito's
formula, Gronwall's inequality, and condition A.22) will yield the following
estimate
A.24) E\X,{t)\^<C{l + \x\^), Vte[0,T], ee@,l].
Here and in what follows, the constant C > 0 will be a generic one,
depending only on L and T, and may change from line to line. By A.24) and
(LI5), we obtain
E\Y,iT)\ < E\9iX,iT))\ + E\Y,{T) - g{X,(T))\
■ <C(l + kl)+e<C(l + kl).

§2. Dynamic programming method and the HJB equation 57
Next, let (a;) = a/T+T^- It is not hard to check that both D {x) and
D^ {x) are uniformly bounded, thus applying Ito's formula to (¥^{1)), and
note A.22) and A.24), we have
E{Y,iT))-E{YS))
1
+
^.MI-1-.M^^n}
ds
A.26) ' 2L' ^^ ^' \ '^ ' {Y,{s))
>-LeJ {l + \X,{s)\ + {Y,{s)))ds
>-C{l + \x\)-LE f {Y,{s))ds, Vte[0,T].
Now note that \y\ < {y) < I + \y\, we have by Gronwall's inequality and
A.25) that
A.27) E{Y,{t))<C{l + \x\), Vie[0,T], ee@,l].
In particular, A.27) leads to the boundedness of the set {|y£@)|}£>o. Thus,
along a sequence we have Y^,^ @) -> y, as fc -> oo. The A.13) will now follow
easily from the continuity of V{x, •) and the following equalities:
A.28) 0 < Vix, n, @)) < EfiX,, (T), n, (T)) < e,.
Finally, if A.23) holds, then redoing A.25) and A.26), we see that
A.27) can be replaced by E {Ye{t)) <C,\/t G [0,T], e e @,1]. Thus the
same conclusion holds. D
We will see in §3 that if (HI) holds, then V{-, •) is continuous.
§2. DynEimic ProgrEimming Method and the HJB Equation
We now study the optimal control problem associated with @.1) via the
Bellman's dynamic programming method. To this end, we let s S [0,T) and
consider the following controlled system (compare with A.1)):
{dX{t) = h{t,X{t),Y{t),Z{t))dt + a{t,X{t),Y{t),Z{t))dW{t),
dY{t) = h{t,X{t),Y{t),Z{t))dt + Z{t)dW{t), t£ [s,T],
X{s) = X, Y{s) = y,
Note that under assumption (HI) (see the paragraph containing A.5)), for
any {s,x,y) e [0,T) x ]R" x JR"* and Z(-) e Z[s,T] = L%{s,T;Br'"^),
equation B.1) admits a unique strong solution, denoted by, {X{-),Y{-)) =
{X{-; s,x,y,Z{-)),Y{- ;s,a;, y,Z(-))). Next, we define the cost functional as
follows:
B.2) J{s,x,y;Z{-))^Ef{X{T;s,x,y,Z{.)),Y{T;s,x,y,Z{.))),

58 Chapter 3. Method of Optimal Control
with / defined by A.6). Similar to Problem{OC), we may pose the
following optimal control problem.
Problem (OC)^. For any given is,x,y) e [0,T) x R" x WT, find a
Z(-) e Z[s,T], such that
B.3) Vis, X, y) ^ ^^mi^^ ^.J{s, x, y; Z(-)) = J(s, x, y; Z(-)).
We also define
B.4) ViT,x,y) = f{x,y), (rr, y) S R" x R'".
Function V{-, •, •) defined by B.3)-B.4) is called the value function of the
above family of optimal control problems (parameterized by s e [0,T)). It
is clear that when s = 0, Prohlem{OC)s is reduced to Prohlem{OC) stated
in the previous section. In another word, we have embedded Problem{OC)
into a family of optimal control problems. We point out that this family of
problems contains some very useful "dynamic" information due to allowing
the initial moment s e [0,T) to vary. This is very crucial in the dynamic
programming approach. From our definition, we see that
B.5) V{0,x,y) = V{x,y), W{x,y) eWC xWT.
Thus, if we can determine V{s,x,y), we can do so for V{x,y). Recall
that we called V{x,y) the optimal cost function oi Problem{OC), reserving
the name value function for V{s,x,y) for the conventional purpose. The
following is the well-known Bellman's principle of optim,ality.
Theorem 2.1. For any 0 < s < ?< T, and {x,y) e R" x R"*, it holds
B.6) V{s,x,y) = ini EV{s,X{s;s,x,y,Z{-)),Y{s;s,x,y,Z{-))).
A rigorous proof of the above result is a little more involved. We present
a sketch of the proof here.
Sketch of the proof. We denote the right hand side of B.6) by V{s, x, y).
For any z{-) € Z[s,T], by definition, we have
V{s,x,y) < J{s,x,y;Z{-))
= EJis, Xis; s, X, y, Z{-)),Y{s; s, x, y, Z(-)); Z{-)).
Thus, taking infimum over Z(-) e 2^[s,T], we obtain
B.7) V{s,x,y)<V{s,x,y).
Conversely, for any e > 0, there exists a Zs{-) G Z[s,T], such that
Vis, x,y)+e> J{s, x, y; Zei-))
= EJis, X{s; s, X, y, Z,(-)), Y{s; s, x, y, Z,(-)); Z,(-))
^ ■ ' >EVis,Xis;s,x,y,Z,i-)),Yis;s,x,y,Z,i-)))
> V{s,x,y).

§2. Dynamic programming method and the HJB equation 59
Combining B.7) and B.8), we obtain B.6). D
Next, we introduce the Hamiltonianioi the above optimal control
problem:
^(....Q,.)^{(.(£-;:]))
B.9) +ltr[g("(^'"/'^))("(^'"/'^))'^]},
V(s, X, y, q, Q, z) e [0, T] X R" X WT
X 11"+"* X 5"+"* X WT""^,
and
B.10)
H{s, X, y, q, Q) = inf His, x, y, q, Q, z),
V(s, X, y, q, Q) e [0, T] X R" X R"* X ]R"+"* x S
n+m ^ cn+m
where §"+"* is the set of all (n + m) x {n + m) symmetric matrices. We
see that since R"*^'^ is not compact, the function H is not necessarily
everywhere defined. We let
B.11) V{H) ^{(s, X, y, q, Q) \ H{s, x, y, q, Q) > -oo}.
From above Theorem 2.1, we can obtain formally a PDE that the value
function V{- ,■ ,•) should satisfy.
Proposition 2.2. Suppose V{s,x,y) is smooth and H is continuous in
IntD(i?). Tiien
B.12) V,is, X, y) + H{s, x, y, DV{s, x, y), D^V{s, x, y)) = 0,
for aJi {s,x,y) G [0,T) x R" x R"*, such tiiat
B.13) {s,x,y,DV{s,x,y),D^V{s,x,y)) G IntP(//),
wiiere
Proof. Let {s,x,y) e [0,T) x R" x R"* such that B.13) holds. For any
^ g j^mxd^ let (X(-), ¥{■)) he the solution of B.1) corresponding to (s, x, y)
and Z(-) = z. Then, by B.6) and Ito's formula, we have
■V{s,X{s),Y{s))-V{s,x,y)
B.14) "^-"\ §_g
o<e|
-> Vs{s,X,y) + n{s,X,y,DV{s,x,y),D'^V{s,x,y),z).
Taking infimum in 2; e R"*^"^, we see that
B.15) V,{s, X, y) + H{s, x, y, DV{s,x, y), D^Vis, x, y)) > 0.

60 Chapter 3. Method of Optimal Control
On the other hand, for any e > 0 and s £ is,T), by B.6), there exists
a Z(-) = Zs{-) e Z[s,T], with the corresponding state being {X{-),Y{-)),
such that
B.16)
g > ^(Vis,X{s),Y{s))-V{s,x,y)^
~ \ s - s /
= j^^El^'{v,{t,X{t),Y{t))
+ n{t,X{t),Y{t),DVit,X{t),Yit)),D'V{t,X{t),Y{t)),Z{t)))]dt
^--^E1^' {v,{t,X{t),Y{t))
H{t, X{t), Yit), DV{t, X{t),Y{t)),D^V{t,X{t), Y{t))) ]dt
s
+
-> Vs{s, X, y) + H{s, X, y,DV{s, x, y), D'^V{s, x, y)).
Here, we have used B.13) and the assumption that H is continuous in
IntD(i?). Combining B.15)-B.16), we obtain B.12). D
Equation B.12) is called the Hamilton-Jacobi-Bellman (HJB for short)
equation associated with our optimal control problem. In principle, one
can determine the value function V{-,-,-) through solving B.12)-B.13)
together with the terminal condition B.4). However, since !>(-/?) might
be a very complicated set, solving B.12)-B.13) together with B.4) is very
difficult. Thus, much more needs to be done in order to determine the value
function V.
§3. The Value Function
In this section, we are going to study the value function V introduced in
the previous section in some details.
§3.1. Continuity and semi-concavity
We first look at the continuity of the value function V{s,x,y). Note that
since the control domain R"*^'^ is not compact, we can only prove the
right-continuity of V{s,x,y) in s G [0,T).
Proposition 3.1. Let (HI) hold. Then V{s,x,y) is right-continuous in
s e [0, T) and there exists a constant C > 0, such that
0<V{s,x,y)<Cil + \x\ + \y\),
^ ■ ' V(s,a;,2/) e [0,T] x R" x R"*,
\V{s,x,y) - V{s,x,y)\ < C{\x -x\ + \y- y\),
^'' \/se[0,T], x,x€WC, y,y£Wi"'.
Proof. It is clear that for any (s, x, y) G [0, T] x R" x R"*, we have
0 < V{s,x,y) < J{s,x,y;0) < CA + |a:| + \y\).

§3. The value function 61
This proves C.1).
Next, let s £ [0,T] and {x,y),{x,y) G R" x R™ be fixed. Then, for
any Z{-) e Z[s,T], by Ito's formula and Gronwall's inequality, using (HI),
we have
E\Xit;s,x,y,Zi-)) - X{t;s,x,y,Zi-))f
C.3) + E\Y{t;s,x,y,Z{.)) -Y{t;s,x,y,Z{-))f
<C{\x-x\^ + \y-y\^}, te[s,T],
with C > 0 only depending On L and T. Then C.2) follows from A.8),
which implies the (Lipschitz) continuity of V{s,x,y) in {x,y).
We now prove the right-continuity of V{s,x,y) in s £ [O,^]. First of
all, it is clear that for any Z{-) £ Z[0,r], the function
{s,x,y) i-> j{s,x,y;Z\^^,j,^{-))
is continuous. Thus, by the definition of V, it is necessary that V{s,x,y)
is upper semi-continuous. On the other hand, by B.6) and C.2), taking
Z{-) ■= 0, we have
C.4) y(s,a;,y) <y(s,a;,2/)-|-C(s-s)^/^ VO<s<s<r.
Thus, by the upper semi-continuity of V, we must have
limy(?,a;,y) = V{s,x,y),
which gives the right-continuity of V in s £ [0, T). D
Prom B.5) and C.2), we see that under (HI), the function y(a;,y) is
continuous, the assertion that we promised to prove in §1.
Next, we would like to establish another important property for the
value function. To this end, we introduce the following definition.
Definition 3.2. A function 93 : R" —>^ R is said to be semi-concave if there
exists a constant C > 0, such that the function $(a;) = ip{x) — C\x\'^ is
concave On R", i.e.,
C.5) ^{\x -F A - X)x) > X^{x) -F A - A)$E), VA £ [0,1], x,x e R".
A family of functions ips : R" —>^ R is said to be semi-concave uniformly in
e if there exists a constant C > 0, independent of e, such that ipe{x) — C\x\'^
is concave for all e.
We have the following result.
Lemma 3.3. Function ip : R" —^ R is semiconcave if and only if
Xifiix) + A - XMx) - ifiiXx + A - X)x) < CAA --A)la; - x\\
^ ' ' VA£ [0,1], x,xeR".
In the case that tp £ W^^l AR-"), it is semiconcave if and only if
C.7) D^ip{x)<CI, a.e.a;£R",

62 Chapter 3. Method of Optimal Control
where D'^ip is the (generalized) Hessian ofip (i.e., it consists of second order
weak derivatives ofip).
Proof. We have the following identity:
X\x\^ + A - A)|Sp - |Aa; + A - \)x\^ = A(l - A)|a; - x\^,
VAe [0,1], x,xeW.
Thus, we see immediately that C.5) and C.6) are equivalent.
Now, if 93 is C^, then, by the concavity of ip{x) — C\x\^, we know that
C.7) holds. By Taylor expansion, we can prove the converse. For the
general case, we may approximate ip using moUifier. D
It is easy to see from the last conclusion of Lemma 3.3 that if ip has a
bounded Hessian, i.e., ipx is uniformly Lipschitz, then, it is semi-concave.
This observation will be very useful below.
Let us make some further assumptions.
(H3) Functions b, a, h and g are differentiable in {x, y) with the
derivatives being uniformly Lipschitz continuous in {x,y) £ R" x R™, uniformly
in it,z)e[0,T]xWC"'''^.
We easily see that under (H3), the function / defined by A.6) has a
bounded Hessian, and thus it is semi-concave.
Now, we prove the following:
Theorem 3.4. Let (HI) and (H3) hold. Then the value function V{s,x,y)
is semi-concave in {x,y) £ ]R" x JR™ uniformly in s £ [0,r].
Proof. Let s £ [0,r), a;o,a;i £ R" and yo,yi £ R™. Denote
C.8) XX = Xxi + {1- \)xo, yx = Xyi + A - \)yo, A £ [0,1].
Then, for any e > 0, there exists a Zs{-) £ Z[s,T] (which is also depending
on A), such that
C.9) J{s,xx,yx;Z,{-)) <V{s,xx,yx) + e.
We now fix the above Z^i-) and let {Xx{-),Yx{-)) be the solution of B.1)
corresponding to {s,xx,yxi Zs{-)). We denote
^Q1n^ f V\{r) = {Xx{r),Yx{r)), x ^ rn 11 ^ r ti
^•^^ W^^ \ f\^n ^^ ^ ^ A £ 0,1 , r £ [s,r].
{ ^\{r) = Ar7i(r) -I- A - A)r7o(r),
Using Ito's formula and Gronwall's inequality, we have
C.11) E\Xi{t) - Xo{t)f + E\Yi (t) - Yo{t)\' < C{\xi - xo\' + \yi - yo\').

C.12)
§3. The value function 63
Since f{x,y) is semi-concave in {x,y) (by (H3)), we have
XV{s,xi,yi) + A - \)V{s,xo,yo) -V{s,xx,yx) -e
< \J{s,Xi,yi;Zs{-)) + A - X)J{s,xo,yo;Ze{-))
- J{s,xx,yx;Zs{-))
= E[xf{r„{T)) + A - A)/(r,o(r)) - f{rjx{T))}
< CE{xil - A)|r,i (D - r,o(r)p + \UT) - VxiT)]'}.
Let us now estimate the right hand side of C.12). For the first term, we
have
C.13) E\r,i{t)-r,o{t)\^<C{\xi-xo\^ + \yi-yo\^), te[s,T].
To estimate the second term on the right hand side of C.12), let us denote
C.14)
'bx{r)^b{r,rjxir),Z,ir)),
hxir) = h{r,rjx{r),Z,{r)), A £ [0,1], r £ [s,T\.
^(Jx{r) =a{r,r]x{r),Zsir)),
Then, applying Ito's formula, one has (we suppress r in the integrand below)
E\^xit)-vxit)\'
= 2E f (AXi + A - A)Xo - Xx,\bi + A - ANo -bx)dr
C.15) ' ft
+ 2E (AYi + A - A)ro - Yx,Xhi + A - X)ho -hx)dr
J s
+ E I |AGi + A - A)Go - cTApdr, t e [s,T].
J s
Note (we suppress r and Zs{r) from the second line on)
|A6i(r) + (l-ANo(r)-6A(r)|
= |A6(r,i) + (l-AN(r,o)-6(r7A)|
< |A[6(r7i) - 6F)] + A - A)[6(r7o) - 6(Ca)]| + L\^x - Vx\
C.16)
|a / F^(eA + a(l-A)(r7i-r7o))da,(l-A)(r7i-r7o))
I Jo
+ A - A) / F^F - aA(r7i - r]o))da, -X{r]i - %))
Jo
+ L\^x-r]x\
<CX{l-X)\rii-riof+L\^x-Vx\-
We have the similar estimates for the terms involving h and a. Then it

64 Chapter 3. Method of Optimal Control
follows from C.13) and C.15) that
EMt) - Vx{t)\' < CA2A - AJ(|xi - xo\' + \yi - yoP)'
+ C f E\Ur)-Vx{r)?dr, yte[s,T].
J s
By applying Gronwall's inequality, we obtain
C.17) E\^^{t) - r,x{t)\^ < CX\l - X)Wx, - xo\' + \yi - yo\').
Combining C.12), C.13) and C.17), we obtain the semi-concavity of
V{s,x,y) in {x,y), uniformly in s £ [0,r]. D
§3.2. Approximation of the value function
We have seen that due to the noncompactness of the control domain R™^'*,
it is not very easy to determine the value function V through a PDE (the
HJB equation). In this subsection, we introduce some approximations of
the value function, which will help us to determine the value function
(approximately) .
First of all, let W{t) = {Wi{t),W2{t)) be an (n-I-m)-dimensional Brow-
nian motion which is independent of W{t) (embedded into an enlarged
probability space, if necessary) and let {^t}t>o be the filtration generated
by W{t) and W{t), augmented by all the P-nuU sets in T. Define
'Zo[s,T]^Z[s,T],
C.18) I Zo[s,T]={Z : [s,r] X n ^ R™^'* | Z is {j't}t>o-adapted ,
J^E\Z{t)\'dt < oo }.
Next, for any S > 0, we define
( Zs[s,T]={Z e Z[s,T]\\Z{t)\ < ^, 3..e.te[s,T], a.s. },
C.19) { ^ ^ ^ %
[Zs[s,T]^{Z e Zo[s,T]\\Zit)\ < -, a.e.te[s,r], a.s.}.
The following inclusions are obvious.
Zo[s,T] D ZsAs,T] D Zs,[s,T]
C.20) _ n _ n n vS2>Si>o.
Zo[s,T] D ZsAs,T] D Zs,[s,T]
In what follows, for any Z £ Zo[s,r] (resp. Zo[s,r]) and S > 0, we
define the j-truncation of Z as follows:
C.21) Zs{t,w) = {
Z{t,w), ii\Z{t,ij)\<^,
(S\Z(t,u)\' ' " "' r

53. The value function
65
Clearly, Z& £ Z&{s,T\ (resp. Z&\s,T\).
We now consider, for any e > 0, the following regularized state equation
(compare to B.1)):
C.22)
' dX{t) = b{t, X{t),Y{t), Z{t))dt + a{t, X{t),Y{t),Z{t))dW{t)
+ V2edWi{t),
dY{t) = h{t,X{t), Y{t), Z{t))dt + Z{t)dW{t)
+ ^/2edW2{t), te[s,Tl
[X{s) = x, Y{s) = y.
Define the cost functional by J^'^{s,x,y\Z{-)) (resp. J^'^{s,x,y;Z{-)))
which has the same form as B.3) with the control being taken in Z5[s,r]
(resp. Z5[s,r]) and the state satisfying C.22), indicating the dependence
on E > 0 and e > 0. The corresponding optimal control problem is called
Problem @0)°^'^ (resp. Problem {OC)^ ). The corresponding
(approximate) value functions are then defined as, respectively.
C.23)
V^^'{s,x,y)^ irif J^''{s,x,y;Z{-)),
V'''{s,x,y)= mi j''%s,x,y;Z{.)).
z()eZs[s,i\
Due to the inclusions in C.20), we see that for any {s,x,y) £ [0,r] x R" x
R™,
V^''{s, X, y) > V^''{s, X, y) > 0, V<5, e > 0,
C.24) { V^-'%s,x,y) > V^''%s,x,y), ^62 >Si>0, e>0,
V^'''is,x,y) > V^'''{s,x,y), ^5^ > <5i > 0, e > 0.
Also, it is an easy observation that V^'^{s,x^y) = V{s,x,y), y{s,x,y).
Note that for E > 0 and e > 0, the corresponding HJB equation for the
value function V^'^{s,x,y) takes the following form:
C.25)
' y/>^ + eAV^'' + H^{s,x,y,DV^'',D'^V^'') = 0,
{s,x,y) e @,r) xR" xR™;
. V^''{T,x,y) = f{x,y), {x,y) £ R" x R™,
where A is the Laplacian operator in R""*"™, and H^ is defined by the
following:
H\s,x,y,q,Q)= inf I {q,
Ks,x,y,z)
h{s,x,y,z)
)
+
-tr \q ('^^*'^'y'^^\ (^^^'2;,y,2:) \ -11

66 Chapter 3. Method of Optimal Control
for (s,a;,y,g,Q) £ [0,r] x R" x R™ x ]R"+'" x 5'"+'", where 5'"+'" is the
set of all {n + m) x (n + m) symmetric matrices. We observe that for e > 0,
C.25) is a nondegenerate nonlinear parabolic PDE; and for e = 0, however,
C.25) is a degenerate nonlinear parabolic PDE. The following notion will
be necessary for us to proceed further.
Definition 3.5. A continuous function v : [0, T] x ]R" x R™ ^ R is called
a viscosity subsolution (resp. viscosity supersolution) of C.25), if
f3 26) ^iT,x,y)<fix,y), V(a;,2/) e R" x R™,
(resp. viT,x,y) > f{x,y), ^{x,y) £ R" x R™),
and for any smooth function ip{s,x,y) whenever the map v — ip attains a
local maximum (resp. minimum) at {s,x,y) £ [0,r) x R" x R™, it holds:
iPs{s,x,y)+eAip{s,x,y)
+ H\s,x,y,Dipis,x,y),D'^ipis,x,y))>0 (resp. < 0).
If V is both viscosity subsolution and viscosity supersolution of C.25), we
call it a viscosity solution of C.25).
We note that in the above definition, v being continuous is enough.
Thus, by this, we can talk about a solution of differential equations without
its differentiability. Furthermore, such a notion admits the uniqueness.
The following proposition collects some basic properties of the
approximate value functions.
Proposition 3.6. Let (HI) hold. Then
(i) V^'^{s,x,y) and V^'^{s,x,y) are continuous in {x,y) £ R" x R™,
uniformly in s £ [0, T] and S,e> 0; For fixed S>0 ande>0, V^'^{s, x, y)
and V^'^{s,x,y) are continuous in {s,x,y) £ [0,T] x R" x R™.
(ii) For S > 0 and e >0, V^'^{s, x, y) is the unique viscosity solution of
C.25), and for S,e > 0, V^'^{s,x,y) is the unique strong solution of C.25).
(iii) For S > 0 and e > 0, V^'^{s,x,y) is a viscosity super solution of
C.25), V^'°{s,x,y) is the unique viscosity solution of C.25) (with e = 0).
The proof of (i) is similar to that of Proposition 3.1 and the proof of
(ii) and (iii) are by now standard, which we omit here for simplicity of
presentation (see Yong-Zhou [1] and Fleming-Soner [1], for details).
The following result gives the continuous dependence of the
approximate value functions on the parameters S and e.
Theorem 3.7. Let (HI) hold. Then, for any s £ [0,r], there exists a
continuous function rjs : [0,oo) x [0,oo) -^ [0, oo), with r]s{0,r) = 0 for all
r > 0, such that
\V''^s,x,y) - V''Hs,x,y)\ < rjUS -S\ + \e- e\, \x\ + \y\),
C.28) \V'''{s,x,y) - V''Hs,x,y)\ < rj{\6 - S\ + \e- i\, \x\ + \y\),
y{s,x,y)e [0,r]xR" xR™, S,S,e,e e [0,1].

53. The value function
67
Proof. Fix {s,x,y) £ [O.T] x R" x R™, 5,6,e,e > 0, and Z e Z[s,T].
Let Zs (resp. Z^) be the 1/S- (resp. l/<5-) truncation of Z; and (X, F) (resp.
{X,Y)) the solution of C.22) corresponding to {e,Zs) (resp. (e, Z^)). By
Ito's formula and Gronwall's inequality,
E{\X{T) - X(T)|2 + |r(r) - Y{T)\^}
<C{E j '\Zs{t)~Z^{t)\''dt + \yfe-Vi\^],
where C > 0 depends only on L and T. Thus, we obtain
<C\y/e-y/i\, V(s,a;,y), E,e,e>0.
C.29)
C.30)
Combining with Proposition 3.6, we see that V^'^{s,x,y) is continuous in
{e,x,y) e [0,oo) x R" x R™ uniformly in E > 0 and s £ [0,r].
Next, for fixed {s,x,y) £ [0,r] x R" x R™, e > 0, and ^ > E > 0, by
C.24), we have
C.31)
0<V'''{s,x,y)~V^^'{s,x,y).
On the other hand, for any 6 > 0, and eo > 0; we can choose Z^° e Zs[s,T]
so that
C.32) V^''{s,x,y)+eo > J^''is,x,y;Z''').
Let Zi° be the t-truncation of Z^°, and denote the corresponding solution
0 0 ^^
of C.22) with Z^° (resp. Zf") by iX'°,Y^°) (resp. {X^°,Y^°)). Setting
(X,r) = {X'°,Y'°), {X,Y) = (X^",?^"), e = e,Zs = Z'\ and Z-^ ■= Zf"
in C.29), we obtain
C.33)
£;{|x^''(r) - x^''(r)|2 -^ |r«(r) - r^''(r)|2}
<C£;
/
lz^°@-^f (Opd^-
We consider the following two cases:
Case 1. 5>0. In this case, note that \Z^°it) - Zl°{t)\ < \1/S - 1/S\,
a.e.t e [s,T], a.s. By A.8) and (HI), one easily checks that
j'>%s,x,y-Z''')>j''^s,x,y;Zl'^)-C
>V^''{s,x,y)-C
C.34)
1 _ 1
1 _ 1
s s
Combining C.31), C.32) and C.34), we obtain (note eo > 0 is arbitrary)
C.35)
0 <V^''{s,x,y) -V^''{s,x,y)\ <C
Vis,x,y), S,S>0, e>0,
1 1

68 Chapter 3. Method of Optimal Control
where C is again an absolute constant.
Case 2. E = 0. Now let ^ > 0 be small enough so that the right side of
C.33) is no greater than e^. Then, similar to C.36), we have
C.36) J0'^(s,a:,2/;Z^°) > v'^''{s,x,y) - e^.
Combing C.31), C.32) and C.36), one has 0 <V^^^{s,x,y)-V^^^{s,x,y) <
2eo, which shows that
C.37) V'''{s,x,y)lV''''{s,x,y), HO.
Since V'^'^{s,x,y) is continuous in {e,x,y) (see C.30) and Proposition 3.6-
(i)), by Dini's theorem, we obtain that the convergence in C.37) is uniform
in (e, X, y) on compact sets. Thus, for some continuous function rjs : [0, oo) x
[0, oo) -^ [0, oo) with r7s@,r) = 0 for all r > 0, one has
^3 3g) 0 < V'^'{s,x,y) - y°'^(s,a;,2/) < r^^il \x\ + \y\),
V(s,a;,y), ee[0,l], S>0.
Combining C.30), C.35) and C.38), we have that V^'^{s, x, y) is continuous
in {5,e,x,y) £ [0,oo) x [0,oo) x R" x R™. The proof for V^'^ is exactly
the same. D
Corollary 3.8. Let (El) hold. Then
C.39) V^'°is,x,y) = V^'°{s,x,y), y{s,x,y) £ [0,r] x R" x R™, S> 0.
Proof. If E > 0, then both V^'^ and V^'^ are the viscosity solutions of
the HJB equation C.25). Thus, C.39) follows from the uniqueness. By the
continuity of V^'° and V^'° in E > 0, we obtain C.39) for S = 0. D
Corollary 3.9. Let V{0,x,y) = 0. Then, for any e > 0, there exist 5,e > 0
and Z^'^i-) e Zs[0,T] satisfying
C.40) J^''{0,x,y;Z^''{-))<i,
such that, if{X^'^{-),Y^'^{.)) is the solution of B.1) with Z{-) = Z^''{-),
then the triplet {X^'^,Y^'^,Z^'^) is an approximate solution of A.1) with
accuracy
Proof. Let V{0,x,y) = 0. Since V = y°'°, by Theorem 3.7, there
exist 6,e > 0, such that V^'^{0,x,y) < i. Now by C.9) we can find a
Z^'^ e Zs[0, T] such that C.40) is satisfied. Let (X''•^ Y^'^) be the solutions
of B.1) with s = 0, and Z = Z^'^. Then we have (see A.21))
E\Y^'%T)-g{X^''iT))\
< 3EfiX^''{T),Y^''iT)) + ^ZEf{X^^^{T),Y^'^{T))
= 3J^'^@,X,y; Z'''{-)) + ^3J^'^{0,x,y; ZS<^{.))
< 3e + VM.

§4. A Class of Approximately Solvable FBSDEs 69
This proves our assertion. D
To conclude this section, we present the following result.
Proposition 3.10. Let (HI) and (H3) hold. Then V^'^{s,x,y) is semi-
concave uniformly in s £ [0,T], S £ @,1] and e £ [0,1]. In particular, there
exists a constant C > 0, such that
C.41) AyV^^'{s,x,y)<C, y{s,x,y) e[0,T] xK" xWT, S,e e @,1],
where Ay = j:^^,dl.
Proof. The proof of the first claim is similar to that of Theorem 3.4.
To show the second one, we need only to note that by C.7),
A,V-=„{(° »)d'V':[1 ;)}<C.
which gives C.41). D
§4. A Class of Approximately Solvable FBSDEs
We have seen from Proposition 1.4 that if A.13) holds, then @.1) is
approximate solvable. Further, from §3 we see that if {x,y) £ R" x R™ is
such that V{0,x,y) = 0, then one can actually construct a sequence of
approximate solutions to @.1). Finally, A.13) is also an important step of
solving @.1) (see Proposition 1.1). In this section, we look for conditions
under which A.13) holds. Moreover, we would like to construct the nodal
set Af{V) for some special and interesting cases.
In what follows, we restrict ourselves to the following FBSDE:
D.1)
' dXit) = bit,Xit),Y{t))dt + a{t,X{t),Yit))dW{t),
dY{t) = h{t,X{t),Y{t))dt + Z{t)dW{t), te [0,T],
^X{0)=x, Y{T)=g{X{T)).
The difference between @.1) and D.1) is that in D.1), the functions b, a
and h are all independent of Z. To study the set Af{V), we introduce the
nodal set of value function V{s,x,y):
D.2) Af{V) = {{s,X,y) e [0, T] x R" x R™ | V(s,a;,y) = 0 }.
Clearly,
D.3) {0} X M{V) = M{V) fl [{0} X R" X R™].
We will study Miy) below, which will automatically give the information
on M(y) that we are looking for.
Let us now first make an observation. Suppose there exists a function
e : [0,r] X R" ^- R™, such that
D.4) V{s, X,e{s,x)) = 0, V(s,x) e [0, T] X R",

70 Chapter 3. Method of Optimal Control
then it holds
D.5) {{s,x,9{s,x)) I {s,x) e[0,T]xB"}CM{V).
In particular,
D.6) {x,e{0,x)) eAf{V), Va;e]R".
This gives the nonemptiness of the nodal set M{V). Thus, finding some
way of determining 6{s,x) is very useful. Now, let us assume that both V
and 6 are smooth and we find an equation that is satisfied by 6 (so that
D.4) holds). To this end, we define
D.7) w{s,x) = V{s,X, e{s,x)), V(s, x) e [0, T] x R".
Differentiating the above, we obtain
(w, = v, + {Vy,e,),
WxiXj =VxiXj + {Vxiy.Oxj ) + {yyXj,Oxi )
+ {Vy,ex,xj) + {Vyyex^,ex,), i<i,j<n.
Clearly,
D.9) tr [aa'^Wxx] = tr {aa'^ [Vxx + V^xy^x + ^l^yy^x + (Vy, 6*^^ ) ] },
where we note that Yxy is an (n x m) matrix and dx is (m x n) matrix.
Then it follows from B.12) that (recall D.1) for the form of functions 6, a
and K)
0 = y, + itr[G<7^K,] + F,K ) + (/i, Vj,)
D.8)
2
1
2 zeR^x"
+ - inf tr ^Jycz'^ + Yxyzu'^ + yyyzz'^\
D.10)
= «;,-( Vy, ^, ) +-tr \uu'^(Wxx - l^xy^x - elVyyOx)]
+ {h,Wx-6lVy) + {h,Vy)-\{traa'^exx,Vy)
+ - inf tr [V^yGz' + Vxyzu' + VyyZz' ]
= {ws + xtr [aa'^Wxx] + {b,Wx)}
-{Vy,e, + itr [aa^Oxx] + Oxb~h)}
+ - inf tr [2{z - exa)G'^Vxy + {zz'^ - ex(Ta^e'^)Vyy].
2 zgR'"X<'
Thus, if we suppose ^ to be a solution of the following system:
{es + ^tT[aa'^exx] + Oxb-h = 0, (s,a;) £ [0,r) x R",

§4. A Class of Approximately Solvable FBSDEs 71
Then we have
, . \ Ws + -tr [GG^W;xx] + {b,Wa:)>0,
Hence, by maximum principle, we obtain
D.13) 0 > w{s, x) = V{s, X, 6{s, x)) > 0, V(s, x) e [0, T] x R".
This gives D.4). The above gives a proof of the following proposition.
Proposition 4.1. Suppose the value function V is smooth and 9 is a
classical solution of D.11). Then D.4) holds.
We know that V{s,x,y) is not necessarily smooth. Also since aa^
could be degenerate, D.11) might have no classical solutions. Thus, the
assumptions of Proposition 4.1 are rather restrictive. The goal of the rest
of the section is to prove a result similar to the above without assuming
the smoothness of V and the nondegeneracy of aa^. To this end, we need
the following assumption.
(H4) Function g{x) is bounded in C^+"(]R") for some a £ @,1) and
there exists a constant L > 0, such that
D.14) |6(s,a;,0)| + |G(s,a;,0)| + |/i(s,a;,0)| <L, V{s,x) e [0,T] xWl".
Our main result of this section is the following.
Theorem 4.3. Let (H1)-(H3) hold. Then, for any x £ R", A.13) holds,
and thus, D.1) is approximately solvable.
To prove this theorem we need some lemmas.
Lemma 4.4. Let (H1)-(H3) hold. Then, for any e > 0, there exists a
unique classical solution 9^ : [0,T] xWl" ^ R™ of the following (nondegen-
erate) parabolic system:
D 15) I ^' ^ ^^^' ^ t' W^'^^L] +Olb-h = 0, (s, x) e [0, T) X R",
with 9^, 91. and 9%.^. all being bounded (with the bounds depending on
e > 0, in general). Moreover, there exists a constant C > 0, independent
of e £ @,1], such that
D.16) \9'{s,x)\<C, V(s,a;) e [0,r]xR", ee @,1].

72 Chapter 3. Method of Optimal Control
Proof. We note that under (H1)-(H3), following hold:
D.17)
\ 0 < {(J(j'^){s,x,y) <C{1 + \y\^)I,
\{a^,a'^){s,x,y)\ + \{ay,a'^){s,x,y)\ < C{1 + \y\),
l<i<n, l<A;<m,
\h{s,x,y)\<L{l + \y\),
I -{h{s,x,y),y)<L{l + \y\'').
Thus, by Ladyzenskaja, et al [1], we know that for any e > 0, there exists
a unique classical solution 6^ to D.15) with 6^, 6^. and 6^.^. all being
bounded (with the bounds depending on e > 0). Next, we prove D.16). To
this end, we fix an e e @,1] and denote
D.18)
Set
D.19)
AeW = eA'w + -ti[aa'^{s,X,6^{s,x))'Wxx] + {b{^,x,6^{s,x)),Wx )
n n
1 1 _ ,
wis,x)^^\9%s,x)\' = ^Yl^^'''is,xr.
i=l
Then it holds that (note D.17))
m m
Ws =J2^''''^V'' = Yl^''''[-Aee'''' +h''{s,X,9')]
k=l fc=l
m n n
k=\ i,j=i 2=1
m n -
k=l i,j=l
m n ^ m
k=l 2=1
> -AeW - 2Lw - L.
k=l
Thus, w; is a bounded (with the bound depending on e > 0) solution of the
following:
D.20)
Ws+AeW + 2Lw>-L, {s,x) e[0,T)xB",
By Lemma 4.5 below, we obtain
D.21) w{s,x)<C, y{s,x)e[0,T]xWC,

§4. A Class of Approximately Solvable FBSDEs 73
with the constant only depending on L and ||5||oo (and independent of
e > 0). Since w is nonnegative by definition (see D.19)), D.16) follows.
n
In the above, we have used the following lemma. In what follows, this
lemma will be used again.
Lemma 4.5. Let As be given by D.18) and w be a, bounded solution of
the following:
( Ws + AsW + Xow>-ho, {s,x) e[0,T) xWC,
D.22) i
for some constants ho,go > 0 and Aq G H, with the bound of w might
depend on e > 0, in general. Then, for any A > Ao V 0,
D.23) w{s,x)<e^^\goW^^^], V(s,a;) £ [0,r] x R".
L A — Aq -I
Proof. Fix any A > Aq V 0. For any /3 > 0, we define
D.24) ^s,x)=e^'w{s,x)-l3\x\\ V(s,a;) e [0,T] x R".
Since w{s,x) is bounded, we see that
D.25) lim ^{s,x) = -oo.
|x|—^oo
Thus, there exists a point (s,ac) £ [0,r] x R" (depending on P > 0), such
that
D.26) ^s,x) < ^{s,x), y{s,x) e [0,T] x R".
In particular,
D.27) e^'w{s,x) - /3|S|2 = ^{s,x) > $(r,0) = e^'^w{T,0),
which yields
D.28) p\xf < e^'^wis,x) - e^'^w{T,0) < Cs-
We have two cases. First, if there exists a sequence P\-0, such that l = T,
then, for any {s,x) £ [0,r] x R", we have
w{s,x) <e-^'[p\x\'^ +^{T,x)]
D.29) <e-^^[p\x\'+e^^go~P\x\^]
< /3|a;p + e^^5o ^ e^^5o, as P ^ 0.
We now assume that for any P > 0,1 <T. In this case, we have
0>{^s + As^){s,x)
D.30) = Xe'-'^w + e'-^lws + AsW] - pAsHx^^^^
> (A - Ao)e^^«; - e^% - pAs{\x\')l^-.

74 Chapter 3. Method of Optimal Control
Note that (see D.28))
-4-(l^l')L=x = 2ne+\a{-s,x,9%s,x))\^ + 2{b{s,x,9'{s,x)),x)
< 2ne + Ce+ Ce\x\ < C^ + CeP~'^^^.
Hence, for any {s,x) e [0,T] x ]R", we have
e^'w{s, x) - P\x\^ = $(s, x) < $(s, x) = e^'w{s, x) - P\x\^
e^Th XT
A — Ao A — Ao
Sending /3 —>^ 0, we obtain
pATt
D.31) wis,x) < ~, V(s,x) e [0, T] X R".
A — Ao
Combining D.29) and D.31), one obtains D.23). D
Proof of Theorem 4.3. We define (note C.24))
w^''{s,x)^v^^'{s,x,e'{s,x)) >o, y{s,x) e [o,t] xR".
Then we obtain (using C.25), C.29) and D.15))
0 = y/'^ + eAV''^ + itr [<7<7^y//] + (b, Y^' ) + {h, V^'' )
= [wi'^ + eAw'>' + ^tr [<7<7^«;f>|] + F, «;f'^) } + sAyV''^
D.32) ^ 2
- (y/'Sei + eAr + -tr [aa^eU + O^b - h)
+ \ ^mi^^tr [2{z - eioWv!^^ + {zz^ - ^^<7<7^(^^)^)y;/]
< {«;f-^ + eAw'^' + itr [<7<7^«;f'|] + F, «;f>^ ) } + eC.
The above is true for all e,5 > 0 such that \9l{s,x)a{s,x,9^{s,x))\ < |,
which is always possible for any fixed e, and S > 0 sufficiently small. Then
we obtain
J wl^' + AeW^'' > -eC, V(s, x) e [0, T] x R",
On the other hand, by (HI) and (H3), we see that corresponding to the
control Zs{-) = 0 £ Zs[s,T], we have (by Gronwall's inequality) \Y{T)\ <

§5. Construction of Approximate Adapted Solutions 75
C(l + \y\), almost surely. Thus, by the boundedness of g, we obtain (using
Lemma 4.5)
0 < w^''{s,x) = V^-'{s,x,e'{s,x))
< J^''{s,x,e'{s,x);0) < CA + \e'{s,x)\) < C.
Next, by Lemma 4.5 (with Aq = go = 0, A = 1 and ho = eC), we must have
w^'^{s,x) < eCe^, V(s,a;) e [0,r] x R". Thus, we obtain the following
conclusion: There exists a constant Cq > 0, such that for any e > 0, one
can find a. S = S{e) with the property that
D.33) 0 < V^''{s,x,e'{s,x)) < eCo, VE < S{e).
Then, by C.28), C.39) (with 6 = 0) and D.33), we obtain
0<V{0,x,e'{0,x)) < \V°'°{0,x,e'{0,x)) -V^^'{0,x,e'{0,x))\+ eCo
<r]oie + S,\x\ + \e'{0,x)\)+eCo.
Now, we let E —>^ 0 and then e —>^ 0 to get the right hand side of the above
going to 0. This can be achieved due to D.16). Finally, since 6^{s,x)
is bounded, we can find a convergent subsequence. Thus, we obtain that
V{0,x,y) = 0, for some y e BT. This implies A.13). D
§5. Construction of Approximate Adapted Solutions
We have already noted that in order that the method of optimal control
works completely, one has to actually find the optimal control of the
Problem (OC), with the initial state satisfying the constraint A.13). But on the
other hand, due to the non-compactness of the control set (i.e., there is no
a priori bound for the process Z), the existence of the optimal control itself
is a rather complicated issue. The conceivable routes are either to solve the
problem by considering relaxed control, or to figure out an a priori compact
set in which the process Z lives (it turns out that such a compact set can
be found theoretically in some cases, as we will see in the next chapter).
However, compared to the other methods that will be developed in the
following chapters, the main advantage of the method of optimal control lies
in that it provides a tractable way to construct the approximate solution
for fairly large class of the FBSDEs, which we will focus on in this section.
To begin with, let us point out that in Corollary 3.9 we had a scheme
of constructing the approximate solution, provided that one is able to start
from the right initial position {x,y) £ AfiV) (or equivalently, V{0,x,y) =
0). The draw back of that scheme is that one usually do not have a way
to access the value function V directly, again due to the possible
degeneracy of the forward diffusion coefficient a and the non-compactness of the
admissible control set Z[0,r]. The scheme of the special case in §4 is also
restrictive, because it involves some other subtleties such as, among others,
the estimate D.16).
To overcome these difficulties, we will first try to start from some initial
state that is "close" to the nodal set M{V) in a certain sense. Note that

76 Chapter 3. Method of Optimal Control
the unique strong solution to the HJB equation C.25), V^'^, is the value
function of a regularized control problem with the state equation C.22),
which is non-degenerate and with compact control set, thus many standard
methods can be applied to study its analytical and numerical properties, on
which our scheme will rely. For notational convenience, in this section we
assume that all the processes involved are one dimensional (i.e., n = m =
d —V). However, one should be able to extend the scheme to general higher
dimensional cases without substantial difficulties. Furthermore, throughout
this section we assume that
(H4) g £ C^; and there exists a constant L > 0, such that for all
it,x,y,z)e[0,T]x]Ee,
\ \b{t,x,y,z)\ + \a{t,x,y,z)\ + \h{t,x,y,z)\ < L(l + |a;|);
^ ■ ' \\g'ix)\ + \g"ix)\<L.
We first give a lemma that will be useful in our discussion.
Lemma 5.1. Let (HI) and (H4) hold. Then there exists a constant C > 0,
depending only on L and T, such that for all 5,e > 0, and {s,x,y) £
[0,r] X B?, it holds that
E.2) V''^s,x,y)>fix,y)-C{l + \x\^),
where f{x,y) is defined by A-6).
Proof. First, it is not hard to check that the function / is twice
continuously differentiable, such that for all {x, y) e ]R^ the following hold:
E.3) <^
(\fA^,y)\<\9'{x)\, \fyix,y)\<l,
s u^.. (g(^)-j/)g"(^) , 9'{xf
Jxx\X,y) — _/_x\911/9 '^
[l + (l/-5(a;)J]l/2 [1 + (y - g)J]3/2 '
fyy{x,y) = ; , \,„,3 > 0, fxyix,y) = -g'ix)fyy{x,y).
[I + (y - gix))^]2
Now for any S,e>0, (s, x,y) e [0, T] x R^ and Z e Zs[s, T], let (X, Y) be
the corresponding solution to the controlled system C.22). Applying Ito's
formula we have
?''{s,x,y;Z)=Ef{X{T),Y{T))
E.4) fT
= f{x,y)+E / Uit,Xit),Yit),Zit))dt,
J s

§5. Construction of Approximate Adapted Solutions 77
where, denoting {fx - fx{x,y), fy = }y{x,y), and so on),
U{t,x,y,z) = fxb{t,x,y,z) + fyh{t,x,y,z)
+ o [/xxcr^(^, X, y, z) + 2fxy(j{t, x, y, z)z + fyyz'^]
f
Jx
tyy-
>-C{l + \x\%
E.5)
1 f^
> fxb{t,x,y,z) + fyh{t,x,y,z) + -Ifxx - ~-\(T^{t,x,y,z)
^ '- Jyy -■
where C > 0 depends only on the constant L in (H4), thanks to the
estimates in E.3). Note that (H4) also implies, by a standard arguments using
Gronwall's inequality, that E\X{t)\'^ < C(l + \x\'^), Vt e [0,T], uniformly
in Z{-) e Zs[s,T], S>0. Thus we derive from E.4) and E.5) that
V^''{s,x,y)= inf ?''{s,x,y;Z)
ZeZs[s,T]
f{x,y)+ inf E f \i{t,X{t),Y{t),Z{t))dt
ZeZs[s,T]
>f{x,y)-C{l + \x\^),
proving the lemma. D
Next, for any a; £ R and r > 0, we define
Qx{r) ={yen: f{x,y) <r + C{l + \x\^)},
where C > 0 is the constant in E.2). Since lim|y|_>oo f{x,y) — +oo, Qx(f)
is a compact set for any a; e R and r > 0. Moreover, Lemma 5.1 shows
that, for all <5, e > 0, one has
E.6) {l/elR:y*'^@,a:,y)<r}CQ,(r).
Prom now on we set r = 1. Recall that by Proposition 3.6 and Theorem
3.7, for any p > 0, and fixed a; £ R, we can first choose E, e > 0 depending
only on x and Qx(l), so that
E.7) 0<y*'^@,a;,2/)<y@,a;,y) + p, for all y £ Q^(l).
Now suppose that the PBSDE A.1) is approximately solvable, we have
from Proposition 1.4 that infygRy@,a;,2/) = 0 (note that (H4) implies
(H2)). By E.6), we have
0= inf y@,a;,y) = min y@,a;,y).
yeR y€Q»(i)
Thus, by E.7), we conclude the following
Lemma 5.2. Assume (HI) and (H4), and assume that the FBSDE @.1)
is approximately solvable. Then for any p > 0, there exist S,e > 0 and
depending only on p, x and Qx{l), such that
0<miV^''{0,x,y)= min V^''{0,x,y) < p.
yeE. y€Q»(i)

78 Chapter 3. Method of Optimal Control
n
Our scheme of finding the approximate adapted solution of @.1)
starting from X{0) — X can now be described as follows: for any integer k, we
want to find {y^*)} C Qx(l) and {Z(*)} C Z[Q,T] such that
E.8) Ef{X^''\T),Y(''\T))<^,
k
here and below C^ > 0 will denote generic constant depending only on L,
T and x. To be more precise, we propose the following steps for each fixed
k.
Step 1. Choose Q < 5 < \ and 0 < e < 5^, such that
miV^''{<d,x,y)= min V^^'{Q,x,y) <\.
yeE. yeQ^(\) k
Step 2. For the given 5 and e, choose y(*) e Qx(l) such that
y^'^@,a;,2/W)< min V'''{0,x,y) + \.
yeQ.(i) k
Step 3. For the given 6, e, and y(^\ find Z(*) e Zs[0,T], such that
J@,a;,2/('=);Z('=)) = Ef {X^''\T),Y(''\T)) < y*'^@,a;,i/(*)) + %,
k
where (X^*', r^*') is the solution to B.1) with r^*' @) = y^*' and Z = Z(*);
and Cx is a constant depending only on L, T and a;.
It is obvious that a combination of the above three steps will serve our
purpose E.8). We would like to remark here that in the whole procedure we
do not use the exact knowledge about the nodal set Af{V), nor do we have
to solve any degenerate parabolic PDEs, which are the two most formidable
parts in this problem. Now that the Step 1 is a consequence of Lemma 5.2
and Step 2 is a standard (nonlinear) minimizing problem, we only briefly
discuss Step 3. Note that V^'^ is the value function of a regularized control
problem, by standard methods of constructing e-optimal strategies using
information of value functions (e.g., Krylov [1, Ch.5]), we can find a Markov
type control Z(*)(t) = a^''\t,X^''\t),Y^''\t)), where a^*) is some smooth
function satisfying sup^j.^ \a^''\t,x,y)\ < | and (X^*',F^*') is the
corresponding solution of D.8) with Y'(*'@) = y^''\ so that
E.9) P'^@,a;,2/W;ZW)<y*'^@,a;,2/W) + i.
k
The last technical point is that E.9) is only true if we use the state
equation C.22), which is different from B.1), the original control problem that
leads to the approximate solution that we need. However, if we denote
(X(*),r(*)) to be the solutions to B.1) with r(*)@) = y^*' and the
feedback control Z(*)(t) = a(*)(X(*)(t),r(*'(t)), then a simple calculation

§5. Construction of Approximate Adapted Solutions 79
shows that
0 < J@, X, 2/(*'; Z(*)) = EfiX^''^ (T), r^*) (T))
E.10) <£;/(x('=)(r),rW(r)) + c„x/2i
thanks to E.9), where Ca is some constant depending only on L, T and
the Lipschitz constant of a^*'. But on the other hand, in light of Lemma
5.1 of Krylov [1], the Lipschitz constant of a^*' can be shown to depend
only on the bounds of the coefficients of the system B.1) (i.e., 6, /i, a, and
g{z) = z) and their derivatives. Therefore using assumptions (HI) and
(H4), and noting that sup^ |Z(*'(t)| < sup|a(*'| < i, we see that, for fixed
E, Ca is no more than C(l + |a;| +1/5) where C is some constant depending
only on L. Consequently, note the requirement we posed on e and 5 in Step
1, we have
E.11) Ca\/2i < C(l + |a;| + \)^/W < 2y/2C{l + \x\M < ^r^,
where C^ = C(l + \x\J^/2 + 1. Finally, we note that the process Z(*)(-)
obtain above is {J^t}t>o-adapted and hence it is in 2'5[0,r] (instead of
Z5[0,r]). This, together with E.10)-E.11), fulfills Step 3.

Chapter 4
Four Step Scheme
In this chapter, we introduce a direct method for solving FBSDEs. Since
this method contains four major steps, it has been called the Four Step
Scheme.
§1. A Heuristic Derivation of Four Step Scheme
Let us consider the following FBSDE:
{dX{t) = b{t,X{t),Y{t),Z{t))dt + a{t,X{t),Y{t),Z{t))dW{t),
dY{t) = hit, Xit),Yit),Zit))dt + Z{t)dWit),
X{0) = x, Y{T)=g{X{T)).
We assume throughout this section that the functions b, a, h and g are
deterministic. As we have seen in the previous chapter that for any given
X e R", the solvability of A.1) is essentially equivalent to the following:
V{0,x,e{0,x)) =0,
where 9{s, x) is the "solution" of some parabolic system and V{s, x, y) is the
value function of the optimal control problem associated with the FBSDE
A.1). Assuming the Markov property (since coefficients are
deterministic!) we suspect that V{t,X{t),e{t,X{t))) = 0, and Y{t) = 9{t,X{t))
should hold for all t. In other words, we see a strong indication that there
might some special relations among the components of an adapted solution
{X, Y, Z), which we now explore.
Suppose that {X, Y, Z) is an adapted solution to A.1). We assume that
that Y and X are related by
A.2) Y{t)=eit,Xit)), \/te[0,T], a.s.P,
where 6 is some function to be determined. Let us assume that 6 £
Ci>2([0,r] X R"). Then by Ito's formula, we have for 1 < A; < m:
dY''it) = d9''it,X{t))
= {e!^it,xit)) + {e',it,xit)),bit,xit),eit,xit)),zit)))
A.3) 1 .
+ -tr [9',,it,X{t))iaa^)it,X(t),^(t,X(t)), Z(t))] ]dt
+ {e',{t, x{t)), a{t, x{t),e{t,x{t)), z{t))dw{t)).
Comparing A.3) and A.1), we see that if 6 is the right choice, it should be

§1. A heuristic darivation of four step scheme 81
that, for A; = 1, • • • ,m,
^h''it,xit),eit,xit))
= e^{t,x{t)) + {e',{t,x{t))Mt,x{t),e{t,x{t)),z{t)))
A.4)
+ itr [e',,it,X{t))iaa'^){t,X{t),e{t,X{t)),Z{t))];
\e{T,X{T))=g{X{T)),
and
A.5) e,{t,x{t))a{t,x{t),e{t,x{t)),z{t)) = z{t).
The above heuristic arguments suggest the following Four Step Scheme for
solving the FBSDE A.1).
The Four Step Scheme:
Step 1. Find a function z{t,x,y,p) that satisfies the following:
V(t,a;,2/,p) e [0,r] X R" X R™ X R^
m . . lomxn
Step 2. Using the function z obtained in above to solve the following
parabolic system for 6{t,x):
' ^t + ^tr [9',^{aa^){t,x,e,z{t,x,e,9,))]
A.7) i +{bit,x,e,zit,x,e,e^)),e^j-h''{t,x,e,z{t,x,e,e^)) = o,
{t,x) e[0,T)xB", l<k<m,
J{T,x) = g{x), xeK".
Step 3. Using 6 and z obtained in Steps 1-2 to solve the following
forward SDE:
fl8) idX{t) = b{t,X{t))dt + a{t,X{t))dW{t), te[0,T],
\x{0)=x,
where
A 9) I bit,x) = b{t,x,e{t,x),z{t,x,e{t,x),ea:{t,x))),
\a{t,x) = a{t,x,9{t,x),z{t,x,6{t,x),ea:{t,x))).
Step 4- Set
A.10) p(')=«(;.^('».
\z{f) = z{t,x(f),e{t,x(t)),e.{t,x{tj)).
If the above scheme is realizable, {X, Y, Z) would give an adapted solution
of A.1). As a matter of fact, we have the following result.

82 Chapter 4. Four Step Scheme
Theorem 1.1. Let A-6) admit a unique solution z{t,x,y,p) which is
uniformly Lipschitz continuous in {x, y,p) with z{t, 0,0,0) being bounded. Let
A-7) admit a classical solution 6{t,x) with bounded 6^ and O^x- Let
functions b and a be uniformly Lipschitz continuous in {x,y,z) with b{t, 0,0,0)
and a{t, 0,0,0) being bounded. Then the process {X{•),¥{■), Z{-))
determined by A.8)-A.10) is an adapted solution to A.1). Moreover, ifh is also
uniformly Lipschitz continuous in {x,y,z), a is bounded, and there exists
a constant C £ @,1), such that
\[(j{s,x,y,z) -(j{s,x,y,z)fe^{s,x)\ < P\z - z\,
y{s,x,y) e [0,r]x]R"x]R'", z,IeK"''"^,
then the adapted solution is unique, which is determined by A.8)-A.10).
Proof. Under our conditions both b{t,x) and a{t,x) (see A.9)) are
uniformly Lipschitz continuous in x. Thus, for any x £ R", A.8) has a
unique strong solution. Then, by defining Y{t) and Z{t) via A.10) and
applying Ito's formula, we can easily check that A.1) is satisfied. Hence,
{X, Y, Z) is a solution of A.1).
It remains to show the uniqueness. We claim that any adapted solution
{X, Y, Z) of A.1) must be of the form we constructed using the Four Step
Scheme. To show this, let {X, Y, Z) be any solution of A.1). We define
A.12) Y{t) = e{t,x{t)), z{t) = z{t,x{t),e{t,x{t)),e,{t,x{t))).
By our assumption, A.6) admits a unique solution. Thus, A.12) implies
A.13) Z{t) = ex{t,X{t))u{t,X{t),Y{t),Z{t)), a.s.te[0,T].
Now, applying Ito's formula to 6{t,X{t)), noting A.7) and A.10), we have
the following (for notational simplicity, we suppress t in X{t), etc.):
dy''{t) = de''{t,x{t))
= [et{t,X) + {6l{t,X),b{t,X,Y,Z))+^^tr[6>^,,{t,X){aa'^){t,X,Y,Z)^dt
+ {el{t,X),a{t,X,Y,Z)dW{t))
= [{6l{t,X),b{t,X,Y,Z)-b{t,X,Y,Z))
+ itr [e'^,,{t,X){{aa'^){t,X,Y,Z) - {aa'^){t,X,Y,Z)]\
+ h''{t,X,Y,Z)^dt+ {el{t,X),a{t,X,Y,Z)dW{t)).

§1. A heuristic derivation of four step scheme 83
Then, it follows from A.1) and A.13) that
E\Yit)-Y{t)\'^ = -E [ y]{2(f*-r*)-
Jt k=i ^
[{e',{s,X),b{s,X,Y,Z)-b{s,X,Y,Z))
A-14) +^tT{e',,{s,X)[{aa^){s,X,Y,Z)-iaa^){s,X,Y,Z)]}
+ h''{s,X,Y,Z)-h''{s,X,Y,Z)^
{a{s,X,Y,Z) - a{s,X,Y,Z)}'^e^{s,X) + Z- z\ jds.
+
Since, by A.11), the boundedness of 9x, and the uniform Lipschitz
continuity of G, we have
{a{s,X,Y,Z)-a{s,X,Y,Z)fe^^{s,X) + Z-Z
>\Z - Z\' - \a{s,X,Y,Z) - a{s,X,Y,ZMf{s,X)\'
>{1 - p)\Z - Z\''- C\Y - Y\\
here and in the sequel C > 0 is again a generic constant which may vary
from line to line. Thus A.14) leads to that
E\Y{t) - Y{t)\^ + {1-P) j E\Z{s) - Z{s)\''ds
A.15) <Cj E{\Y{s)-Y{s)\' + \Y{s)-Y{s)\\Z{s)-Z{s)\]ds
<Ce f E\Yis)-Yis)\^ds+e f \Zis) - Z{s)\^ds,
where e > 0 is arbitrary and Cs depends on e. Since /3 < 1, choosing
e < I — C and applying Gronwall's inequality, we conclude that
A.16)
Y{t) = Y{t), Z{t) = Z{t), a.s., a.e. te[0,r]
Thus any solution of A.1) must have the form that we have constructed,
proving our claim.
Finally, let {X,Y,Z) and {X,Y,Z) be any two solutions of A.1). By
the previous argument we have
A.17)
Y{t) = e{t,x{t)), z{t) = z{t,x{t),e{t,x{t)),e,{t,x{t))),
Yit) = eit,xit)), z{t) = z{t,x{t),e{t,x{t)),6S,x{t))).
Hence X{t) and X{t) satisfy exactly the same forward SDE A.8) with the
same initial state x. Thus we must have
x{t) = x{t), vte [o,r], a.s.p,

84 Chapter 4. Four Step Scheme
which in turn shows that (by A.17))
Y{t) = Y{t), Z{t) = Z{t), yt e [0,T], a.s.P.
The proof is now complete. D
Remark 1.2. We note that the uniqueness of FBSDE A.1) requires the
condition A.11), which is very hard to be verified in general and therefore
looks ad hoc. However, we should note this condition is trivially true if a
is independent of z\ Since the dependence of a on variable z also causes
difficulty in solving A.6), the first step of the Four Step Scheme, in what
follows to simplify discussion we often assume that a — a{t, x, y) when the
generality is not the main issue.
§2. Non-Degenerate Case — Several Solvable Classes
Prom the previous subsection, we see that to solve FBSDE A.1), one needs
only to look when the Four Step Scheme can be realized. In this subsection,
we are going to find several such classes of FBSDEs.
§2.1. A general case
Let us make the following assumptions.
(Al) d = n; and the functions 6, a, h and g are smooth functions taking
values in R", R™, R"''", R™''" and R™, respectively, and with first order
derivatives in x,y,z being bounded by some constant L > 0.
(A2) The function a is independent of z and there exists a positive
continuous function z/(-) and a constant /i > 0, such that for all {t, x, y, z) £
[0, T] X R" X R™ X R"^™
B.1) u{\y\)I < a{t,x,y)a{t,x,yf < fil,
B.2) \b{t,x,0,0)\ + \h{t,x,0,z)\<fi.
(A3) There exists a constant C > 0 and a £ @,1), such that g is
bounded in C2+«(R'").
Throughout this section, by "smooth" we mean that the involved
functions possess partial derivatives of all necessary orders. We prefer not to
indicate the exact order of smoothness for the sake of simplicity of
presentation.
Since a is independent of z, equation A.6) is (trivially) uniquely
solvable for z. In the present case, FBSDEs A.1) reads as follows:
r dX{t) = b{t,Xit),Y{t),Z{t))dt + a{t,X{t),Y{t))dW{t),
B.3) l^dY{t) = hit,Xit),Yit),Zit))dt + Z{t)dWit), te[0,T],
[X{0)=x, Y{T) = g{X{T)\

§2. Non-degenerate ceise — several solvable classes
85
and A.7) takes the following form:
1
2
' ^t + Jtr [e',,iaa'^)it,x,e)] + {h{t,x,e,z{t,x,e,e,)),el)
B.4)
-h''{t,x,9,z{t,x,e,e^)) = 0,
it,x) eiO,T)xB", l<k<m,
^e{T,x)=g{x), a;e]R".
Let us first try to apply the result of Ladyzenskaja et al [1]. Consider
the following initial boundary value problem:
B.5) I
6t+ Yl ciijit,x,9)e,,,.+J2biit,x,9,zit,x,9,9,))e',^
-h''{t,x,9,z{t,x,9,9^)) = 0,
{t,x) e [0,T] xBr, l<k<m,
^ IsBh^ 9{x), \x\ = R,
[9iT,x)=gix), xEBr,
where Br is the ball centered at the origin with radius R> 0 and
{aij{t,x,y)) = -a{t,x,y)a{t,x,yf,
{bi{t,x,y,z),---,bn{t,x,y,z))'^ ■= b{t,x,y,z),
. {h^{t,x,y,z),---,h^it,X,y,z))'^ = h{t,x,y,z).
Clearly, under the present situation, the function z{t,x,y,p) determined by
A.6) is smooth. We now give a lemma, which is an analogue of
Ladyzenskaja et al [1, Chapter VII, Theorem 7.1].
Lemma 2.1. Suppose that all the functions a^j, bi, /i* and g are smooth.
Suppose also that for all {t,x,y) e [0,T] x R" x R™ and p £ WC"""", it
holds that
B.6)
B.7)
B.8)
i^i\y\)I<{aijit,x,y)) <lii\y\)I,
\bit,x,y,z{t,x,y,p))\ < /i(|i/|)(l + \p\),
9 ,
—aij{t,x,y)
+
dy
i^aij{t,x,y)
<K\y\),
for some continuous functions fi{-) and z/(-), with ^{r) > 0;
B.9) \hit,x,y,zit,x,y,p))\ < [ei\y\) + F(b|, \y\)]il + \p\'),
where P{\p\, \y\) —^ 0, as |p| —^ oo and e{\y\) is small enough;
m
B.10) Y.^\t,x,y,z{t,x,y,p))y'' > -L{1 + ly^,
fc=i

86 Chapter 4. Four Step Scheme
for some constant L > 0. Finally, suppose that g is bounded in C^"'""(]R")
for some a £ @,1). Then B.5) admits a unique classical solution. D
In the case g is bounded in C^+"(]R"), the solution of B.5) and its
partial derivatives 9{t,x), 9t{t,x), 9x{t,x) and 9xx{t,x) are all bounded
uniformly in i? > 0 since only the interior type Schauder estimate is used.
Using Lemma 2.1, we can now prove the solvability of B.4) under our
assumptions.
Theorem 2.2. Let (A1)-(A3) hold. Then B.4) admits a unique classical
solution 9{t,x) which is bounded and 9t{t,x), 9x{t,x) and 9xx{t,x) are all
bounded as well. Consequently, FBSDE B.3) is uniquely solvable.
Proof. We first check that all the required conditions in Lemma 2.1 are
satisfied. Since a is independent of z, we see that the function z{t,x,y,p)
determined by A.6) satisfies
B.11) \z{t,x,y,p)\<C\p\, V(t,a;,2/,p)e[0,r]x]R"x]R'"x]R'"^".
Now, we see that B.6) and B.8) follow from (Al) and (A2); B.7) follows
from (Al), B.2) and B.11); and B.9)-B.10) follow from (Al) and B.2).
Therefore, by Lemma 2.1 there exists a unique bounded solution 9{t,x; R)
of B.5) for which 9t{t, x;R), 9x{t,x;R) a.nd9xxit,x;R) together with 9{t,x)
are bounded uniformly in i? > 0. Using a diagonalization argument one
further shows that there exists a subsequence 9{t,x,R) which converges
uniformly to 9{t,x) as R^ oo. Thus 9{t,x) is a classical solution of B.4),
and 9t{t,x), 9x{t,x) and 9xxit,x), as well as 9{t,x) itself, are all bounded.
Noting that all the functions together with the possible solutions are
smooth with required bounded partial derivatives, the uniqueness follows
from a standard argument using Gronwall's inequality.
Finally, by Theorem 1.1, FBSDE B.3) is uniquely solvable. O
§2.2. The case when h has linear growth in z
Although Theorem 2.2 gives a general solvability result of the FBSDE B.3),
condition B.2) in (A2) is rather restrictive; for instance, the case that
the coefficient h{t, x, y, z) is linearly growing in z is excluded. This case,
however, is very important for applications in optimal stochastic control
theory. For example in the Pontryagin maximum principle for optimal
stochastic control, the adjoint equation is of the form that the corresponding
h is afRne in z. Thus we would like to discuss this case separately.
In order to relax the condition B.2), we compensate by considering the
following special FBSDE:
B.12)
' dX{t) = b{t,X{t),Y{t),Z{t))dt + a{t,Xit))dWit),
dY{t) = h{t,X{t),Y{t),Z{t))dt + Z{t)dW{t),
^X{0) = x, Y{T)=g{X{T)).
We assume that a is independent of y and z, but we allow h to have a linear
growth in z. In this case, the parabolic system looks like the following

§2. Non-degenarate case — several solvable cleisses 87
(compare with B.4)):
^t + ^tr {el,cj{t, x)a{t, xf) + ( bit, X, 6, z{t, x,e,e,)),el)
B.13) \ -h''it,x,e,z{t,x,9,9^))=^0,
it,x) e[0,T]xB", l<k<m,
J{T,x)=g{x), a:e]R".
Since now h has linear growth in z, the result of Ladyzenskaja et al [1] does
not apply. We use the result of Wiegner [1] instead. To this end, let us
rewrite the above parabolic system in divergence form:
B.14)
where
^t + E K(^>2:)^,J,. = f''it,x,9,9,),
(t,a;) e [0,r] xR'", 1 < A; < m,
[9{T,x)=g{x), a:e]R",
{aij{t,x)) = -a{t,x)a{t,x)'^.
B.15) <
f''{t,x,y,p) = Y^ aija:.{t,x)p'l -Y^hi{t,x,y,z{t,x,y,p))p'l
+ h''{t,x,y,z{t,x,y,p)).
By Wiegner [1], we know that for any T > 0, B.14) has a unique classical
solution, global in time, provided the following conditions hold:
B.16) z// < {aij{t, x)) < fil, V(t,x) e [0, T] x R",
,0 17^ T.y''fHt,x,y,p) < eolpf + C{1 + \y\'),
B-1') k=i
y{t,x,y,p) e [0,r] X R" X R™ X R"^",
where u,fi,C,eo are constants with eo being small enough. (To fit the
framework of Wiegner [1], we have taken H = |yp, c* = 0 and r* = 0,
A; = 1, ■ ■ ■, m. See Wiegner [1] for details). Therefore, we need the following
assumption:
(A2)' There exist positive constants z/,/i, such that
B.18) z// < ait, x)a{t, xf < /i/, V(t, x) £ [0, T] x R",
B.19)
\b{t,x,y,z)\, \h{t,x,0,0)\ < fi,
y{t, X, y, z) e [0, T] X R" X R™ X R
"T- w Tcymxn

88 Chapter 4. Four Step Scheme
Theorem 2.3. Suppose that (Al), (A2)' and (A3) hold. Then B.12)
admits a unique adapted solution {X, Y, Z).
Proof. In the present case, for the function z{t,x,y,p) determined by
A.6), we still have B.11). Also, conditions B.16) and B.17) hold, which
will lead to the existence and uniqueness of classical solutions of B.14) or
B.13). Next, applying Theorem 1.1, we can show that there exists a unique
adapted solution (X, Y, Z) of B.12). D
Since h{t,x,y,z) is only assumed to be uniformly Lipschitz continuous
in {y,z) (see (Al)), we have
B 20) \Ht,x,y,z)\<C{l + \y\ + \p\),
V(t,a;,2/,^) e [0,r] X R" X R™ X R™^".
In other words, the function h is allowed to have a linear growth in (y, z).
§2.3. The case when m = 1
Unlike the previous cases, this is the case in which the existence of adapted
solutions can be derived from a more general system than B.4) and B.13).
The main reason is that in this case, function 9{t, x) is scalar valued, and
the theory of quasilinear parabolic equations is much more satisfactory than
that for parabolic systems. Consequently, the corresponding results for the
FBSDEs will allow more complicated nonlinearities. Remember that in the
present case, the backward component is one dimensional, but the forward
part is still n dimensional.
We can now consider A.1) with m = 1. Here W is an n-dimensional
standard Brownian motion, 6, cr, h and g take values in R", R"^", R
and R, respectively. Also, X, Y and Z take values in R", R and R",
respectively. In what follows we will try to use our Four Step Scheme to
solve A.1). To this end, we first need to solve A.6) for z. In the present
case, using the convention that all the vector are column vectors, we should
rewrite A.6) as follows:
B.21) z = a{t,x,y,zfp.
Let us introduce the following assumption.
(A2)" There exist a positive continuous function z/(-) and constants
C, /3 > 0, such that for all (t, x, y, z) e [0, t] x R" x R x R",
B.22) i^{\y\)I < a{t,x,y,z)u{t,x,y,zf < CI,
{[a{t,x,y,z)'^] ^z-[(j{t,x,y,^'^] ^z,z-z)
>p\z-z\''
B.24) \b{t,x,0,0)\ + \h{t,x,0,0)\ < C.

§3. Infinite horizon case 89
We note that condition B.23) amounts to saying that the map z \-^
\a(t,x,y^z)^\~^z is uniformly monotone. This is a sufficient condition for
B.21) to be uniquely solvable for z. Some other conditions are also possible,
for example, the map z \-^ —\a{t^x,y,z^'^\''^z is uniformly monotone.
We have the following result for the unique solvability of FBSDE A.1)
with m = 1.
Theorem 2.4. Let {Al) with m = I, (A2)" hold. Then there exists a
unique smooth function z{t,x,y,p) that solves B.21) and satisfies B.11).
In addition, if (A3) also holds, then FBSDE A.1) (with m = 1) admits an
adapted solution determined by the Four Step Scheme.
The proof is omitted here.
We should note that the well-posedness of A.7) in the present case
(m = 1) follows from Ladyzenskaja [1, Chapter V, Theorem 8.1]. We see
that the condition B.24) together with (Al) means that the functions b
and h are allowed to have linear growth in y and z. Also, note that we do
not claim the uniqueness of adapted solutions since a condition similar to
A.11) is not easy to be made explicit.
§3. Infinite Horizon Case
In this section, we are concerned with the following FBSDE:
f dX{t) = b{X{t), Y{t))dt + a{X{t),Y{t))dW{t), t e [0, oo),
dY{t) = [h{X{t))Y{t)-l]dt-(Z{t),dW{t)), te[0,oo),
X{0) = X,
, Y{t) is bounded a.s., uniformly in t E [0, oo).
C.1)
Note that the time duration here is [0, oo). Thus, C.1) is an FBSDE in an
infinite time duration. In this section, we only consider the case m = 1,
i.e., Y{-) is a scalar-valued process. Hence, Z{-) is valued in R'*. Note that
X{-) is still taking values in R".
§3.1. The nodal solution
First of all, let us introduce the following notion.
Definition 3.1. A process {{X{t),Y{t), Z{t))}t>o is called an adapted
solution of C.1) if for any T > 0, (X, Y, Z)L ^eM[0,T], and
X{t)=x+ f b{X{s),Y{s))ds+ f aiX{s),Yis))dW{s),
Jo Jo
Y{t)=Y{T)-f [h{X{s))Y{s)~l]ds + J {Z{s),dW{s)),
0<t<T <oo,
C.2) {
such that 3M > 0, \Y{t)\ < M, Vt, P-a.s. Moreover, if an adapted solution
(X, Y, Z) is such that for some 6 e C^iIi")nCl (R"), the following relations

90 Chapter 4. Four Step Scheme
hold:
C.3) I^W-^(^@)> te[o,oo),
\z{t) = a{x{t),e{x{t))ye,{x{t)),
then we call (X, Y, Z) a nodal solution of C.1), with the representing
function 6.
Let us now make some assumptions.
(HI) The functions a,b,h are C^ with bounded partial derivatives and
there exist constants A,/i > 0, and some continuous increasing function
V : [0, oo) —>■ [0, oo), such that
C.4) \I < aix,y)aix,y)^ < fil, (x,y) eK" x R,
C.5) Hx,y)\<u{\y\), {x,y) eU" xR,
C.6) inf h{x) = S > 0, sup h{x) = 7 < oo.
The following result plays an important role below.
Lemma 3.2. Let (HI) hold. Then the following equation admits a classical
solution e e C2+«(]R"):
C.7) ^tT(e^Mx,o)a^{x,e)^ + {b{x,9),e^)-h{x)e + i = o, xewi".
such that
C.8) -<0(x)<\, xeWC.
7 0
Sketch of the proof. Let Bij(O) be the ball of radius R > 0 centered
at the origin. We consider the equation C.7) in Bij(O) with the
homogeneous Dirichlet boundary condition. By [Gilbarg-Trudinger, Theorem
14.10], there exists a solution 6^ e C2+"(Bi?@)) for some a > 0. By the
maximum principle, we have
C.9) 0 < e'^ix) <\, xe Br{0).
0
Next, for any fixed xq e R", and R > \xo\ + 2, by Gilbarg-Trudinger [1,
Theorem 14.6], we have
C.10) \e^ix)\<C, xeBiixo),
where the constant C is independent of i? > |a;o| +2. This, together with
the boundedness of a and the first partial derivatives of a,b,h, implies
that as a linear equation in 6 (regarding a{x,6{x)) and b{x,6{x)) as known

§3. Infinite horizon case 91
functions), the coefficients are bounded in C^. Hence, by Schauder's interior
estimates, we obtain that
C.11) ||^^||c2+»(B,(,„)) <C, VR> \xo\ + 2.
Then, we can let i? —>^ oo along some sequence to get a limit function 9{x).
By the standard diagonalization argument, we may assume that 9 is defined
in the whole of R". Clearly, 9'e C^+"(]R") and is a classical solution of
C.7). Finally, by the maximum principle again, we obtain C.8). D
Now, we come up with the following existence of nodal solutions to
C.1). This result is essentially the infinite horizon version of the Four Step
Scheme presented in the previous sections.
Theorem 3.3. Let (HI) hold. Then there exists at least one nodai
solution {X, y, Z) of C.1), with the representing function 9 being the solution
of C.7). Conversely, if (X, Y, Z) is a nodal solution of C.1) with the
representing function 9. Then 9 is a solution of C.7).
Proof. By Lemma 3.2, we can find a classical solution 9 e C^+^iU")
of C.7). Now, we consider the following (forward) SDE:
f3 121 idXit)=biXit),9iXit)))dt-^aiXit),9iX{t)))dW{t), t > 0,
^ ' ^ \x{0)=x.
Since 9^ is bounded and b and a are uniformly Lipschitz, C.12) admits
a unique strong solution X{t), t £ [0,oo). Next, we define ¥{■) and Z{-)
by C.3). Then, by Ito's formula, we see immediately that {X,Y,Z) is an
adapted solution of C.1). By Definition 3.1, it is a nodal solution of C.1).
Conversely, let {X, Y, Z) be a nodal solution of C.1) with the
representing function 9. Since 9 is C^, we can apply Ito's formula to Y{t) = 9{X{t)).
This leads to that
dYit)=[{biXit),9iXit))),9AXit)))
C-13) + itr {9,,{X{t))aa^{X{t),9{X{t))))]dt
-^ (9,iXit)),aiXit),9{X{t)))dW{t)).
Comparing C.13) with C.1) and noting that Y{t) = 9{X{t)), we obtain
that
C.14)
{biX{t),9iX{t))),9,iXit)))-^^ti[9,,iXit))aa^iXit),9iXit)))]
= h{X{t))9{X{t)) - 1, yt> 0, P-a.s.
Define a continuous function F : ]R" —> R by
C 15) Fix)^{bix,9ix)),9,ix))-^^tT[9,A^)aa^{^:0{x))]
- h{x)9{x) + 1.

92 Chapta: 4. Four Step Scheme
We shall prove that F = 0. In fact, process X actually satisfies the following
FSDE
C.16) idX{t) = b{X{t))dt + a{X{t))dWit), t>0;
\Xo=x,
where b{x) = b{x,6(x)) and a{x)=a{x,6{x)). Therefore, X is a time-
homogeneous Markov j)rocess with some transition probability density
p{t,x,y). Since both b and a are bounded and satisfy a Lipschitz
condition; and since a = GG^ is uniformly positive definite, it is well known
(see, for example, Friedman [1,2]) that for each y £ ]R", p{-,■ ,y) is the
fundamental solution of the following parabolic PDE:
I,J—I ■' i=l
and it is positive everywhere. Now by C.14), we have that F(X(t)) = 0 for
all t > 0, P-a.s., whence
C.18) ^ = E^^^{F(X(t)f\= \ v(t,x,y)F{yfdy, Vt > 0.
By the positivity of p(t,a;, y), we have F{ij) = 0 almost everywhere under
the Lebesgue measure in R". The result then follows from the continuity
ofF. D
Theorem 3.3 tells us that if C.7) has multiple solutions, we have the
non-uniqueness of the nodal solutions (and hence the non-uniqueness of the
adapted solutions) to C.1); and the number of the nodal solutions will be
exactly the same as that of the solutions to C.7). However, if the solution
of C.7) is unique, then the nodal solution of C.1) will be unique as well.
Note that we are not claiming the uniqueness of adapted solutions to C.1).
§3.2. Uniqueness of nodal solutions
In this subsection we study the uniqueness of the nodal solutions to C.1).
We first consider the one dimensional case, that is, when X and Y are
both one-dimensional processes. However, the Brownian motion W{t) is
still d-dimensional (d > 1). For simplicity, we denote
C.19) a{x,y) = \\a(x,y)\^, {x,y) ^V.^.
Let us make the some further assumptions:
(H2) Let m = n = 1 and the functions a, 6, h satisfy the following:
C.20) h{x) is strictly increasing in a; £ R.

§3. Infinite horizon case
93
C.21) <
a{x,y)h{x)-{h{x)y-l) / ay{x,y + I3{y-y))dl3 >r] > 0,
JO
/ [aix,y)by{x,y +p{y-y))
Jo
-ay{x,y + p{y-y))b{x,y)]dp>0, y,ye [^,\],xeWi.
Condition C.21) essentially says that the coefficients b,a and h should
be somewhat "compatible." Although a little complicated, C.21) is still
quite explicit and not hard to verify. For example, a sufficient conditions
for C.21) is
C.22)
' a{x,y)h{x) — {h{x)y — l)ay{x,w) > »? > 0,
a{x,y)by{x,w) - ay{x,w)b{x,y) > 0,
y,we[i-,^], xeK.
It is readily seen that the following will guarantee C.22) (if (HI) is
assumed) :
C.23) ayix,y)=0, by{x,y)>0, (a;,y) £ R x [i, i].
In particular, if both a and b are independent of y, then C.21) holds
automatically.
Our main result of this subsection is the following uniqueness theorem.
Theorem 3.4. Let (H1)-(H2) hold. Then C.1) has a unique adapted
solution. Moreover, this solution is nodal.
To prove the above result, we need several lemmas.
Lemma 3.5. Let h be strictly increasing and 6 solves
C.24) a{x, 6N^^ + b{x, 6N^ - h{x)e -1-1 = 0, a; £ R.
Suppose Xm is a local maximum of 9 and x^ is a local minimum of 9 with
G{xm) < G{xm). Then x^ > xm-
Proof. Since h is strictly increasing, from C.24) we see that 9 is not
identically constant in any interval. Therefore Xm 7^ xm- Now, let us look
at Xm- It is clear that 9x{xm) = 0 and 9xx{xm) < 0. Thus, from C.24) we
obtain that
1
C.25)
Similarly, we have
C.26)
9{xm) <
9{Xm) >
Since 9{xm) < 9{xm), we have
C.27)
1
<
Kxm) '
KXm) '
1
hiXm) Kxm) ''

94 Chapter 4. Four Step Scheme
whence xm < x^ because h{x) is strictly increasing. Q
Lemma 3.6. Let (H1)-(H2) hold. Then C.24) admits a unique solution.
Proof. By Lemma 3.2 we know that C.24) admits at lease one classical
solution 9. We first show that 6 is monotone decreasing. Suppose not,
assume that it has a local minimum at Xm- Since 6 £ C^ and 6 is not
constant on any interval as we pointed out before, ^x > 0 near x^- Using
Lemma 3.5, one further concludes that 9x > 0 over (a;„,oo). In other
words, 6 is monotone increasing on (aj^jOo). The boundedness of 6 then
leads to that lima;_>oo ^i^) exists.
Next we show that
C.28) lim e^ix) = lim O^^ix) = 0,
To see this, we first apply Taylor's formula and use the boundedness of Oxx
to conclude that there exists M > 0 such that for any x £ (a;„,oo) and
/i>0
e{x + h)- 9{x) - Mh^ < 9x{x)h < e{x + h)- e{x) + Mh^.
Since lima;_>oo 6{x + h) — 6{x) = 0, we have
-Mh^ < lim ex{x)h < Hin ex{x)h < Mh^.
Dividing h and letting /i —> 0 we derive \\Ta.x^oo^x{x) = 0. Further, note
that
e.x = -r^[-Kx,o)ex + h{x)e -1].
The boundedness of 6,9x, and 9xx and the assumption (HI) then show that
Oxxx exists and is continuous and bounded as well. Thus apply Taylor's
expansion to the third order and repeat the discussion above one shows
further that limx^oo Oxx{x) = 0 as well, proving C.28). Consequently, by
C.24) we have
C.29) lim e(x) = — r.
^ ' x^oo ^ ' h{+oo)
On the other hand, by C.20), we see that
C.30) lim e{x) > e{xm) > -TT^ > wT-^,
x->oo h[Xm) n[ + 0O)
which contradicts C.29). This means that 6 has no local minimum.
Similarly one shows that 6 can not have any local maximum either, hence it
must be monotone on R. Finally, since
C.31) ^(-oo) = -^-- > —^ = ^(+oo),
^ h{-oo) h{+oo)
it is necessary that 6 is monotone decreasing.

§3. Infinite horizon case 95
Next, let 6 and 6 be two solutions of C.24).. Then, w = 9 — 9 satisfies
0 = a{x, 9)wxx + b{x, 9)Wx
/I
[ayix,9 + pw)9xx + by{x,9 + pw)9x]dpy
= a{x, 9)wxx + b{x, 9)wx - c{x)w,
]'W
where
c{x) ■= h{x) - / [ay{x,9 + p
Jo
Mx)9- l-b{x,9)9x
w) —
C.33)
a{x,9)
+ by{x,9 + pw)9x]dp
a{x, 9)h{x) - {h{x)9 - 1) J^ ay(x, 9 + p{9 - 9))dp
a{x,9)
+ \9x\ f [a{x,9)byix,9 + p(9-9))
- ay{x,p9 + p{9- 9))b{x,9)\dp > ^.
J fi
Here, we have used the fact that 9x{x) ■= —\9x{x)\ (since 9 is decreasing
in x) and C.21) as well as C.7). Prom (HI), we also see that a{x,9) > 0
and b{x,9) are bounded. Thus, by the lemma that will be proved below,
we obtain w = 0, proving the uniqueness. D
Lemma 3.7. Let w be a bounded classical solution of the following
equation:
C.34) a{x)wxx + b{x)Wx — c{x)w = 0, a; £ R,
with c(x) > Co > 0, a{x) > 0, x £ R", and with a and b bounded. Then
w{x) = 0.
Proof. For any a > 0, let us consider $a(a;) = 'w{x) — a\x\'^. Since w
is bounded, there exists some Xa at which $„ attains its global maximum.
Thus, ^'^{xa) = 0 and ^'^{xa) < 0, which means that
C.35) WxiXa) = 2aXa, Wxx{Xa) < 2a.
Now, by C.34),
c{Xa)w{Xa) = a{Xa)Wxx{Xa) + b{Xa)Wx{Xa)
C.3d)
< 2a[a{xa) +b{xa)xa)-
For any a; £ R, by the definition of Xa, we have (note the boundedness of
a and b)
'w{x) — a\x\^ < wixa) — a\xa.^
<^{2a{xa) + 2b{xa)xa-\xa?) <Ca.
Co

96 Chapter 4. Four Step Scheme
Sending a —> 0, we obtain w{x) < 0. Similarly, we can show that w{x) > 0.
Thus w{x) =0. D
Proof of Theorem 3.4- Let {X,Y,Z) be any adapted solution of C.1).
Under (H1)-(H2), by Lemma 3.6, equation C.24) admits a unique classical
solution 6 with 6^ <0. We set
C.38) <^ ^ -, te[0,oo).
[z{t) = a{x{t),e{x{t))ye,{x{t)),
By Ito's formula, we have (note C.19))
C 39) dYit)=[^^iXitmXit),Yit))+e,,iXit))aiXit),Yit))\dt
+ (aiX{t),Y{t)ye,{X{t)),dW{t)).
Hence, with C.1), we obtain (note C.24)) that for any 0 < r < t < oo,
E[Y{r) - Y{r)f - E[Y{t) - Y{t)f
= -E f [2[Y -Y][e^{X)b{X,Y) + 9^^{X)a{X,Y)
-h{X)Y + l\ +|G(X,rN'^(X)-Z|2}(is
< -2E J\y -Y][9,{X){b{X,Y) -b{X,Y))
^^•^°) + ^xx(X) {a{X, Y) - a{X, Y)) - h{X){Y - f)] ds
= -2EJ'{[Y-Y]-'[\e,{X)\l by{X,Y + P{Y-Y))dp
-^xx(X) f ayiX,Y + p{Y -Y))dp + hiX)\^ds
= -2E f cis)\Y{s)-Yis)\^ds,
J r

§3. Infinite horizon case 97
where (note the equation C.24))
cis)=h{X) + \e,{X)\ [ by{X,Y + p{Y--Y))dC
Jo
b{X,Y)e^{X)-h{X)Y+l
a{X,Y)
■J ay{X,Y + C{Y~Y))dC
/•I ^
-[h{X)Y-l] ayiX,Y + piY-Y))dp
Jo
+ \9,{X)\ [ [aiX,Y)by{X,Y + p{Y^Y))
Jo
- b{X, Y)ay{X, Y + C{Y- Y))] d/3 > ^.
Denote ip{t) = E[Y{t)-Y{t)f and a = ^ > 0. Then C.40) can be written
as
C.42) ip{r) < ip{t) - a / ip{s)ds, 0 <r <t<oo.
Jr
Thus,
C.43) ('""* / '^^'^'^')' " '""* ("^^^^ ~ " / ^^'^'^')
>e~"VW, te[r,oo).
Integrating it over [r,T], we obtain (note Y and Y are bounded, and so is
„-ar _ „-aT i-T
C.44) 93(r) < e-«^ / 93(s)(is < CTe'"'^, T > 0.
Therefore, sending T —> oo, we see that ip{r) = 0. This implies that
C.45) Y{r) = Y{r) = e{X{r)), r £ [0,oo), a.s.w £ ft.
Consequently, from the second equality in C.40), one has
C.46) Zis) = Z{s) = a{X{s),e{X{s))fe^{X{s)), Vs £ [0,oo).
Hence, {X,Y,Z) is a nodal solution. Finally, suppose {X,Y,Z) and
{X,Y,Z) are any adapted solutions of C.1). Then, by the above proof,
we must have
C.47) Y{t)=e{Xit)), Y{t)=eiXit)), te[0,oo).

C.48)
98 Chapter 4. Four Step Scheme
Thus, by C.1), we see that X{-) and X{-) satisfy the same forward SDE
with the same initial condition (see C.12)). By the uniqueness of the strong
solution to such an SDE, X = X. Consequently, Y = Y and Z = Z. This
proves the theorem. Q
Let us indicate an obvious extension of Theorem 3.4 to higher
dimensions.
Theorem 3.8. Let (HI) hold and suppose there exists a solution 6 to C.7)
satisfying
h{x) - [ [J2 4'(^' A - /^)^(^) + /^^)^---i (^)
n
i=l
a;e]R",^e[i,i].
Then C.1) has a unique adapted solution. Moreover, this solution is nodal
with 6 being the representing function.
Sketch of the proof. First of all, by an equality similar to C.32), we
can prove that C.7) has no other solution except 0{x). Then, by a proof
similar to that of Theorem 3.4, we obtain the conclusion here. Q
Corollary 3.9. Let (HI) hold and both a and b be independent ofy. Then
C.1) has a unique adapted solution and it is nodal.
Proof. In the present case, condition C.48) trivially holds. Thus,
Theorem 3.8 applies. D
§3.3. The limit of finite duration problems
In this subsection, we will prove the following result, which gives a
relationship between the FBSDEs in finite and infinite time durations.
Theorem 3.10. Let (H1)-(H2) hold and let 6 be a solution of C.7) with
the property C.48). Let {X,Y,Z) be the nodal solution of C.1) with the
representing function 6, and (X^,Y^,Z^) e M[0,K] be the adapted
solution of C.1) with [0, oo) replaced by [0, K], and Y^{K) = g{X{K)) for
some bounded smooth function g. Then
C.49) lim E{\X^{t)-Xit)\'^ + \Y^it)-Y{t)\'^+E\Z^it)-Z{t)\'^}=0,
K—^oo
uniformly in t on any compact sets.
To prove the above result, we need the following lemma.

§3. Infinite horizon case
Lemma 3.11. Suppose that
99
C.50)
( \I < {a'^{t,x)) <fil,
\b'it,x)\<C, l<i<n,
c{t,x) > rj > 0,
I \woix)\ < M,
{t,x) e [0,oo) xR",
with some positive constants A, /i, rj, C and M. Let w be the classical
solution of the following equation:
C.51)
Then
C.52)
Wt-^ d''{t,x)WxiXj - ^&'(^a;)w;a;i +c{t,x)w = 0,
i=\
{t,x) e [0,oo) xR",
\w{t,x)\ < Me""*, {t,x) e [0,oo) X R".
Proof. First, let i? > 0 and consider the following initial-boundary
value problem:
C.53)
w:
, - J2 a'^(^^)<x, -Ylb\t,x)w^, +c{t,x)w^ = 0,
{t,x)[0,Oo)x e Br,
,w\^g = wo{x)x^{x),
where Br is the ball of radius R > 0 centered at 0 and x^ is some
"cutoff" function. Then we know that C.53) admits a unique classical solution
^R g (j2+a,i+a/2(^Q^ ^ [0, oo)) for some a > 0, where C^+^-i+^Z^ is the
space of all functions v{x,t) which are C^ in x and C^ in t with Holder
continuous Vx^xj and Vf of exponent a and a/2, respectively. Moreover, we
have
C.54) \w'^{t,x)\<M, {t,x) e[0, go) X Br,
and for any xq G R" and T > 0, @ < a' < a)
C.55) w'^Aw, mC^+°'''^+°''/'^{[0,T]xBi{xo)), as i? ^ oo,
where w is the solution of C.51). Now, we let ipit, x) = Me~(''~^>* (e > 0).

100
Then
C.56)
Chapter 4. Four Step Scheme
= {c{t, x)-r] + e)M-(''-^>* > eM-f''-^^* > 0,
'f'laBn > 0 =
w
idBf
Thus, by Friedman [1, Chapter 2, Theorem 16], we have
C.57)
w
\t, x) < ip{t,x) = M-(''-^>*, {t, x) e [0, oo) X Br.
Similarly, we can prove that
C.58)
w
\t, x) > -Me-(''-^>*, (t,x) e [0, oo) X Br.
Since the right hand sides of C.57)-C.58) are independent of R, we see
that
C.59) \w{t,x)\<Me-'^''-'^\ {t,x) e[0,Go) xW.
Hence, C.52) follows by sending e -> 0. D
Proof of Theorem 3.10. By the result from §2, we know that
(X^,r^,Z^) satisfies
C.60)
fY^{t)=e^{t,X^{t)),
te [0,K], a.s.wen,
where 6^ is the solution of the parabolic equation:
C.61) <
9^ + J2 a'^ix,9^)9^^,^+J2b'{x,9^)9^^ - h{x)9^ + 1 = 0,
{x,t) e]R"x [0,r),
[9^1^^ = g{x),
with a = jcrc'^- Next, we define ip to be the solution of
C.62)
(fit- "^ a'' {x, v)VxiXi - X] V{x, ip)ip:ci + h{x)ip -1 = 0,
{t,x) e [0,oo) xR",
Clearly, we have
C.63) 9^{t,x)=ip{K-t,x), {t,x) e[0,K]xW.

§3. Infinite horizon case
Now, we let w{t,x) = v{t,x) — 6{x). Then
101
C.64)
Wt- Y^ a'-'{x,ip)w:ciXj ~ Y^h{x,ip)w^.
/•I '^
-[hix)-J E]a;^(a;,^ + /3«;)^,,,.
n
+ Y, &y(^' ^ + M^^i) dp] W = 0,
i=l
= g{x)-9{x).
t=o
We note that both ip{x,t) and 9{x) lie in [i i]. Thus, by condition C.48)
and Lemma 3.11, we see that
C.65) '^''^^'''^ ~ ^^''^' " '"^^^ ~ ^'""^ ~ ^^"'^l - r""'"''*^'
{t,x) e [0,K] X R" X [0,if], K>0.
Now, we look at the following forward SDEs:
- dX^{t) = biXitf,9^it,X{t)^))dt
C.66) i +a{X{t)'^,e^{t,X{t)'^))dW{t),
^X^{0)=x.
C.67)
f (iX@ = 6(X@,6'(X(t)))(it + G(X@,6'(X@))(iW@,
\x@) = a;.
By Ito's formula, we have
E\X^it)-X{t)f
= El'[2{x^-x,b{x'<,e^{s,x''))-b{x,e{x)))
+ tr ([<7(X^,^^(s,X^)) - a{X,e{X))]-
[<7(X^,^^(s,X^))-<7(X,^(X))]^)]ds
<CE /" [|X^-X|(|X^-X| + |6i^(s,X^)-6i(X^)|)
+ (|x^ - x| + |^^(s,x^) - e{x^)\y]ds
<C f [E\X^{s)-X{s)\^+e-^'>^'^-'^]ds
<C f \X^{s)-X{s)\'^ds + Ce-^''^^-'K
Jo
C.68)

102 Chapter 4. Four Step Scheme
Applying Gronwall's inequality, we obtain that
C.69) E (|X^@ - X{t)\^) < Ce-^^^^-^K t e [0,K], K > 0.
Furthermore,
E {\Y^{t) - Y{t)\') = E (|^^(t,X^(t)) - e{Xm')
<2E{\e^{t,x^it))-eix^it))\'^)
C.70) +2E{\9{X^{t))-e{X{t))\'')
< Ce-2''(-^-*) + CE {\X^{t) - X{t)\^)
^^^-2v{K-t)^ te [0,K], K>0.
Similarly, we have
C.71) E{\Z^it)-Z{t)\'^) <Ce-^''^^-'\ te[0,K], K>0.
Finally, letting if —> oo, the conclusion follows. D

Chapter 5
Linear, Degenerate Backward Stochastic
Partial Differential Equations
§1. Formulation of the Problem
We note that in the previous chapter, all the coefficients b, a, h and g
are deterministic, i.e., they are all independent of w £ ft. If one tries to
apply the Four Step Scheme to FBSDEs with random coefficients, i.e., b,
G, h and g are possibly depending on w £ ft explicitly, then it will lead
to the study of general degenerate nonlinear backward partial differential
equations (BSPDEs, for short). In this chapter, we restrict ourselves to the
study of the following linear BSPDE:
A.1)
1
du= {-- V-iADu) -{a,Du)~cu- V-(Bg) - {b,q) -f}dt
+ {q,clW{t)), {t,x)e[0,T]xB.",
<'>J-W^
t=T
where Du is the gradient of u,
n
,v-$„)^, v$ = (#i,---,$„)eci(]R";]R""'"),
i=l
l^ V-$ = (V-$i
and
A.2)
S",
nxd
y4:[0,T]x]R"xft
B:[0,T]xK"xn^WC
a:[0,r]x]R"xft-)-]R",
6:[0,r]x]R"xft-)-]R'^,
c, / : [0, T] X R" X ft ^ R,
^ 5 : R" X ft ^^ R,
are random fields (S" is the set of all (n x n) symmetric matrices). We
assume that W ■= {W{t) : t £ [0,r]} is a d-dimensional Brownian motion
defined on some complete filtered probability space (ft,T, {Tt}t>0:^): with
{^t}t>o being the natural filtration generated by W, augmented by all the
P-nuU sets in T.
In our discussions, we will always assume that A and B are differen-
tiable in x. In such a case, A.1) is equivalent to an equation of a general
form. To see this, we note that
A.3)
ti [AD^u] = V-iADu) ~(V-A,Du);
tr[B^Dq] = ViBq)-(V-B,q),

104 Chapter 5. Linear, Degenerate BSPDEs
where D^u is the Hessian of u and
Dq={Dq,,---,Dqa)^
Therefore, if we define
A.4) a = a + -V-A;
it
then A.1) is the same as
( dx^qi
.
\^x„gi
b =
■■■ dx^qa
■■■ 9x„gd
6 + V-B,
A.5)
du= {- -tr [AD^u] -{a,Du) ~cu - tr [B'^Dq]
-(b,q)-f}dt+{q,dW{t)), (t,a;)e[0,r]x]R",
Since A.1) and A.5) are equivalent, all the results for A.1) can be
automatically carried over to A.5) and vice versa. For notational
convenience, we will concentrate on A.1) for well-posedness (§§2-5) and on A.5)
for comparison theorems (§6).
Next, we introduce the following definition.
Definition 1.1. If yl and B satisfy the following:
A.6) A{t, x) - B{t, x)B{t, x)'^ > 0, a.e. {t, x) £ [0, T] x R", a.s.,
we say that equation A.1) is parabolic; if there exists a constant 5 > 0, such
that
A.7) A{t,x)-B{t,x)B{t,x)'^ >5I, a.e. (t,a;) e [0,r] xR", a.s.
we say that A.1) is super-parabolic; whereas, if A.6) holds and there exists
a set G C [0, T] x ]R" of positive Lebesgue measure, such that
A.8) det [A{t,x) - B{t,x)B{t,x)^] = 0, V(t,a;) £ G, a.s.
we say that A.1) is degenerate parabolic.
We see that in the above definition, only A and B are involved. Thus,
the above three notions are adopted to equation A.5) as well.
Note that if A.1) is super-parabolic, it is necessary that A{t,x) is
uniformly positive definite, i.e.,
A.9) A{t,x) > E/ > 0, a..e.{t,x) £ [0,r] x R", a.s.
However, if A{t,x) is uniformly positive definite and A.6) holds, we do
not necessarily have the super-parabolicity of A.1). As a matter of fact, if
A{t,x) satisfies A.9) and
A.10) A{t,x)=B{t,x)B{t,xf, a.e.(t,a;) e [0,r] xR", a.s.

§1. Formulation of the problem 105
then A.1) is degenerate paraboUc. This is the case if we have the BSPDE
from the Four Step Scheme for FBSDEs with random coefficients (see
Chapter 4, §6).
Now, we introduce the notion of solutions to A.1). In what follows, we
denote Br = {x eB." \ \x\ < R} for any R > 0.
Definition 1.2. Let {{u{t,x;Lj),q{t,x;Lj)),{t,x,ij) £ [0,r] x R" x ft} be
a pair of random fields.
(i) {u,q) is called an adapted classical solution of A.1) if
A.11) r '\r ''"" vi?>o,
ueC^{[0,T]-LHn;C\BR))),
such that almost surely the following holds for all {t,x) £ [0, T] x R":
u{t,x) = g{x) + / l-V-[A{s,x)Du{s,x)] + {a{s,x),Du{s,x))
+ c(s, x)u{s, x) + V-[B{s, x)q{s, x)]
A.12) -,
+ {b{s,x),q{s,x)) +f{s,x)jds
j {q{s,x),dWis)) .
(ii) {u,q) is called an adapted strong solution of A.1) if
ueCA[0:nL\n;H\Bn))),
qeL^:^{0,T;H\Bn;K'')),
A-13) { _;:::„,;.^:. ^^^' vi?>o.
such that almost surely A.12) holds for allt e [0,T], a..e.x £ R".
(iii) {u,q) is called an adapted weak solution of A.1) if
ueC:Fi[0:T];L\n;H\BR))),
qeLl{0,T-L\Bii-n'')),
A-14) :; 2,: ,; vi?>o,
such that almost surely for all ip £ C^(R") and all te[0,T],
I u{t,x)ip{x)dx — j g{x)ip{x)dx
= / / {-l{Ms,x)Duis,x),Dipix))
a 15)
^ ' + {a{s,x),Du{s,x)) ip{x) + c{s,x)u{s,x)ip{s,x)
- (B{s, x)q{s, x), Dip{x)) -|- (b{s,x), q{s, x)) ip{x)
+ f{s,x)ip{x)>dxds — I ( / q{s,x)ip{x)dx,dW{s)) .
' Jt JR"

106 Chapta: 5. Linear, Degenerate BSPDEs
We note that in the definition of adapted classical solution, we need A
and B to be C^ in x; in the definition of adapted strong solution, we need
A and B to be differentiable in x almost everywhere; and in the definition
of adapted weak solution, we need only the coefficients {A, B, a, b, c} to be
bounded and / and g to be locally square integrable.
It is clear that for A.1), if {u,q) is an adapted classical solution, it is
an adapted strong solution; if (u, q) is an adapted strong solution, it is an
adapted weak solution. The following result tells the other way around,
which will be useful later. In the following proposition, by "the coefficients
are regular enough" we mean that all the coefficients have the required
differentiability, continuity and integrability.
Proposition 1.3. Let the coefficients of A.1) be regular enough. Let
{u,q) be an adapted weak solution of A.1). If in addition, A.13) holds,
then {u,q) is an adapted strong solution of A.1). Further, if A.11) holds,
then {u,q) is an adapted classical solution of A.1).
Proof. Let {u,q) he an adapted weak solution of A.1) such that A.13)
holds. Then, from A.15), by integration by parts, we have
/ {u{t,x) - g{x)}ipix)dx
= / {/ {-V-[A{s,x)Du{s,x)] + (a{s,x),Du{s,x))
+ c{s,x)u{s,x) + V-[B{s,x)q{s,x)] + {b{s,x),q{s,x))
+ f{s,x)}ds — I {q{s,x),dW{s)) \ip{x)dx.
The above is true for all tp e Cg°{Wl"). Then, A.12) follows, proving that
(u, q) is an adapted strong solution. The other assertion is obvious. Q
Although the parabolicity condition A.6) is not necessary in Definition
1.2 and Proposition 1.3, we will see later that such a condition is very
crucial for our studying the well-posedness of BSPDE A.1).
§2. Well-posedness of Linear BSPDEs
In this section, we state the results of well-posedness for BSPDE A.1). The
proofs of them will be carried out in later sections.
To begin with, let us introduce the following assumption concerning
the coefficients of equation A.1)- Let m > 1.
(H)„ Functions {A,B,a,b,c} satisfy the following:
'^eL^@,r;C™+i(E";S")),
B e L^@,r;C™+i(E";]R"^'*)),
B.1) i aeL^@,r;C6'"(lR";]R")),
6eL^@,r;C6'"(lR";]R'*)),
.ceL^(o,r;Cr(]R")).
A.16)

§2. Well-posedness of linear BSPDEs 107
We note that (H)„ implies that the partial derivatives of A and B in
X up to order (m + 1), and those of a, b and c up to order m are bounded
uniformly in (t, x, w) by a constant K^ > 0. This constant will be referred
in the statements of Theorems 2.1, 2.2 and 2.3.
In what follows, we let
a={ai,- ■ ■ ,a„), .a^'s are nonnegative integers,
n
A
al^Xl"- 9"^d^l---dS:, x^^xT
Any a of the above form is called a multi-index. li C = (/3i, ■ ■ ■, /?«) is
another multi-index, by /3 < a, we mean that /J, < a, for each i = 1, ■ ■ ■, n,
and by /3 < a, we mean P < a and at least for one i, one has Ci < ai.
Now, we state the following result concerning the well-posedness of
BSPDE A.1).
Theorem 2.1. Suppose that the parabolicity condition A.6) holds and
(H)m holds for some m > 1. Suppose further that the coefficient B{t,x)
satisfies the following "symmetry condition":
B.2) [B{d..B^)f =B{d.,B^),
a.e. (t, x) e [0, r] X R", a.s., 1 < i < n.
Then for any random fields f and g satisfying
r/eL^(o,r;H'»(]R")),
Uei^,(fi;H'"(iR")),
BSPDE A.1) admits a unique adapted weak solution {u,q), such that the
following estimate holds:
maxE\\u{t,-)\\%r,.+E f \\q{t,-)\\jir.-^dt
+ Yl ^ [ [ {{{A-BB'^)D{d"u),D{d"u))
B.4) |a|<m -^^ •^*^"
+ \B'^[D{d°'u)\+d°'q\ ^dxdt
<Ce[J'^ \\f{t,-)\?H„dt + \\9\\^H^},
where the constant C > 0 only depends on m, T and Km.
Fijrthermore, if m > 2, the weak solution {u,q) becomes the unique
adapted strong solution of A.1); and if m > 2 + n/2, then {u,q) is the
unique adapted classical solution of A.1).
The symmetry condition B.2) is technical. It will play a very important
role in proving the existence of adapted solutions. However, we point out

108 Chapter 5. Linear, Degenerate BSPDEs
that such a condition is not needed for the uniqueness of adapted weak (and
hence strong and classical) solutions. See §3.1 for details. Several examples
satisfying such a condition are listed below:
' d = n = 1; B is a scalar;
B.5) < B is independent of x;
_ B{t,x) = ip{t,x)Bo(t), where 93 is a scalar-valued random field.
The following result tells us that the symmetry condition B.2) can be
removed if the parabolicity condition A.6) is strengthened.
Theorem 2.2. Suppose A.6) holds and (H)m with m > 1 is in force.
Suppose further that for some eo > 0, either
B.6) A - BB'^ > sqBB'^ > 0, a.e. it,x) e [0,T] x R", a.s.,
or
A - BB'^ >soYl {d"B){d°'B'^) > 0,
B.7) |a| = l
a..e.{t,x) e [0,T] x R", a.s.
Then the conclusion of Theorem 2.1 remains true and the estimate B.4) is
improved to the following:
ma.xE\\u{t,-)\\jj„+E f \\q{t,-)\\jir„dt
B 8) + Yl ^ f f {AD{d"u),D{d°'u))dxdt
\a\<m ■'° ■'^'^
<CE{l^\\fit,-)fHr..dt + \\grH„],
where the constant C > 0 only depends on m, T, K^ and eq.
In addition, if A is uniformly positive definite, i.e., A.9) holds for some
S > 0 (this is the case if A.1) is super-parabolic, i.e., A.7) holds), then
B.8) can further be improved to the following:
max
B.9) *^
i^xE\\u{t,-)\\j,^+E f {\\u{t,-)\\j,^^,+Mt,.)\\j,^}dt
[O.-'J Jo
<CE{l^\\f{t,-)\\],„..dt + \\9\\],^].
We note here that conditions B.6) and B.7) together with A.9) are
still weaker than the super-parabolicity condition A.7). For example, if
n > d and B is an (n x d) matrix, then BB^ is always degenerate. We can

§2. Well-posedness of linear BSPDEs 109
easily find an A such that B.6), B.7) and A.9) hold but A.7) fails. Let us
also note that if B.6) or B.7) holds, we have
B.10) \B'^^\^ <{A^,^), y^eWi", a.e.{t,x)e[0,T]xB", a.s.
Thus, B.8) follows from B.4) easily.
In the above theorems, we have assumed that / and g are square inte-
grable in a; £ R" globally. This excludes the case that / and g approach
infinity as |a;| goes to infinity. In some important applications, such a case
happens very often. Thus, in the rest of this section, we would like to
extend the above theorems a little further so that / and g are allowed to have
certain growth as |a;| —> oo. To this end, let us make an observation.
Suppose {u,q) is an adapted classical solution of A.1). Let A > 0 and denote
(x) = ^/\x\^ + l. Set
B.11) ' _ y ' it,x)e[0,T]xK".
Then, by a direct computation, we see that {v,p) satisfies the following
BSPDE: (compare with A.1))
dv = { -^V-iADv) - (a,Dv)-cu - V-(Bp) - {b,p) -J}dt
B-12) < +{p,dWit)), it,x)e[0,T]xK",
with
B.13)
%=T
5 = c+^(^[^],[^])+^V-(^[^])+A(a,[^]),
[7 = e-^^^f{t,x), g = e-<^^g{t,x).
Conversely, if {v,p) is an adapted classical solution of B.12), then {u,q),
which is determined through B.11), is an adapted classical solution of A.1).
Clearly, the same equivalence between A.1) and B.12) holds for adapted
strong and weak solutions, respectively.
On the other hand, from B.13) we see easily that the group
{A, B,a,h, c} satisfies (H)„ if and only if {A, B,a,b, c} satisfies (H)„. This
is due to the fact that for any multi-index a, it holds that
\d°'{x)\<C, \/xeK",
with the constant C > 0 only depending on \a\. Hence, from Theorems 2.1
and 2.2, we can derive the following result.

110 Chapter 5. Linear, Degenerate BSPDEs
Theorem 2.3. Let m > 1 and (H)m hold for {A,B,a,b,c}. Let A.6) and
B.2) hold. Let A > 0 such that
U-^<-^feLlriO,T;H'-OR")),
le-^<>5ei^,(fi;H'"(iR")).
Then BSPDE A.1) admits a unique adapted weak solution {u, q), such that
the following estimate holds:
B.15) |a|<'n
+ \B'^{D[d"ie-^''-^u)]} + d°'{e-^<-U)f\dxdt
<CE{l^^\\e-'(-^f{t,.)\\j,^dt + \\e-'<-^9\\j,^},
where the constant C > 0 only depends on m, T and Km-
Furthermore, if m > 2, the weak solution {u, q) becomes the unique
adapted strong solution of A.1); and if m > 2 + n/2, then {u,q) is the
unique adapted classical solution of A.1).
In the case that B.2) is replaced by B.6) or B.7), the above conclusion
remains true and the estimate B.15) can be improved to the following:
max^E\\e-^<K{t,-)\\lr„+El \\e-^<U{t,-)\\lr„dt
B16) + Yl ^ [ [ {AD[d"ie-^<-^u)],D[d"{e-^<-^u)])dxdt
Finally, if in addition, A.9) holds for some S > 0, then B.16) can further
be improved to the following:
max^E\\e-'<-^uit,-)\\]j„
B.17) +E r {\\e-^(-)uit,W„r^^^+\\e-'^-^qit,WH'^}dt
JO
< Ce{ j^ \\e-^(-) fit,-)\\l„.,dt+\\e-'()g\\%„}.
Clearly, B.14) means that / and g can have an exponential growth as
I a; I —> oo. This is good enough for many applications.

§3. Uniqueness of adapted solutions 111
We note that {A,B,a,b,c} satisfies (H)^ if and only if {A,B,a,b,c}
satisfies (H)^, where a and b are given by A.4). Thus, we have the exact
statements as Theorems 2.1, 2.2 and 2.3 for BSPDE A.5) with a and b
replaced by a and b.
§3. Uniqueness of Adapted Solutions
In this section, we are going to establish the uniqueness of adapted weak,
strong and classical solutions to our BSPDEs. From the discussion right
before Proposition 1.3, we see that it suffices for us to prove the uniqueness
of adapted weak solutions.
§3.1. Uniqueness of adapted weak solutions
For convenience, we denote
2 L J \ ' /
Mq = ViBq] + (b,q).
Then, equation A.1) is the same as the following:
j du = -{Cu + Mq + f}dt + (q,dW{t)), {t,x) e[0,T] x R",
In this section, we are going to prove the following result.
Theorem 3.1. Let B.3) hold and the following hold:
■ AeLf{0,T;L°°{B";S")),
B e L^@,r;L°°(R";R"^'*)),
C.3) i aeL^@,r;L°°(R";R")),
6eL^@,r;L°°(R";R'*)),
.ceL^@,r;L°°(R")).
Then, the adapted weak solution (u, q) of C.2) is unique in the class
UeCH[o,r];L2(ft;Hi(]R»))),
\qeLlr{0,T-L^B"-B.'')).
To prove the above uniqueness theorem, we need some preliminaries.
First of all, let us recall the Gelfand triple H^ (R") ^ L^iB") ^ H'^ (R").
Here, if~^(R") is the dual space of if^R"), and the embeddings are
dense and continuous. We denote the duality paring between H^ (R") and
if~^(R") by (■,-)o, and the inner product and the norm in L^(R") by

112 Chapter 5. Linear, Degenerate BSPDEs
(■,-)o and I ■ |o, respectively. Then, by identifying L^(]R") with its dual
L^(]R")* (using Riesz representation theorem), we have the following:
C.5)
and
C.6)
(^>¥')o = (t/',¥')o
= [ iP{x)ipix)dx, VipeL'^iK"), ipeH'iK"),
J2dii;i e H-'{B"), \/A e L\K"), l<i<n,
i=l
{J2dii^i,v)o = -f A{x)dMx)dx, v<^eHi(]R")
Next, let {u,q) be an adapted weak solution of C.2) satisfying C.4).
Note that in C.4), the integrability of {u,q) in x is required to be global.
By C.5)-C.6), we see that
C.7) £u + 7WgeL^@,r;if-i(]R")).
In the present case, from A.15), for any 93 £ H^{WC) (not just C^(]R")),
we have
{d{u,^)o = -{Cu + Mq + f,^)o + {{qMo,dW{t)), te[Q,T],
Here, {q,^p)o={{q\,v)o,-•■ A<ld,^)o) and q = {qi,-■ ■ ,qd)- Sometimes, we
say that C.2) holds in if-i(]R") if C.8) holds for all 93 £ ifi(]R").
In proving the uniqueness of the adapted weak solutions, the following
special type of Ito's formula is very crucial.
Lemma 3.2. Let (, £ L^@,r;if-i(]R")) and {u,q) satisfy C.4), such that
C.9) du = ^dt + {q,dW{t)), te[0,T].
Then
C.10)
Ht)\l = HO)\l + f {2 (C(s),u(s) )o + \q{s)\l}ds
Jo
+ 2 [\{q{s),u{s))o,dW{s)), te[0,T].
Jo
Although the above seems to be a very special form of general Ito's
formula, it is enough for our purpose. We note that the processes u, q and ^
take values in different spaces if ^(R"), L^(]R") and H~^{Wl"), respectively.
This makes the proof of C.10) a little nontrivial. We postpone the proof of
Lemma 3.2 to the next subsection.

§3. Uniqueness of adapted solutions 113
Proof of Theorem 3.1. Let (u,g) be any adapted weak solution of C.2)
with / and g being zero, such that C.4) holds. We need to show that
{u,q) = 0, which gives the uniqueness of adapted weak solution. Applying
Lemma 3.2, we have (note C.7))
E\u{t)\l = E j {2{Cu{s)+Mq{s)Ms))o - \q{s)\l}ds
= EI I \-(ADu,Du) + (a,D{u^))+2cu'^
- 2 (q,B'^Du) +2 (bu,q) -\qf\dxds
^^"^^^ =Er f {-{iA-BB^)Du,Du)
-\q + B'^Du - bu\^
+ [b^ + 2c - V-(a + Bb)]u'^^ds
<C I E\u{s)\lds, te[0,T].
By Gronwall's inequality, we obtain
E\{t)\l = o, te[0,T].
Hence, u = 0. By C.11) again, we must also have g = 0. This proves the
uniqueness of adapted weak solutions to C.2). D
§3.2. An Ito formula
In this subsection, we are going to present a special type of Ito's formula
in abstract spaces for which Lemma 3.2 is a special case.
Let V and H be two separable Hilbert spaces such that the embedding
y "^ if is dense and continuous. We identify H with its dual H' (by Riesz
representation theorem). The dual of V is denoted by V. Then we have
the Gelfand triple V "—^ H = H' "-^V. We denote the inner product and
the induced norm of ii by (■, Oo and | ■ |o, respectively. The duality paring
between V and V is denoted by (■, ■ )o, and the norms of V and V are
denoted by || -1| and || ■ ||*, respectively. We know that the following holds:
C.12) (u,u)o = (u,u)o, yueH,veV.
Due to this reason, H is usually called the pivot space. It is also known (see
[Lions]) that in the present setting, there exists a symmetric linear operator
Ae £(V,V')> such that
C.13) {Av,v)q<-\\v\\^, yveV.
Now, let us state the following result which is more general than Lemma
3.2.

114
Lemma 3.3. Let
C.14)
satisfying
C.15)
Then
C.16)
Chapter 5. Linear, Degenerate BSPDEs
■ueCA[0,T];V),
du = ^dt + {q,dW{t)), te[0,T].
Ht)\l = HO)\l + f {2{mMs))o + \q{s)\l]ds
Jo
+ 2 [ ({q{s),u{s))o,dW{s)), te[0,T].
Jo
In the above, q £ L'^jr{0,T\HY means that q = (gi,■ ■ ■ ,gd) with g, £
L'^{0,T;H). In what follows, we will see the expression q e L'jr{0,T;Vy
whose meaning is similar. Before giving a rigorous proof of the above result,
let us try to prove it in an obvious (naive) way. Prom C.16), we see that
the trouble mainly comes from ^ since it takes values in V. Thus, it is
pretty natural that we should find a sequence ^^ £ L^@,r;if), such that
C.17) Cfc^C, inL^@,r;y'), (A; ^ oo),
and let Uk be defined by
C.18) Uk{t)=u{0)+ f ^k{s)ds+ f {q{s),dW{s)), t e [0,T].
Jo Jo
Since the processes Uk, ^k and q are all taking values in H, we have
\ukit)\l = \uiO)\l + [ {2Ukis),Ukis))o + \qis)\l}ds
Jo
+ 2 / {{q{s),Uk{s))o,dW{s)), te[0,T].
Jo
This can be proved by projecting C.18) to finite dimensional spaces, using
usual Ito's formula, then pass to the limit. Having C.19), one then hopes
to pass to the limit to obtain C.16). This can be done provided one has
the following convergence:
Uk^u, inL^@,r;y).
However, C.17)-C.18) only guarantees
Uk^u, in L%iO,T;V').
Thus, the convergence of Uk to u is not strong enough and such an approach
does not work! In what follows, we will see that to prove C.16), much more
has to be involved.
C.19)

C-20) , . _2,
§3. Uniqueness of adapted solutions 115
Let us now state two standard lemmas for deterministic evolution
equations whose proofs are omitted here (see Lions [1]).
Lemma 3.4. Let v : [0,T] —> V be absolutely continuous, such that
(veL\0,T;V),
[i) eL'^{0,T;V').
Thenv eC{[0,T];H) and
C.21) j^\v{t)\l = 2(v{t),v{t))o, a.e.te[0,T].
Lemma 3.5. Let A £ C{V,V') be symmetric satisfying C.13). Then for
any vq e H and f e L^{0,T;V'), the following problem
C.22) lv = Av + f, te[0,T],
[ v{0) = vo,
admits a unique solution v satisfying C.20) and
C.23) \v{t)\l + [ \\v{s)fds < \vo\l + f \\f{s)\\lds, t £ [0,r].
Jo Jo
Moreover, it holds
C.24) \vit)\l = \vo\l+2 f {Avis) + fis),vis))ods, te[0,T].
Jo
Now, we consider stochastic evolution equations. We first have the
following result.
Lemma 3.6. Let v be an {Tt}t>o-3,dapted V'-valued processes which is
absolutely continuous almost surely and q bean {Tt}t>o-s^dapted H-valued
process such that the following holds:
C.25)
■veL'^iO,T;V),
i e L%iO,T;V'),
^qeLlr{0,T;Vf.
Let
C.26) M{t)= f {q{s),dW{s)), t£[Q,T].
Jo
Then, M £ Cjr{[0,T];V) and
C 27) \M{t)\l = 2 j\{M{s),q{s))o,dW{s)) + j^ \q{s)\lds,
te [0,T], a.s.

116
C.28)
Chapter 5. Linear, Degenerate BSPDEs
d{v{t),M{t))o = {v{t),M{t))odt + ({v{t),q{t))o,dW{t)),
a.e.i e [0,r], a.s.
Proof. First of all, it is clear that M e Cjr{[0,T];V) and C.27) holds
since we may regard both M and q as if-valued processes. We now prove
C.28). Take a sequence of absolutely continuous processes Vk with the
following properties:
C.29)
f«fceL^(o,r;y),
VkeL%iO,T;H),
Vk^v, inL^@,r;y),
Now, in H, we have (note C.12))
f3 30) d{^k{t),M{t))o = ivk{t),Mit))odt+{{vk{t),q{t))o,dWit))
= {vk{t),M{t))odt + {{vk{t),q{t))o,dW{t)).
Pass to the limit in the above, using C.29), we obtain C.28). D
Lemma 3.7. Let A £ £(y, V) be symmetric satisfying C.13). Then, for
any /, q, Uq satisfying
C.31)
the following problem
C.32)
-/eL^(o,r;y'),
qeLlr{Q,T;HY,
^uo eH,
du={Au +f)dt + {q,dW{t)), t e [0,T],
u@) = uo,
admits a unique solution u £ L^{0,T; V) fl C^([0, T];H), such that
C.33)
K*)lo = ko|o+ / {2{Auis) + fis),uis))o + \qis)\l}ds
JO
+ 2 f {{q{s),u{s))o,dW{s)), Vte[0,r], a.s.
Jo
Proof. We first let q £ Llr{0,T;V)^ and define M{t) by C.26).
Consider the following problem:
C.34)
v = Av + f + AM, te[0,T],
v{0) = uq.

§3. Uniqueness of adapted solutions 117
By Lemma 3.5, for almost all weft, C.34) admits a unique solution v.
Obviously (by the variation of constants formula, if necessary), v is {J-'t}t>o-
adapted. Thus, we have
UeL^(o,r;y),
[v e L%iO,T,V'),
which implies (by Lemma 3.4) v e Cjr{[0,T];H) and (by C.24))
C 35) Ht)\l = \Ml + ^j {Avis) + fis)+AMis),vis))ods,
Vte [0,T], a.s.
Set uit) = v{t) + M{t). Then, we see that u e LjriO, T; V) n Cjr{[0, T]; H)
is a solution of C.32). We now combining C.27)-C.28) and C.34)-C.35)
to obtain the following:
Ht)\l = \v{t)\l + \Mit)\l + 2{v{t),M{t))o
= |uo|o+2 / {Av{s) + f{s) + AM{s),v{s))ods
Jo
+ 2 [ {iMis),qis))o,dWis))+ f \q{s)\lds
Jo Jo
+ 2 [ (v{s),M{s))ods + 2 [ {{v{s),q{s))o,dW{s))
Jo Jo
C.36) =\uo\l+2 (Au{s) + f{s),v{s))ods
+ 2 [ ({u{s),q{s))o,dW{s))+ [ \q{s)\lds
Jo Jo
+ 2 f {Auis)+fis),Mis))ods
Jo
= Klo+ f {2{Auis) + fis),uis))o + \q{s)\l}ds
Jo
+ 2 [ ({u{s),q{s))o,dW{s)).
Jo
Next, we claim that solution to C.32) is unique (for any /, q and uq
satisfying C.31)). As a matter of fact, if u is another solution to C.32), then
u - u is a solution of C.32) with /, q and uq all being zero. Applying C.36)
to u — u, we obtain (see C.13))
\u{t)-u{t)\l = 2 {A[u{s)-u{s)],u{s) -uis))ods < 0,
Jo
which results in u = u. Thus, we have proved our lemma for the case
q e LlriO,T;Vy. Now, for general case, i.e., q £ Ljr{0,T;H)'^, we take a

118 Chapter 5. Linear, Degenerate BSPDEs
sequence Qk £ L'jr{0,T;Vy with
Qk^q, mLl{0,T;HY.
Let Uk be the solution of C.32) with q being replaced by g^. Then applying
C.36) to Uk - ui, we have (note C.13))
E\ukit) - utit)\l +2E f \\ukis) - utis)\\''ds
Jo
C.37)
<E [ \qk{s) - qtis)\lds ^ 0, k,i
Jo
oo.
This means that the sequence {uk} is Cauchy in ^^^@, T; V)r)Cjr{[0, T]; H).
Hence, there exists a limit u of {uyk in this space. Clearly, u is a solution
of C.32). Also, we have a similar equality C.33) for each Uk- Pass to the
limit, we obtain the equality C.33) for u (with general q £ L'jr{0,T;H)'^).
n
Now, we are ready to prove Lemma 3.3.
Proof of Lemma 3.3. Set
(uo= u@) e H,
\f^^-AueLUO,T;V').
Then u is a solution of C.32) with C.31) holds. Hence, C.33) holds, which
yields C.16). D
Now, by taking V = H^{K"), H = L^{B.") and V = if-i(lR"), we see
that Lemma 3.2 follows immediately from Lemma 3.3.
§4. Existence of Adapted Solutions
The proofs of existence of adapted solutions is based on the following
fundamental lemma.
Lemma 4.1. Let the parabolicity condition A.6) and the symmetry
condition B.2) hold. Let (H)m hold for some m > 1. Then there exists a
constant C > 0, such that for any u e C§°{Wl") and q £ C§°{Wl";Bf), it
holds
L
{ Y. {{iA-BB^)Did"u),D{d"u))
*^ l«l<
+ \B'^D{d°'u) + d^ql^} + Yl \d"'l\^}dx
D.1) |a|<m-l
<C f Yl {-'2{d"u)d"iCu + Mq) + \d"q\'^ + \d°'u\'^yx,
•'^^ \a\<m
a.e.t e [0, T], a.s.

§4. Existence of adapted solutions 119
If B.6) or B.7) holds instead of B.2), the above can be replaced by the
following:
L
{ Y. {AD{d'^u),D{d'^u))+ Y. \d'^q\''}dx
\ot\<m \a\<m
D.2) ^^ j Y [-2{d"u)d°'{Cu + Mq) + \d"q\^+ \d"u\^'^dx,
" |a|<m
a.e.i e [0,r], a.s.
Furthermore, if B.6) or B.7) holds and A{t, x) is uniformly positive definite,
then D.2) can be improved to the following:
i
\a\<m+l l«l<'n
D.3) ^^ r Y^ i^-2{d"u)d"{Cu + Mq) + \d"q\'^ + \d"u\'^yx,
|a|<m
a.e.i e [0,r], a.s.
We note that the square root of the left hand side of D.1) is a norm
in the space C^(]R") x C^{B"; WC^). Thus, if we denote the completion of
the space C^(]R") x C^(]R";]R'*) under this norm by 'H"'{t,u)) (note that
it depends on (i, w) £ [0,r] x ft), then we have the following inclusions:
Co°°(lR") X Co°°(]R";]R'*) cW^it,^) C ^'"(IR") x H"'-\K";K'^).
It is clear that estimate D.1) also holds for any {u, q) £ W^{t,ijj). A similar
argument holds for D.2) and D.3).
Since the proof of the above lemma is rather technical and lengthy, we
postpone its proof to the next section.
Before going further, let us recall the following fact concerning the
differentiability of stochastic integrals with respect to the parameter. Let
h e L^@,r; C^(R";Wl^)). Then it can be shown that the stochastic
integral with parameter: Jg (h{s, x, ■),dW{s)) has a modification that belongs
to L^jr{0, T; C™"^(]R"; R™)) and it satisfies
M.^ ^" I {h{s,x,-),dW{s)) = f {d"h{s,x,.),dW{s)),
14.4J Jo Jo
for \a\ = 1,2, • • •,m — 1.
Consequently, ii h e LJriO, T; C^), then
/ {his,- ,-),dWis)) e L':^{0,T;Cn,
Jo
and D.4) holds for all multi-index a.

120
Chapter 5. Linear, Degenerate BSPDEs
In the rest of this section, we prove the existence of adapted weak
solutions to A.1) (or equivalently C.2)) under conditions of Theorems 2.1
or 2.2.
We first assume that conditions of Theorem 2.1 hold.
Let us take an orthonormal basis {ipk}k>i C Co°(lR") for the Hilbert
space if" = if'"(R"), whose inner product is denoted by
\a\<m
The induced norm is denoted by
{pi,---,Pd) e {H"'Y, we denote
When q = {qi,---,qd), P =
iq,p)m = Yl^li'P'^'
i=l
which should not be misunderstood from the context. As a usual
convention, ii° = L^(]R"). Let A; > 1 be fixed. Consider the following Unear
BSDE (not BSPDE):
D.5) {
du'^it) = {-Yl [i^VuVjUu^'it) - {{M^u^jUq^'it))]
-if,V>,)m}dt + {q''Ht),dWit)),
U'=^(r) = E,(^,-)m, l<j<k.
By the result of Chapter 1, we know that there exists a unique adapted
solution
D.6)
We define
D.7) <
u''^i-)eCA[0,T];K),
g'=^(-)eiMo,r;]R'^),
1 < i < A-
k
q''{t,x,Lj) = Y^q''^{t,Lj)ipj{x),
{t,x,Lj) e [0,T] xR" X ft.
Then we see that for any fixed {t,Lj) £ [0, T] x ft,
u''it, •, w) e Co°°(]R"), q''it, • ,w) e C^iK";^).
Also, the following holds:
( du'' = { - PkiCu'' + Mq''] - f''}dt + {q'',dW{t)),
D.8)
\t=T
= 9 ,

§4. Existence of adapted solutions 121
where
Ffc : H™ ^ span{931,•••,<^fc}^H,™,
is the orthogonal projection (in if"), and
/' = Pkf, g" = Pk9.
Note that, as processes, u* and gf,• • • ,9* ^^'^ taking values in if™, where
g* = (gf, • • • ,qj). Thus, in particular,
D.9) FfcU* = u*. A; > 1.
Next, we want to derive a proper estimate for (u*,g*). By Lemma 4.1, we
have the following:
I { Y. {{{A-BB'^)D{d"u'^),D{d"u''))
•'^" \a\<m
+ \B'^D{d"u'')+d°'q''\^]+ Yl \d°'(l''?}dx
D.10) |a|<m-l
<C j Yl {-2(9«u*)a«(£u* + 7Wg*) + |aV|^
+ |a"u*|2}(ia;.
On the other hand, applying Ito-Ventzel's formula to |9"u*p, we have from
D.8) that
•'^^ \a\<m
+ \d°'q''\^}dxds
D.11) =£;y {_2(u^Ffc(£u'=+A/lg'=) + /'=)„ + |g'=|^}ds
= e/ / y] {-2{d°'u'')d"iCu''+Mq'')
+ |a"g*|2 - 2{d°'u''){d°'f'')}dxds.
The third equality in above is due to D.9). Then combining D.10)-D.11),

122
we obtain
Chapter 5. Linear, Degenerate BSPDEs
[ { Yl {{iA-BB^)D{d"u''),D{d"u'
'))
\a\<m
+ \B^D{d"u'')+d"q''\'^}+ Yl I^Vl^jda;
|a|<ra-l
D.12) <c{E\g>'\l-E\uHt)t
+ ^ Yl [ [ ['2id"u'')id"f'') + \d"u''\'^]dxds}
< c{E\g>'t - E\u''it)t + j'^ E[\u>'{s)\l + \f''{s)\l]ds}.
By Gronwall's inequality, we obtain
m^ E\u\t)\l + E f \q\t)\l_,dt
+ Y^f f {{{A-BB'^)D{d"u''),D{d"u''))
D.13) W\<m ■'° ■'^'^
+ B^[Z)(a"u*)]+aV| }dxdt
<CE{j\f''it)t + \g''t}.
Note that the constant C > 0 in D.13) only depends on T, m and K„
Prom D.13), we may assume that
ju''^u, weak* mL'^iO,T;L^in;H^)), 0<i<m,
[q'' ^q, weakly in L^@, T; hY, 0<i<m-l,
and for any \a\ < m,
{A - BB'^y/'^D{d"u'') -^{A- BB'^f/'^D{d°'u),
D.15)
B^[Z)(a«u*)] + d°'q'' -^ B'^[D{d°'u)] + d^q,
weakly in L^@,r;if°).
By taking limits in D.13), we see that {u,q) satisfies the estimate B.4)
with the constant C > 0 only depending on T, m and K^n- We are going
to prove that (u, g) is a weak solution of C.2). To this end, let us take
peH^ @, T) such that
D.16)
■p@)=0, p(r) = l,
o</5(i)<i, te[o,T\.

§4. Existence of adapted solutions
123
D.18) 2
{{a{t),Du\t)) +c{t)u\t) + {b{t),q\t)) +f''{t),^)m]}dt
a.s.
Let ^ > 0 be fixed and k>£. For any 93 e iff C C§°{Wi"), from D.8) and
the fact Pkip = ip, we have
D.17) - pmCu'it) + Mq\t) + f'it), ^U}dt
+ f p{t){{q\t),ip)m,dW{t)).
Jo
By the definition of £ and M, using integration by parts, we obtain
- Pit) [ - liAit)Du''it) + B{t)q\t), D^)r
+
+ [\{t){{qHt),¥>)m,dW{t)),
JO
If we denote
F{x,w)=g- f {muHt) - pit)[{a{t),Du''{t)) +c{t)u''it)
Jo
+ {bit),q''{t))+f>'{t)]}dt- f {p{t)q\tUW{t)),
Jo
G{x,Lj)= [ p{t)[lAit)Du''{t) + B{t)q''{t)]dt,
Jo ^
then D.18) reads
D.20) {F,^U = {G,D^)^, V<^ e Co°°(]R"), a.s.
By the lemma below, we must have
D.21) {F,ip)o = {G,Dip)o, V(^ e Co°°(]R"), a.s.
This tells us that
{9\^)o = J [m{u''{t),^)o
D.22) - p{t){Cu\t) + Mq\t) + f''it),ip)o}dt
+ f p{t){{q\t),^)o,dW{t)),
Jo
D.19) <
a.s.
We want to pass to the limit in D.22) to obtain a similar equality for (u, q).
By D.14) with ^ = 1 for u* and ^ = 0 for g*, together with the convergence

124 Chapter 5. Linear, Degenerate BSPDEs
of (/*, 5*) to (/, g), we can pass to the limit in D.22) weakly in L^(ft) for
all terms except the last term which is involving the Ito integral. To treat
this last term, we define K : Llr{0,T;H°)^ -^ L^{n) by
D.23) Kp= [ p{t){{p{t),^)o,dW{t)), ypeL'^{0,T;H°
Jo
Then
D.24)
E\Kp\^ = E I \p{i){p{t),^U^dt
JO
<MlE j \p{t)\ldt, VpeLlriO,T;HY-
Jo
This means that if is a bounded linear operator. Thus, for any rj £ L^(ft),
one has
D25) ^('^i pit){iq''{t)-q{t),^)o,dW{t)))
= {'n,K{q'' -q))mn) = (^*»7,9* -9)L2.(o,T.ffO)d ^ 0.
Thus, we obtain
i9,V>)o=l \^pit)iu{t),ip)o
D.26) - p{t){Cu{t) + Mq{t) + f{t),ip)o]dt
+ I p{t){{q{t),^)o,dW{t)), a.s.
Jo
Now, fixed any t e @, T). For any e > 0, we let
'0, s<t- e/2,
D.27) p,(s) = ii + ^, t-e/2<s<t + e/2,
.1, s>t + e/2.
Choosing p = p^ in D.26) and letting e —> 0, we obtain
D.28)
{9,v)o = iuit),v)o - f {Cu{t)+Mq{t) + fit),^)odt
I ({q{t),^)o,dW{t)), V93 e Co°°(]R"), a.s
+
It
This means that {u,q) is an adapted weak solution of C.2). By Theorem
3.2, it is unique.
In the case that m > 2, from B.4), we see that A.13) holds and thus,
by Proposition 1.3, {u,q) is an adapted strong solution C.2). In the case
m > 2 + n/2, by Sobolev's embedding theorem, A.11) holds and therefore,
(u, q) is an adapted classical solution of C.2) by Proposition 1.3 again.

§4. Existence of adapted solutions 125
Finally, let us look at the case when conditions of Theorem 2.2 hold.
Suppose B.6) or B.7) holds instead of B.2). By Lemma 4.1, we still have
D.1), which is now equivalent to D.2). Then all the proof that we have
presented above remains true. Moreover, we have estimate B.8). In the
case that A is uniformly positive definite, a little more careful estimate leads
to B.9). In fact, for the present case, in D.12), we can use integration by
parts to get
rT
D.29) l«l^™ *
E y] [ [ 2{d°'u'')id°'f'')dxds
<CeI^ {In" is) t^. + lf is) t_,}ds.
We leave the details of the proof to the interested readers. D
We now prove the following lemma which has been used in the above
proof.
Lemma 4.2. Let F e ^'"(R") and G e ^'"(R")", such that
D.30) iF,^U = iG,D^U, V93 e Co°°(]R").
Then
D.31) (F, ip)o = (G, Dv)o, Vip e Co°°(]R").
Proof. Let S = S{B") be the set of all 93 £ C°°(]R"), such that
D.32) ^a,0i^)= sup Ix^d^ifiix)] < 00, Va,/3.
Under the family of semi-norms ^a,0, >S is a Prechet space. Also, Cq° (R") is
a dense subset of S. Thus, D.30) holds for all 93 £ 5. Next, by Hormander
[1, p. 161], Fourier transformation 93 1-4 ^ is an isomorphism of S onto itself.
Applying Parseval's formula to D.30), we obtain
D.33) f [f(o-(g@,o]( E n')^(OdC = o, y^ec^iu").
" |a|<m
Now, for any t/; £ Co°°(]R"), we have $iO{E\a\<J^"?) ' £ ^- Thus,
there exists a. tp E S, such that
D.34) (pio = m{ E i^"i')"'-
\a\<m
Combining D.33) and D.34), using Parseval's formula again, we obtain
(F,t/,)„ = {G,Di,U, Vt/; e Co°°(]R").
This proves our lemma. D

E.1)
126 Chapter 5. Linear, Degenerate BSPDEs
§5. A Proof of the Fundamental Lemma
In this section, we are going to prove Lemma 4.1, which has played an
essential role in the proof of well-posedness theorems for A.1).
Proof of Lemma 4-1- Let ^=|a| < m. For any u £ Co°(lR") and
q e Co°(lR."; H'^), by definition of £ and M, and differentiation, we have
X« ^ /" I - 2{d°'u)d°'{Cu + Mq) + \d°'q\^^dx
= [ {-2{d°'u)d°'\lv-iADu) + (a,Du)+cu
+ V-[Bq] + {b,q)j+\d"q\''yx
= / I - 2id"u) \l V-[AD{d"u)] + (a, D{d"u)) +c(a"u)
+ V-[B(a«g)] + F,a«g)] +\d"q\''
O<0<a
+ (a«-^a,z)(a^u))+(a«-^c)(a^u)
+ V-[{d"-'^B)id>^q)] + {d"-H,d^q) }] jda;
where Cap is a positive integer depending on a and /3, and
Io« = /" I - 2(a«u) [i V-[ylZ)(a"u)] + (a, Z)(a«u)) +c{d"u)
E.2) -^R" •• '-2
+ V\B{d"q)] + {h,d"q)]+ \d"q\'']dx,
E.3) *^" o<^<«
+ (d^-^a, D{d^u)) +(a"-^c)(a^u) + (d^-^b, d^q) ] dx,
E.4) I^ = -2f E C„0(a«u)V.[(a«-^B)(a^g)]da;,
•'E-" 0</3<c<
l/3|<l»|-l
E.5) l3« = -2/ E C„0(a«u)V.[(a«-^B)(a^g)]d.T:.
•'E-" 0</3<c<
We note that in the case ^ = 0,1", I" and I" are all absent. We now treat
Xq*) Xf, I" and X", separately.

127
§5. A proof of the fundamental lemma
Since A and B are C™"*"^ in x, we see immediately that
E.6) \I^\ + \I^\<C{\u\j + \q\U)-
Now, let us look at Iq and I^. Using integration by parts, we have
I«= f \ {ADid"u),Did"u)) +2{d"q,B^D{d"u)) +\d"q\^
- (a, D[{d"uf]) -2c{d"u)^ - 2 F(a«u), d"q) jda;
E-'^) = [ \{{A-BB^)Did"u),D{d"u))+\B^Did"u)\^ + \d"qf
+ 2 (a«g, B^D{d"u)) -2 (b{d"u),d"q)
- 2{B'^D{d"u), h{d"u)) +[V-(a - B6) - 2c](a«uJ}(ia;.
In the meantime, let us look at each term in I^. For /3 < a with |/3| = |a| —!>
using integration by parts, we have
- / {d"u)V-[{d"-'^B){d'^q)]dx
Jr"
= j (d'^u) v-{d"->^[{d"-^B)id^q)]}dx
E.8) •^*^"
= - / {Did^u),{d"-'^B)d"q + id^'^°'-^^B)d^q)dx
/R"
=- / [({d"-^B'^)D{d^u),d"q) + {D{d>^u),{d''^"-'^^B)d<^q))]dx.
Jr"
Thus, it follows that
I«+l3«= f \{{A- BB'^)D{d°'u),D{d°'u))
^R" *•
B^Z)(a«u)+a«g-6(a«u)- ^ Ca0{d"-'^B^)D{d>^u)
+
O<0<a
l/3| = |a|-l
-|6(a"u)|2-| J2 Ca0{d"-^B^)D{d'^u)\
0<l3<a
E.9) +2 ^ C„0((a«-^B^)Z)(a^u),B^Z)(a«u))
0</3<c<
-2 ^ c„0((a«-^B^)z)(a^u),6(a«u))
0</3<c<
+ [V-ia-Bb)~2c]id°'uf
-2 Yl Ca0{D{d>'u),{d^^"->'^B){d>'q))}dx.
0<l3<a
l/3| = l»|-l

128
Note that
Chapter 5. Linear, Degenerate BSPDEs
L
B'^D{d"u) + d^q - b{d"u)
J2 Cc0id"-^B^)D{d^u)
E.10)
0</3<c.
I/3| = I»|-1
dx
B'^D{d"u)+d°'q\ dx
JR" 0</3<c<
>
s/r"
0</3<<J
l/3| = l»|-l
dx
B'^D{d"u)+d"q dx-C\u\j.
|2
Next, by the symmetry condition B.2), for P < a, \C\ = \a\ — 1, we have
/ {{d"-f^B'^)D{df^u),B'^D{d"u))dx
= { {iBd"->^B^)D{d>^u),Did"u))dx
^R"
E.11) = f l\d"-^ ({Bd"-^B^)D{d'^u),D{d<^u))
JR" 2 L
- (d°'-^iBd"-^B'^)Did^u), D{d^u)) ] dx
= -\ I {d"~'^{Bd"->^B^)Did^u),D{d^u))dx > -qui:
2 Jr"
Combining E.6), E.9)-E.11) yields
I" = 2o" + Ii" +12" +13"
E.12) ^ j^\{{A- BB'^)D{d"u),D{d"u))
+ \\B'^D{d"u) + d'^q\^}dx - C{\u\\ + |g||„i).
Now, we sum E.12) up for all \a\ < ^ to get the following:
^e=J2f {' 2(a«u)a«(£« + Mq) + \d"q\^}dx
\a\<( ^"
E.13) ^IYI [ {{iA-BB'')Did"u),Did"u))
l«|<r
+ |B^Z)(a«u) + d"q\^]dx - C{\u\\ + |g|Li)-

§5. A proof of the fundamental lemma 129
Thus, it follows that
^e=Yl [ {{{A~BB^)D{d"u),D{d"u))
^^•^^) + \B^D{d"u) + d"q\'^yx
<c[^e + \u\j + \q\U).
Note that
E.15) \d"q\'^ < 2\B^Did"u) + a"g|2 + 2\B^Did"u)\'^.
Using the parabolicity condition A.6) and the definition of $^ (see E.14)),
we have
E.16) \q\j_, < C{^t-i + \u\j).
Consequently, from E.14), we obtain
E.17) ^e<C{^t + ^e^i + \u\j), l<i<m.
On the other hand, for ^ = 0 (i.e., a = 0), we have
/ \^-2uiCu+Mq) + \q\^\dx
= \-2u\-V-[ADu] + {a,Du)+cu
+ ViBq] + {b,q)]+\q\^]dx
E.18) ^ /■ \{(A-BB^)Du,Du)+\B^Duf + \qf
+ 2(q,B'^Du)-2{bu,q)
- 2 (B^Du,6u) +[V-(a - Bb) - 2c]u'^\dx
> [ {({A-BB'^)Du,Du)+\B'^Du + qfdx-C\u\l.
This implies
E.19) $0 < / [-2uiCu + Mq) + \q\'^^dx + C\u\l.
Hence, it follows from E.17) and E.19) that
E.20) $„<C(*„ + H^),
which is the same as D.1).

130 Chapter 5. Linear, Degenerate BSPDEs
In the case that B.6) holds, we use the following estimate:
f {{d°'-'^B'^)D{d^u),B'^D{d"u))dx
E.21) •^*^"
> -e / \B'^D{d"u)\^dx - C I \D{d'^u)\^dx,
for small enough e > 0 to get
/ {{{A- BB'^)D{d°'u),D{d°'u))
f5 22) +^ ^ Ca0{id"-^B^)Did^u),B^Did'^u))
^ ' ' O<0<a
l/3| = |a|-l
>~f {{A- BB'^)D{d"u), D{d"u)) dx - C\u\
2 ^R"
Then, we still have E.14) and finally have E.20) which is the same as D.1).
In the case B.7) holds, we use the following estimate:
f {{d"-^B'^)Did^u), B'^D{d°'u)) dx
= f ({d"-^B^)D{d'^u),d"~>^[B^Did>^u)]
Jr"
- {d"-^B'^)D{d^u)) dx
E.23) ^_/" {{^d^{-0)B^)D{d^u),B^D{d^u))
+ {id^"-^^B^)Did"u), B^Did^u))
+ \{d"-^B^)D{d'^u)\'^}dx
>-e f {d"-^B'^)D{d°'u)\^dx - C\u\\
Jr"
|2
for £ > 0 small enough to obtain E.22) and finally to obtain E.20).
Note that in the case B.6) or B.7) holds, we have B.10). Then, D.2)
follows from D.1) easily. Finally, if in addition, A.9) also holds, then, D.3)
follows from D.2). This completes the proof of Lemma 4.1. D
§6. Comparison Theorems
In this section, we are going to present some comparison theorems on the
solutions of different BSPDEs. For convenience, we consider BSPDEs of

§6. Comparison theorems 131
form A.5). Let us denote (compare C.1))
Cu = -tr [AD^u] -{a,Du) -cu
F.1)
Mq^tl[B^Dq]-(b,q),
Zu = -tr [AD^u] - (S, r>u) -cu,
{Mq = tl[B'^Dq]-(b,q).
We assume that (H)„ holds for {A,B,a,b,c} and {A,B,a,b,c}. Consider
the following BSPDEs:
( du = -{Cu + Mq + f}dt + {q,dW{t)), it,x) e [0,r]x]R",
F.2) < ,
( du= -{Cu + Mq + J}dt + {q,dW{t)), it,x)e[0,T]xB",
F-3) < _,
Note that F.2) and C.2) are a little different since the operators £ and
M are defined a little differently. However, by the discussion at the end
of §2, we know that Theorems 2.1, 2.2 and 2.3 hold for F.1). Throughout
this section, we assume that the parabolicity condition A.6), the symmetry
condition B.2) and (H)„ (for some m > 1) hold for F.2) and F.3). Then
by Theorem 2.3, for any pairs (/, 5) and (/,g) satisfying B.14), there exist
unique adapted weak solutions {u,q) and {u,q) to F.2) and F.3),
respectively. We hope to establish some comparisons between u and u in various
cases.
Our comparison results are all based on the following lemma.
Lemma 6.1. Let A.6), B.2) and (H)^ with m > 1 hold. Let {u,q)
be the unique adapted weak solution of F.2) corresponding to some (/, g)
satisfying B.14) for some A > 0. Then there exists a constant /i £ R, such
that
E [ e-^^(^>|u(t,a;)-p(ia;
F.4) <e>'^^-'^E [ e~^^^^\g{x)-fdx
+ E f e''(*-*) / e-^^^^f{s,xy\'dxds, Vte[0,r].
Proof. We first assume that (/, g) satisfies B.14) with A = 0. Let
93: R —> [0,00) be defined as follows:
r^ r <-1,
F.5) ip{r) = { {6r^ + 8r* + 3r^f, -1 < r < 0,
0, r > 0.

132
Chapter 5. Lineai, Degenerate BSPDEs
We can directly check that <~p is C^ and
F.6)
<^@) = <^'@) = <^"@) = 0,
<^(-l) = l, <^'(-l) = -2, <^"(-l) = 2.
Next, for any e > 0, we let (/'^(r) = e^^p^^). Then, it holds
F.7)
Denote
F.8)
lim <{>e{r) = \r p, lim vU*") = ~2r , uniformly,
l</5"WI <C, Ve>0, r eR;
B, r<0,
lim <^;'(r) = I
^^0 ^ lo, r>0.
a = o--V-A, h = h-V-B.
Then by A.3), we have
1
F.9)
ti[AD'^u] + {a,Du) = - V-[Ai:>w] + {o,i:'w),
tr [B' Dq] + {b,q) = V-iBq) + {b,q) .
Applying the Ito's formula to ipeiu), we obtain (let Qt = [i,T] x R")
E ips{g{x))dx - E ipe{u{t,x))dx
= £ / |<^U«)[- ^ V-(AD«) - V-(Bg) - {a,-D«)
- cw - F, g) -/] + -(^;'(w)|g|2}da;ds
= £; / \^^':iu)[{ADu,Du)+2{B''Du,q)+\q\']
JQt "-^
- V3^ (w) [ {o, Dw) +CW + { ^ g) +/] jdxds
= E f {]-ip'^{u)[{{A-BB'^)Du,Du)+\B'^Du + q-hu\^
JQt "-^
+ ^¥'"(w)[-WV + 2 {B^£>w,6w)+2 {k,g)]
- {a,Dipe{u)) -ip'^{u)[cu + {b,q) +/] \dxds
>E f {- \v':{u)\h?u'' + {B%D r^':{r)rdr)
JQt •• ^ "'0
+ ["^"(w)" - V'^W] ( ^9) +(V-o)(/?,(w)
— (/3^(w)[c« + /] >dxds.
F.10)

§6. Compaiison theorems 133
We note that
F.11) / ip'^{r)rdr = ip'^{u)u - ipe{u),
Jo
and
F.12) lim[v3;'(w)w - ip'^{u)] = 2w7(„<o) + 2u- = 0.
Thus, let e -> 0 in F.10), we obtain
e[ \g{x)-\^dx-E [ \u{t,x)'\'^dxds
>E f {- /(„<o)|&P«' - ViBb)[-2u'u - |«-p]
JQt ••
F.13) +iV-a)\u'\^+2u-[cu + f]\dxds
>E I U-\b\^ -ViBh) + V-a-2c)\u-\^ -2u-f-\dxds
>-fiE I \u-\'^dxds~E f \f-fdxds,
JQt JQt
where
F.14) /i=sup [-V-o + V-(B6) + |6|2+2c+l] < oo.
Then by Gronwall's inequality, we obtain F.4) for the case A = 0. The
general case can be proved by using transformation B.11) and working on
{v,p) for the transformed equations. D
Our main comparison result is the following.
Theorem 6.2. Let A.6), B.2) and (H)m hold for F.2) and F.3). Let
{f,g) and {f,'g) satisfy B.14) with some A > 0. Let {u,q) and {u,q) be
adapted strong solutions of F.2) and F.3), respectively. Then for some
H>0,
e[ e-^<'^^[u{t,x)-u{t,x)]-fdx
JR"
<eM5^-*)£; / e-^^^^[g{x)-g{x)]-fdx
F.15) r^ r
+ E e^(*-*M e-^<^>|[(£-£)w(s,a;)
+ {M - M)q{s,x) + f{s,x) - f{s,x)]^\ dxds,
Vie[o,r],

134
Chapter 5. Lineai, Degenerate BSPDEs
In the case that
' g{x) - g{x) > 0, Va; e R", a.s.
F.16) I (£ - C)u{t, x) + {M- M)q{t, x) + f{t, x) - /(i, x) > 0,
y{t,x) e[0,T] xR", a.s.
it holds
F.17) u{t,x) > u{t,x), V(i,a;) £ [0,r] x R", a.s.
This is the case, in particular, if C = C, M = M and
{g{x) > 'g{x), a.e.x £ ]R", a.s.
fit, a;) > Jit, x), a.e. (i, x) G [0, T] x R", a.s.
F.18)
Proof. It is clear that
' d(w - w) = -{£(w - w) + A^(g - g)
+ (£ - C)u + iM- M)q + f - J}dt
+ {q-q,dWit)),
Au-u)l^^=g-g.
Then, F.15) follows from F.4). In the case F.16) holds, F.15) becomes
F.19)
F.20)
e[ e-^<'^^[u{t,x)-uit,x)]-fdx<0, Vie[0,r].
Jr"
F.21)
This yields F.17). The last conclusion is clear. D
Corollary 6.3. Let the condition of Lemma 6.1 hold. Let
j gix) > 0, a.e.x £ R", a.s.
\ fit, x) > 0, a.e. it, x) e [0, T] x R", a.s.
and let {u, q) be an adapted strong solution of F.2). Then
F.22) u{t, x) > 0, a.e. (i, x) £ [0, T] x R", a.s.
Proof. We take 1 = C, M ^ M, J = 0 and g = 0. Then {u,q) =
@,0) is the unique adapted classical solution of F.3) and F.18) holds.
Consequently, F.22) follows from F.17). D
Let us make an observation on Theorem 6.2. Suppose (w, g) is an
adapted strong solution of F.3). Then F.16) gives a condition on A, B,
a, b, c, f and g, such that the solution (w, q) of the equation F.2) satisfies
F.17). This has a very interesting interpretation (see Chapter 8). We now
look at the cases that condition F.16) holds.
Lemma 6.4. Let A, B, a, b and c be independent of x. Let f and 'g be
convex in x. Let (w, g) be a strong solution of C.1). Then, u is convex in
X almost surely.

§6. Compaiison theorems 135
Proof. First, we assume that / and g are smooth enough in x. Then,
the corresponding solution (w, g) is smooth enough in x. Now, for any
T] e R", we define
{v{t,x) ^ {D^u{t,x)T],r)),
pit, x) = ip' (i, x), • ■ • ,/(i, x)), V(i, x) e [0, T] X R", a.s.
\^p''{t,x)^{D^t{t,x)r],r]), 1 < k < d,
Then, it holds
{dv = [-2v-Mp-{{D^7)v,v)]dt+{p,dWit)),
By Corollary 6.3 and the convexity of / and g (in x), we obtain
f6 24) {D^u{t,x)ri,r]) = vit,x)>0,
^' ' V(i,a;) e[0,r]x]R", T?e]R", a.s.
This implies the convexity of u{t, x) in x almost surely. In the case that /
and 'g are not necessarily smooth enough, we may make approximation.
D
Proposition 6.5. Let A, B, a, b and c be independent of x. Let f and
'g be convex in x and nonnegative. Let {u,q) be a strong solution of F.3).
Let M = M and let
F.25)
with
( A{t,x) = A{t)+Ao{t,x),
c{t,x) = c{t) +co{t,x),
f{t,x)=Jit,x) + foit,x),
[g{x) = g{x)+go{x),
{t,x) e[0,r] xR", a.s.
F.26) \^oit,x)>0, cit,x)>0, v(t,x)E[0,r]x]R",a.s.
Then F.16) is satisBed and thus F.17) holds.
Proof. By Corollary 6.3 and Lemma 6.4, u is convex and nonnegative.
Thus,
(£ - £)w(t, x) = -tr [AqD^u] + cqu > 0.
Then F.16) follows. D
Next, we have the following.
Proposition 6.6. Let all the functions A, B, a, b, c, f and'g be
deterministic. Let u be the solution of the following equation:
/«, = -£«-/, it,x)e[0,T]x-R",
F-27) < _,

136 Chapter 5. Linear, Degenerate BSPDEs
Further, we assume that u{t,x) is convex in x. Next, let F.25) hold. Then
F.16) is satisBed and F.17) holds.
Proof. In the present case, (w, 0) is an adapted strong solution of F.3).
Then similar to the proof of Proposition 6.5 and note g = 0, we can obtain
our assertion. D
Note that in Proposition 6.6, B and b are arbitrary.

Chapter 6
Method of Continuation
In this chapter, we consider the solvability of the following FBSDE which
is the same as C.16) of Chapter 1 (We rewrite here for convenience):
' dX{t) = b{t, X{t),Y{t),Z{t))dt + a{t,X{t), Y{t), Z{t))dW{t),
@.1) < dY{t) = h{t, X{t}, Y{t), Z{t))dt + Z{t)dW{t),
^X{0)=x, Y{T)=g{X{T)).
Here, functions b, a, h and g are allowed to be random, i.e., they can depend
on w e n. For the notational simplicity, we have suppressed w and we will
do so below.
We have seen that for the case when all the coefficients are
deterministic, one can use the Four Step Scheme to approach the problem (see
Chapter 4), which involving the study of parabolic systems; in the case of
random coefficients, in applying the Four Step Scheme, we need to study
the solvability of BSPDEs (see Chapter 5). In this chapter, we are going
to introduce a completely different method to approach the solvability of
@.1). Such a method is called the method of continuation.
§1. The Bridge
Recall that 5" is the set of all (n x n) symmetric matrices. In what follows,
whenever A is a square matrix, (with A being a scalar), hy A + X, we mean
A + XI. For any A e S", hy A > S, we mean that A ~ S is positive
semidefinite. The meaning oi A < —S is similar. For simplicity of notation,
we will denote M = R" x R™ x R™^'^; a generic point in M is denoted
by 6> = {x,y,z) with a; £ R", y £ R™ and z £ R™""^. The norm in M is
defined by
A.1) \e\^{\x\' + \y\^ + \z\'y/\ ve = {x,y,z)eM,
where \z\^ = tr (zz'^). Similarly, we will use 0 = {X, Y, Z), and so on.
Now, let T > 0 be fixed and let
H[0,T] =L|-@,T; W^''^{M; R" x R"^"^ x R™))
xL2,^(fi;Wi'°°(R";R™)).
Any generic element in H[0,T] is denoted by F = {b,a,h,g). Thus, F =
F, a, h, g) e H[0, T] if and only if
'6eL|-@,r;Wi'°°(M;R")),
a e L|.@,r; Wi'°°(M;R"^'^)),
h e L%-{0,T;W^''^{M;Wr)),
UeL^,(n;Wi'°°(R";R™)),

138
Chapter 6. Method of Continuation
where the space L'^{0,T;W'^'°°{M;'R'^)), etc. are defined as in Chapter 1,
§2. Further, we let
A.3)
n[0,T] = Llr{0,T;WC) x LJr{0,T;WC'"')
X L%{0,T;Wr) X L%{^;WC^).
An element in 'H[0,T] is denoted by 7 = {bo,cro,ho,go) with
f 6oeL2,@,r;]R"),
ho e L^^{0,T;WC^),
We note that the range of the elements in H[0,T] and ?^[0, T] are all in
R" X WC""^ X R™ X R™. Hence, for any T = ib,a,h,g) £ H[0,T] and
7 = F0, o'o, ho, go) G ?^[0, T], we can naturally define
A.4)
T + J = {b + bo,a + ao,h + ho, g + go) e H[0, T].
Now, for any T = {b,a,h,g) £ H[0,T], 7 = {bo,ao,ho,go) £ ?^[0,r]
and X e R", we associate them with the following FBSDE on [0, T]:
' dX{t) = {bit, Q{t)) + bo{t)}dt + {a{t, Q{t)) + aoit)}dWit),
A.5)r,7,x < dY{t) = {h{t, 0(i)) + ho{t)}dt + Z{t)dWit),
^X{0)=x, Y{T)=g{X{T))+go,
with 0(i) = {X{t),Y{t), Z{t)). In what follows, sometimes, we will simply
identify the FBSDEs A.5)r,7,x with (F, 7,0;) or even with F (since 7 and x
are not essential in some sense). Let us recall the following definition.
Definition 1.1. A process 0(-) = {X{■),Y{■),Z{■)) £ M[0,T] is called
an adapted solution of A.5)r,7,x, if the following holds for any t G [0,r],
almost surely.
X{t) =x+ [ {b{t,Q{s)) + bo{s)}ds
Jo
+ [ {a{t,Q{s)) + ao{s)}dW{s),
Jo
Y{t) = g{XiT)) + 90- j {hit, 0(s)) + hois)}ds
(l-6)r,7,2
I
Z{s)dW{s).
When A.5)r,7,x admits a unique adapted solution, we say that A.5)r,7,x is
(uniquely) solvable.
We see that A.6)r,7,a: is the integral form of A.5)r,7,x. In what follows,
we will not distinguish A.5)r,7,x and A.6)r,7,x-

§1. The bridge 139
Definition 1.2. Let T > 0. A T G H[0,T] is said to be solvable if for
any X e R" and 7 £ ?^[0,r], equation A.5)r,7,x admits a unique adapted
solution 0(-) e M[0,T]. The set of all T e H[0,T] that is solvable is
denoted by S[0,T]. Any T G H[0,T] \ S[0,T] is said to be nonsolvable.
Now, let us introduce the following notions, which will play the central
role in this chapter.
Definition 1.3. Let T > 0 and T = {b,a,h,g) £ H[0,T]. A C^ function
^= ("^ ^J y. [0, T] -> 5"+™, with A : [0, T] -^ S"", B : [0, T] -> R"*^"
and C : [0, T] -> 5™, is called a bridge extending from T, (defined on [0, T]),
if there exist some constants K,6 > 0, such that
A.7)
C(r)<o, A{t)>o, Vie[o,r],
and either A.8)-A.9) or A.8)'-A.9)' hold:
<*">i^:; •(::;'>
A.9)
^^ ''U-i//''V''_('.«)-''('.''O
+(*w(<'<'.''):f»)),(
< ~6\x~x\'^, Ve,e e M, a.e.te [0,T], a.s
a{t,e) - ait,e)'
z — z '
(^•«)' <^(^)(,(,):J(,)j>L(,^::(,)))>o, v.,.eR".
<^^^)(::f ' ::^)^
A.9)'
+ 2($(i)('^-^^ (Kt,0)-bit,e)\
+ ($(i)f^^*'^)"^^*'^)yf^^*'^)~^^*'^M)
<-<5{|y-y|^ + k-^n, ye,6£M, a.e.ie[0,r], a.s.
If A.7)-A.9) (resp. A.7) and A.8)'-A.9)') hold, we call $ a type (I) (resp.
type {II)) bridge extending from T (defined on [0, T]). The set of all type

140 Chapter 6. Method of Continuation
G) and type A1) bridges extending from F (defined on [0, T]) are denoted
by BiiT; [0,T]) and -B//(r; [0,r]), respectively. Finally, we let
r ^(F; [0,r]) = BiiT; [0,r]) (J^//(F; [0,r]),
A-10) S ^
[^'(F;[0,r])=^/(F;[0,r])f|^//(F;[0,r]).
Any element $ £ B^{T;[0,T]) is called a strong bridge extending from F
(defined on [0,r]).
Definition 1.4. Let T > 0 and F, F G H[0, T]. We say that they are linked
by a direct bridge if
{^/(F;[o,r])f|^/(r;[o,r])}
A.11) ' '
\J{Bn{T;[0,T])f]Bidr;[0,T])}^<P;
and we say that they are linked by a bridge, if there are Fi, • • •, F^ G H[0, T],
such that with Fq = F and F^+i = F, it holds
{^/(r,;[o,r])f|^/(F,+i;[o,r])}
A.12) ' '
(J {BniTf, [0,T])f]BiiiTi+v, [0,T])} ^ cf,, 0<i<k.
We may similarly define the notion that F and F are linked by a (direct)
strong bridge.
§2. Method of Continuation
In this section, we are going to present the solvability of FBSDEs by the
method of continuation. The notion of bridge plays an important role here.
§2.1. The solvability of FBSDEs linked by bridges
Let us state the following theorem.
Theorem 2.1. Let T > 0 and Fi,F2 G H[0,T] be linked by a bridge.
Then, Fi G S[0,T] if and only if F2 G S[0,T].
The above theorem tells us that if the FBSDE associated with Fi is
solvable, so is the one associated with F2, provided Fi and F2 are linked
by a bridge. In applications, if one wants to prove the solvability of the
FBSDE associated with F2, he/she can start with a known solvable FBSDE
Fi, and try to construct a bridge linking Fi and F2. We will see a detailed
construction of bridges in §5 for an interesting case.
Let us now explain the idea of proving Theorem 2.1. First of all, we
make a simple reduction. By induction, to prove Theorem 2.1, it suffices
to prove it for the case that Fi = F1,o'l,/ii,71) and F2 = (&2,o'2,/i2,ff2)
are linked by a direct bridge. We now assume this. Next, for any 7 =

§2. Method of continuation 141
{bo{-),cro{-),ho{-),go) G 'H[0,T], a; G R" and a G [0,1], we consider the
following FBSDE:
' dX{t) = {A - a)bi (t, 0(i)) + ab2it, 0(i)) + boit)}dt
+ {A - a)aiit, Q{t)) + aa2{t,e{t)) + ao{t)}dW{t),
B.1)^_^ J dY{t) = {il-a)hiit,Q{t))+ah2it,Qit)) + hoit)}dt
+ Z{t)dW{t),
. X@) = X, YiT) = A - a)g,iXiT)) + ag2{X{T)) + g^.
We may give the definition of the (adapted) solutions to above system
B.1)^^ similar to Definition 1.1. It is clear that B.1)^^ and {2.1)\,^
coincide with A.5)ri,7,x and A.5)r2,7,x, respectively. Let us assume that
Ti G <S[0,r], i.e., B.1)^,^ is uniquely solvable for any 7 G 'H[0,T] and
X G R". We want to prove r2 G <S[0,r], i.e., B.1)^^3. is uniquely solvable
for all 7 G ?^[0,r] and x G R". The essence of the method of continuation
is contained in the following claim:
There exists a fixed step-length eo > 0, such that if for some
a G [0,1), B.1)^,. is uniquely solvable for any'y e 'H[0,T] and
X G R", then the same conclusion holds for a being replaced
by a + e < 1 with s G [0,eo].
Once this has been proved, we can start with B.1)"^ with a = 0 which
is solvable by our assumption, increase the parameter a step by step and
finally reach a = 1, which gives the unique solvability of B.1)]^ 3..
In order to prove the above claim, the following a priori estimates for
the adapted solutions of B.1)" 3. will be crucial.
Lemma 2.2_. Let a G [0,1]. Let Q{-)={X{-),Y{-),Z{-)) and ©(•) =
(X(-), F(-), Z(-)) be adapted solutions of B.1)^ ,^ and B.1J_, respectively,
witfi7= (&o,o-o,/io,ffo), 7 = (^0,0^0,/io,ffo) G ?^[0,r] anda;,x G R". Then,
the following estimate holds:
m-)-m\\Uo,T]
= E sup \X{t)-'X{t)\'^+E sup |y(i)-F(i)|2
te[o,T] te[o,T]
B.2) +^1 \Z{t)-Z{t)\'dt
<c[\x- x\^ + E\go - ffol' +E j {|6o(i) - &o(i)|
+ l^o(i) -^o(i)|' + \ho{t)-ho{t)?]dt}.
Since the proof of the above lemma is technical and lengthy, we would
like to postpone it to the next subsection. Based on the above a priori
estimate, we now prove the following result, which we call it the continuation
lemma.
2

142
Chapter 6. Method of Continuation
B.3)
a-\-e
7,x
Lemma 2.3. Let ri,r2 £ H[0,T] be linked by a direct bridge. Then,
there exists an absolute constant eo > 0, such that if for some a £ [0,1],
B.1)" J. is uniquely solvable for any 7 £ ?^[0, T] and x £ R", then the same
is true for B.1)^+^ with e £ [0,eo], a +e < 1.
Proof. Let eo > 0 be undetermined. Let e £ [0,eo]- For k > 0, we
successively solve the following systems for Q''{t) ={X''{t),Y''{t), Z''{t)):
(compare B.1)^+^)
'e°{t)^{X°it),Y°it),Z°it))=0,
dX''+'{t) = {{l-a)bi{t,Q''+\t))+ab2{t,Q''+\t))
- ebiit, 0'=(t)) + eb2{t, 0'=(t)) + boit)}dt
+ {{1 - a)ai{t,e''+'it)) + aa2it,e>'+\t))
- eaiit, 0'=(t)) + eG2(t, 0'=(t)) + aoit)}dWit),
dF'^+i (t) = {A - a)hi {t, 0'=+! (t)) + ah2{t, 0'=+! (t))
- ehiit, 0'=(t)) + 6/12(t, 0'=(t)) + /io(i)}di
+ Z*^+i(t)dW(t),
X''+\0) = x,
¥"+' (T) = A - a)ffi (X'^+i (T)) + aff2(X'=+i (T))
-egiiX\T))+e92{X>'{T))+9o.
By our assumption, the above systems are uniquely solvable. We now apply
Lemma 2.2 to 0*^+i(-) and 0*^(-). It follows that
ll©'+^(-)-0n-)IU[O,T]
< c\^e^E\X''{T) - X''-\T)\'^
+ e^E I |0'=(t)-0'=-i(t)|2dt}
<e2Co||0'=(-)-0'=-^(-)lk[o,T]-
We note that the constant Co > 0 appearing in B.4) is independent of
a and e. Hence, if we choose eo > 0 so that EqCo < 1/2, then for any
e £ [0,eo], we have the following estimate:
B.4)
B.5) \\Q^+\-) - 0'=(-)IIa.[o,t] < \\\Q\-) - Q'-'i-
\mio,t],
yk> 1.
This implies that the sequence {0*^(-)} is Cauchy in the Banach space
A^[0, T]. Hence, it admits a limit. Clearly, this limit is an adapted solution
to B.1)"+^. Uniqueness follows from estimate B.2) immediately. D
Now, we are ready to give a proof of our main result.
Proof of Theorems 2.1. We know that it suffices to consider the case
that Fi and F2 are linked by a direct bridge. Let us assume that A.5)ri,7,i

§2. Method of continuation 143
is uniquely solvable for any 7 £ ?^[0,r] and x £ R". This means that
B.1)^^ is uniquely solvable. By Lemma 2.3, we can then solve B.1).
uniquely for any a € [0,1]. In particular, B.1)i^^, which is A.5)r2,7,x, is
uniquely solvable. This proves Theorem 2.1. D
Note that Lemma 2.2 has the following implication.
Corollary 2.4. Let T £ H[0,T] with ^(r;[0,r]) 7^ cf). Then, for any
7 £ ?^[0,r] and x £ R", A.5)r,7,x admits at most one adapted solution.
Moreover, for any 7,7 £ 'H[0,T] and x,x £ R", the stability estimate B.2)
holds for any adapted solutions 0(-) of A.5)r,7,x and 0(-) of A.5)r,7,5.
Proof. We take Ti = Fa = T in Lemma 2.2. Then, B.2) applies. D
Prom Corollary 2.4 we see that for the F cissociated with example C.3)
in Chapter 1, B{T; [0, T]) = (f) for T = kir + ^, k >0.
§2.2. A priori estimate
In this subsection, we present a proof of the a priori estimate stated in
Lemma 2.2.
Proof. Let 0 and 0 be two adapted solutions of B.1)"^ and B.1)^^,
respectively. Define ^ = ^ — ^ for ^ = X,Y,Z,Q,bo,ao,ho,go, x ~ x —Tc,
and
-hi{t) = hi{t,Q{t))-hi{t,Q{t)),
,^^. \ ^i{t) = cji{t,@{t)) - ai{t,@{t)), .
B.6) < _ _ 2 = 1,2,
hi{t) = hi{t,Q{t))-hi{t,Q{t)),
MT) = 9i{X{T))-gi{X{T)),
Note that Fj £ H]>d,T] implies that all the functions hi,Ui,hi,gi are
uniformly Lipschitz continuous. Suppose the common Lipschitz constant is
-L > 0. Applying Ito's formula to |X(i)p, we obtain that
|X(i)|2 = |J|2 + 2 / {X{s),{l - a)hi{s) + ah-^is) + ho{s)) ds
Jo
+ \{l-a)ai{s)+aa2{s)+ao{s)\^ds
Jo
+ 2 [ {X{s),[{l - a)Ms) + aa2{s) +ao{s)]dW{s))
Jo
< \x\' + C f \X{s)\{\X{s)\ + \Y{s)\ + \Z{s)\ + \Us)\]d'
JO
+ C f {\X{s)\ + \Y{s)\ + \Z{s)\ + \do{s)\fds
Jo
+ 2 / {X{s),[{l-a)di{s)+ad2{s)+do{s)]dW{s)),
Jo
B.7)

144 Chapter 6. Method of Continuation
with some constant C > 0. As before, in what follows, C will be some
generic constant, which can be different in different places. By taking the
expectation and using Gronwall's inequality, we obtain
B.8) E\X{t)\-'<CE{\x\' + l {\Y{t)\-' + \Z{t)\' + \bo{t)? + \Mt)\'}dt},
with some constant C = C{L,T). Next, applying Burkholder-Davis-
Gundy's inequality to B.7) (note B.8)), one has that
E sup \X{t)\'<c{\x\'+ [ {\Y{t)\' + \Z{t)\'
B.9) *G[o>^l Jo
+ \bo{t)? + \Mt)\^}dt}.
On the other hand, by applying Ito's formula to |y(i)p, we have
\Yit)\'+1^ \Zis)\'ds
= |F(r)|2 -2 f {Y{s), A - a)hi{s) + ah2{s) + ho{s)) ds
B.10) -21 {Y{s),Z{s)dW{s))
< C{\X{T)\^ + Iffol' + ^ {|X(.)|2 + \Y{s)\^ + \Ms)\^]ds}
~\[ l^^')!''^' ~'^[ ^ Y{s),Z{s)dW{s)).
Similar to the procedure of getting B.9), we obtain
E sup \Y{t)f ^E [ \Z{t)\'^dt
Jo
CE{\XiT)\' + \9o\' + ^' {|X(i)p + \hoit)\'}dt}.
B.11) '''"''^'
We emphasize that the constants C appeared in B.9) and B.11) only
depend on L and T. Also, in deriving these two estimates, only the condition
Ti e H[0,T] has been used (and we have not used the bridge yet). Now,
we apply Ito's formula to
<*«(fS)'(if:o>-

§2. Method of continuation 145
It follows that
-<*(r)(lS)'(l(?0>-<*'«'D,)'D,)>
-r{<*"Kfs)'(fs)>
B 12) +2(m)(^^*A ( i^~c^)bi{t)+ab2{t)+bo{t)\.
^^^\Y{t)J'\{l~a)hi{t) + ah2{t) + ho{t)J'
^ ^^l m )'
A - a)ai{t) + aa2{t) + do{t) \ ^ \^^
Z{t)
)}'
Let us separate two cases.
Case 1. Suppose $ £ Bj^Ti; [0,r]) (i = 1,2). In this case, we have
( X{T) \
\{l-a)UT)+ocUT))'
B.13) ^ ^ A{T)X{T), X{T)) +2 {B{T)X{T), A - a)g,(T) + a^T))
+ {C{T){{1 - a)UT)+ocUT)], A - a)UT) + aUT))
= aHC{T){UT)-9x{T)],{UT) ~ UT)])
+ a{-.-] + {..-]>5\X{T)\\ Vae[0,l],
where {• • •} are terms that do not depend on a. The above holds because
C{T) < 0 implies that F{a) is concave in a, whereas A.8) tells us that
(recall $e^/(ri;[0,r]), 2 = 1,2)
B.14) F@), F(l) >E|X(r)|2.
Then, B.13) follows easily. Similarly, we have
«">'<*<')(l!:0'(l(':0>
^ ^^U(i)y'V(l-a)/ii(i) + a/i2(i)r
B.15) +(^@(^^-")'|/"'^^^)),
/(l-a)CTi(i)+a52(i)V
V m ) ^
= a^{A{t){a2{t)-ai{t)},a2it)-aiit))
+ a{---} + {---}<-S\Xit)\\ Vae[0,l],

146 Chapter 6. Method of Continuation
since now A{t) > 0 which implies /(a) is convex in a. Then, we have
Left side of B.12) = eUa{T)X{T),X{T))
+ 2 {B{T)X{T), A - a)ffi(r) + ah{T) + % )
+ {CiT){{l - a)MT) + aUT) + 9o},
il-a)giiT) + a92iT)+go)]
>SE\X{T)\'-2\B{T)\E{\X{T)\\go\)
- 2L\CiT)\E{\XiT)\\9o\) ~ \CiT)\E\go\' - K\x\'
>^-E\XiT)\'-C{\x\' + E\go\'}.
Here, the constant C > 0 only depends on K, L, 6, \B{T)\ and |C(r)|.
Similarly, we have the following estimate for the right hand side of B.13).
Right side of B.12) <E f (- S\X{t)\^dt
<-^e| \X{t)\''dt + eE j {|y(i)|2 + |Z(i)|2}dii
Jo
B.17)
fT
+ C,E
with the constant C^ > 0 only depending on the bounds of |$(i)|, as well
as E, L and the undetermined small positive number e > 0. Combining

§2. Method of continuation 147
B.16)-B.17) and note B.11), we have
E\X{T)\'^ + E \X{t)\'^dt
Jo
< a{|xp + ElffoP +EJ {\bo{t)? + \Mt)? + \ho{t)?}dt}
B.18) +f^/ {\Y{t)\'' + \Z{t)\'}dt
< C,{\x\^ + E\go\^ + E j {|6o(i)P + |?o(i)P + \Mt)?]dt]
+ eCE{\X{T)\^ + Iffol' + f {\m\^ + \hoimdt],
with the constant C independent of e > 0, and C^ might be different from
that appeared in B.17). Thus, we may choose suitable e > 0, such that
E\X{T)\^ + E \X{t)\'^dt
B.19) ^«
< Ce{\x\^ + Iffol' + J {|&o(i)P + \Mt)\^ + \hoit)\^}dt].
Then, return to B.11), we obtain
E sup \Yit)\^+E [ \Z{t)\'^dt
B.20) '^"'•^l ^° ^
< Ce{\x\-' + IffoP + j {\bo{t)? + \Mt)? + \ho{t)ndt}.
Finally, by B.9), we have
E sup |X(i)|2<C£;||x|2 + |ffo|'
^ {\bo{t)\' + \Mt)\' + \ho{t)\'}dt}.
te[o,T]
B.21) ^r
+
Hence, B.2) follows from B.20) and B.21).
Case 2. Let $ G ;B//(ri; [0,r]) {i = 1,2) now. In this case, we still
have B.9), B.11) and B.12). Further, we have inequalities similar to B.13)
and B.15) with \X{T)\'^ and |X(i)|2 replaced by 0 and |y(i)|2 + \Z(t)\^,
respectively. Thus, it follows that
B.22) Left side of B.12) > -eE\X{T)\^ - C,{\xf + E\go\^},
with the constant C^ > 0 depending on K,L,S, |-B(r)|, \C{T)\, and the

148 Chapter 6. Method of Continuation
undetermined constant e > 0. Whereas,
Right side of B.12)
B.23) ~A^f {\^it)\^ + \m\-'}dt + eEl \X{t)\-'dt
+ C,E [ {|6o(i)P + \Mt)? + \hoit)\^}dt.
Jo
Now, combining B.22)-B.23) and using B.9), we obtain (for suitable choice
of e > 0)
E [ {\Yit)\' + \Zit)\'}dt
B.24) ^«
< Ce[\x\^ + \go\^ + J {Mt)\^ + \Mtt + \hoit)\^}dt].
Finally, by B.9) and B.11) again, we obtain the estimate B.2). Q
§3. Some Solvable FBSDEs
In this section, we are going to prove the unique solvability of some FBSDEs
by constructing appropriate bridges.
§3.1. A trivial FBSDE
We denote Fq = @,0,0,0) £ H[0,T]. The FBSDE associated with Fo reads
as (compare with A.5)r,7,x)
C.1)
' dX{t) = boit)dt + ao{t)dWit),
dY{t) = hoit)dt + Z{t)dW{t),
^ X@) = X, Y{T) = go.
$ = ( ^ '^^ ) e B^iVo; [0, T]) if and onJy if
Clearly, C.1) is trivially uniquely solvable for all 7 = (bo,ao, ho,go) £
'H[0,T] and x e R". Thus, hereafter, we will refer to the FBSDE associated
with Fo as the trivial FBSDE. Now, let us present the following result.
Proposition 3.1. Let T > 0 and Fq = {0,0,0,0} £ H[0,T]. Then,
0, T]) if and only if
f ^@) < 0, AiT) > 0,
\$(i) <o, Vie[o,r].
Proof. By Definition 1.3, we know that $ £ ;B'(Fo; [0,r]) if and only
if A.7)-A.9) and A.8)'-A.9)' hold. These are equivalent to the following:
f ^@) < -S, AiT) > 6,
U(i)<-<5, Vi£[0,r],

§3. Some solvable FBSDEs 149
for some S > 0. We note that under condition C@) < 0, the second
inequality in A.7) is always true for sufficiently large K > 0. Then, we see
easily that $ € ;B'(ro; [0,r]) is characterized by C.3) since 6 > 0 can be
arbitrarily small. D
From the above, we also have the following characterization:
B'{To;[0,T]) = {Q- j *(s)ds
Qi- f *i(s)ds>0}.
A useful consequence of Proposition 3.1 is the following.
Corollary 3.2. Let T £ H[0,T] admit a bridge $ £ B{T; [0,T]) satisfying
C.2). Then, T e S[0,T].
Proof. Under our assumptions, it holds that
$£^(ro;[o,r])f|^(r;[o,r]).
Since Tq £ S[0, T], Theorem 2.1 applies. D
Next, we would like to discuss some concrete cases.
§3.2. Decoupled FBSDEs
Let F = (b, a, h, g) £ F[0, T] such that
\ h{t,x,y,z) = h{t,x), , . r ,
C.5) <^ ;; ;_ \' '' y{t,x,y,z)e[0,T]xM.
y<j(t,x,y,z) =a{t,x),
We see that the associated FBSDE is decoupled, which is known to be
solvable under usual Lipschitz conditions, by the result of Chapter 1, §4.
The following result recovers this conclusion with some deeper insight.
Proposition 3.3. Let T > 0, Tq = @,0,0,0) £ H[0,T] and T H
{b,a,h,g) £ H[0,T] satisfying C.5). Then,
C.6) ^^(Fo;[0,r])f|^^(F;[0,r]O^^.
Consequently F £ S[0,T].
Proof. We take
C.7) \ ^ ^ I 0 cit)I
ait) = Aoe^o^'^-'\ cit) = -Coe^°\ te[0,T],

150
Chapter 6. Method of Continuation
where Ao,Co > 0 are undetermined constants. We first check that this
$e^*(ro;[0,r]). in fact.
C.8)
' c@) = -Co < 0, aiT) = Ao>0,
a(i) =-A2gAo(T-t> < 0, te[0,T],
^ c(i) =-C^e^o* < 0, te[0,T].
Thus, by Proposition 3.1, we see that $ £ ;B*(ro; [0,r]). Next, we show
that $ e B^{T; [0, T]) for suitable choice of Aq and Co- To this end, we let
L be the common Lipschitz constant for b,a,h and g. We note that C.8)
implies A.7). Thus, it is enough to further have
C.9)
and
C.10)
a{T) + L^c{T) > S,
a{t)\x - x|2 + c{t)\y - yp + c{t)\z - z\''
+ 2a{t){x-x,b{t,x) -b{t,x))+a{t)\a{t,x) - a{t,x)\^
+ 2c(i) {y-y,h{t,x,y,z)- h{t,x,y, z))
<-S{\x-x\' + \y-y\' + \z-z\^},
Vie [o,r], x,xem", y,yewr, z,zewr'"^, a.s.
Let us first look at C.10). We note that
Left side of C.10) < a{t)\x - x\^ + c{t)\y - y\^ + c{t)\z -
+ 2a{t)L\x - x\^ + a{t)L'^\x - xp
C.11) + 2|c(i)|L|y - y\{\x -x\ + \y-y\ + \z- z\}
< {a{t) + 2a{t)L + a{t)L^ + |c(i)|L}|a; - xp
+ {c{t) + 3\c{t)\L + 2LMt)\}\y -y\' + ^\z- z\'.
•:7|2
Hence, to have C.10), it suffices to have the following:
' d(i) + BL + L'^)a{t) + L\c{t)\ < -6,
C.12)
c{t) + {3L + 2L'^)\c{t)\ <-S,
I c(i) < -2S,
Vie [o,r].
C.13)
Now, we take a{t) and c{t) as in C.7) and we require
c{t)+{3L + 2L'^)\c{t)\ = -Co(Co -3L- 2L^)e'^'>\
<-CoiCo-SL-2L^)<-S, Vie[0,r],
and
C.14) c{t) = -Coe^o* < -Co < -2E,- Vi e [0, T].

§3. Some solvable FBSDEs 151
These two are possible if Co > 0 is large enough. Next, for this fixed Co > 0,
we choose ^o > 0 as follows. We want
C.15) a{T) + c{T)L^ = Aoe^°^^-'^ - CoL^e^°' > Aq - CoL^e^°^ > 6,
and
a{t) + BL + L^)a{t) + L\c{t)\
C.16) = -Ao{Ao -2L- L^)e^°^^-'^ + LCoe^"*
<-Ao{Ao-2L-L^) + LCoe^°'^ <-6.
These are also possible by choosing ^o > 0 large enough. Hence, C.9) and
C.12) hold and $ e B'{T; [0, T]). D
Prom the above, we obtain that any decoupled FBSDE is solvable. In
particular, any BSDE is solvable. Moreover, from Lemma 2.2, we see that
the adapted solutions to such equations have the continuous dependence
on the data.
The above proposition also tells us that decoupled FBSDEs are very
"close" to the trivial FBSDE since they can be linked by some direct strong
bridges of To.
§3.3. FBSDEs with monotonicity conditions
In this subsection, we are going to consider coupled FBSDEs which satisfy
certain kind of monotonicity conditions. Let T = {b,a,h,g) £ H[0,T]. We
introduce the following conditions:
(M) Let m > n. There exists a matrix B £ R™'*" such that for some
/3 > 0, it holds that
C.17) {B{x-x),g{x)-g{x))>^\x-xf, Va;,x £ R", a.s.
{B''[h{t,e) - h{t,e)],x -x) + {B[b{t,e) -b{t,e)],y -y)
C.18) + {B[a{t,e) - a{t,e)],z -z) < -l3\x-x\^,
Vie [o,r], e,ee M, a.s.
(M)' Let m <n. There exists a matrix B £ R™^" such that for some
/3 > 0, it holds that
C.17)' {B{x-x),g{x)-g{x))>0, Va;,x e R", a.s.
{B'^[h{t,e) - h{t,e)\,x -x) + {B[h{t,e) - h{t,e)\,y -y)
C.18)' + {B[a{t,e) - <j{t,e)\,z- z) < -(i{\y - y|2 + \z- z\''),
Vie [o,r], e,e e M, a.s.
Condition C.17) means that the function x i-> B'^g{x) is uniformly
monotone on R", and condition C.18) implies that the function 6 i->

152
Chapter 6. Method of Continuation
— {B'^h{t,6),Bb{t,6),Ba{t,6)) is monotone on the space M. The
meaning of C.17)' and C.18)' are similar. Here, we should point out that C.17)
implies m > n and C.17)' implies m < n. Hence, (M) and (M)' overlaps
only for the case m = n.
We now prove the following.
Proposition 3.4. Let T > 0 and T = {b,a,h,g) G H[0,T] satisfy (M)
(resp. (M)'). Then, C.6) holds. Consequentiy, T e S[0,T].
Proof. First, we assume (M) holds. Take
C.19)
A{t) = a{t)I = Se^-^I,
B{t) = B,
_ C{t) = c{t)I = -2ECoe^<'*7,
with E, Co > 0 being undetermined. Since
' C@) = -2SCoI < 0,
A{T) =6I>0,
-Se'^-n 0
0 -2SCie'='°*
te[0,T],
C.20)
$(i) =
<0,
by Proposition 3.1, we see that $ £ ;B*(ro; [0,r]). Next, we prove $ £
B^{T; [0, T]) for suitable choice of S and Cq. Again, we let L be the common
Lipschitz constant for b, a, h and g. We will choose S and Co so that
C.21) a{T)+2l3 + c{T)L^>6,
and
d{t)\x\^ + m\y\^ + c(i)kl' + 2La(i)|a;|(|a;| + |y| + |^|)
C.22) + 2L|c(i)| \y\i\x\ + \y\ + \z\) + L^amx] + \y\ + \z\)'
< B/3 - 6)\x\' - 6{\y\' + \z\'), V(i, 6) £ [0, T] x M.
It is not hard to see that under C.17)-C.18), C.21) implies A.8) and C.22)
implies A.7) and A.9)' (Note A.8) implies A.8)'). We see that the left hand
side of C.22) can be controlled by the following:
C.23)
{a{t) + Ka{t) + K\c{t)\}\x\^ + {c{t) + K\c{t)\ + Ka{t)]\y\
+ {'M+Ka{t)]\z\\
for some constant K > 0. Then, for this fixed iiT > 0, we now choose 6 and
Co. First of all, we require
C.24)
c{t)
+ Ka{t) = -(SCoe^o* + K6e'^-* < -6Co + K6e'^ < -S,

§3. Some solvable FBSDEs
and
c{t) + K\c{t)\ + Ka{t) = -2EC^e^<'* + 2KCoSe'^°'' + K6e'^-^
153
C.25)
< -2ECo(Co -K)+ K6e'^ < -6.
These two can be achieved by choosing Co > 0 large enough (independent
of E > 0). Next, we require
C.26)
and
C.27)
a{t) + Ka{t) + K\c{t)\ = -Ee^-* + K6e'^-* + 26KCoe'='°*
<-6 + KSe'^ + 26KCoe'^°'^ <2/3-S,
a{T) + 2/3 + c{T)L'^ =6 + 2^- 2SCoe'^'>'^L'^ > S.
Since /3 > 0, C.26) and C.27) can be achieved by letting S > 0 he small
enough (note again that the choice of Co is independent of E > 0). Hence,
we have C.21) and C.22), which proves $ e B^{r; [0,T]).
Now, we assume (M)' holds. Take (compare C.19))
C.28)
^^'>-[Bit) C{t)
A{t) = a{t)I = SAoe^°^^-'h,
B{t) = B,
I C{t) = c{t)I = -Se^I,
Vie[o,r],
with S,Ao>0 being undetermined. Note that
' C@) = -SI < 0,
A{T) = Aol > 0,
C.29)
m =
-SAle^o{T-t)j
0
0
-Se^I
<0.
Thus, by Proposition 3.1, we have $ £ B^{To; [0, T]). We now choose the
constants 6 and Aq. In the present case, we will still require C.21) and the
following instead of C.22):
a{t)\x\^ + c(i)|y|2 + c{t)\z\^ + 2La(i)|a;|(|a;| + |y| + |^|)
C.30) + 2L|c(i)||y|(|a;| + |y| + |^|) + L^am^l + \y\ + \z\)'
< -S\x\' + B/3 - S){\y\' + \z\'}, V(i,0) e [0,r] X M.
These two will imply the conclusion $ £ B^{T; [0,r]). Again the left hand
side of C.30) can be controlled by C.23) for some constant K > 0. Now,
we require
a{t) + Ka{t) + K\c{t)\ = -<5A2gAo(T-t> ^ SKAoe^°^^-'^ + K6e'
< -SAoiAo -K)+ SKe'^ < -5,

154 Chapter 6. Method of Continuation
and
a{T) + c{T)L^ = SAoe'^°^'^-^> - SL^e'^
> SiAo - L^e^) > S.
C-32) _. , ., ^
We can choose Aq > 0 large enough (independent oi S > 0) to achieve the
above two. Next, we require
C.33) — + Ka{t) < Ka{t) < SKAoe^"'^ <2P-S,
and
C.34)
c{t) + K\c{t)\ + Ka{t) = -Ee* + KSe^ + KAoSe^"^'^-^^
< 6{Ke'^ + KAoe^°'^) < 2/3 - E.
These two can be achieved by choosing 6 > 0 small enough. Hence, we
obtain C.21) and C.30), which gives $ £ ^*(r; [0,r]).
It should be pointed out that the above FBSDEs with monotonicity
conditions do not cover the decoupled case. Here is a simple example.
Let n = m = 1. Consider the following decoupled FBSDE:
C.35)
' dX{t)=^X{t)dt + dW{t),
dY{t) = X{t)dt + Z{t)dW{t),
^X{0) = x, Y{T) = X{T).
We can easily check that neither (M) nor (M)' holds. But, C.35) is uniquely
solvable over any finite time duration [0,r].
Remark 3.5. Prom the above, we see that decoupled FBSDEs and the
FBSDEs with monotonicity conditions are two different classes of solvable
FBSDEs. None of them includes the other. On the other hand, however,
these two classes are proved to be linked by direct bridges to the trivial
FBSDE (the one associated with To = @,0,0,0)). Thus, in some sense,
these classes of FBSDEs are very "closer" to the trivial FBSDE.
§4. Properties of the Bridges
In order to find some more solvable FBSDEs with the aid of bridges, we
need to explore some useful properties that bridges enjoy.
Proposition 4.1. Let T > 0.
(i) For any F G H[0, T], the set ^/(F; [0, T]) is a convex cone whenever
it is nonempty. Moreover,
D.1) ^/(r;[o,r]) = ^/(r + 7;[o,r]), V7e?^[o,r].
(ii) For any Fi, r2 £ H[0,T], it holds
D.2) ^/(ri;[0,r])f|^Hr2;[0,r])C f| Bi{aTi + ^T2;[0,T]).
a,P>0

§4. Properties of the bridges 155
Proof, (i) The convexity of Bi{T; [0, T]) is clear since A.7)-A.9) are
linear inequalities in $. Conclusion D.1) also follows easily from the definition
of the bridge.
(ii) The proof follows from B.13), B.15) and the fact that Bi{T; [0,r])
is a convex cone. D
It is clear that the same conclusions as Proposition 4.1 hold for
^//(r;[0,r])and^»(r;[0,r]).-
As a consequence of C.2), we see that if ri,r2 £ H[0,T], then
Bi{aTi + /3r2; [0, T]) = cj), for some a, /3 > 0,
^^■^^ => ^/(ri;[o,r])f|^/(r2;[o,r]) = ^.
This means that for such a case, Fi and F2 are not linked by a direct bridge
(of type (/)). Let us look at a concrete example. Let Fj = {bi,ai,hi,gi) G
H[0,T], i = 1,2,3, with
D.4)
bi\ _ (-X 0 \ (x\ fb2\ _ (X 1\ (x
hj \-l -v) \yj' yh2j \0 v) \y
&3^ ( ^\(A ai = G2 = G3 = 0,
9i = 92 = 93 = -X,
with A,!/ e R. Clearly, it holds
D.5) F3=Fi+F2.
By the remark right after Corollary 2.4, we know that ;B(F3;[0,r]) = (j).
Thus, it follows from C.5) and D.3) that Fi and F2 are not linked by
a direct bridge. However, we see that the FBSDE associated with Fi is
decoupled and thus it is uniquely solvable (see Chapter 1). In §5, we will
show that for suitable choice of A and f, F2 £ <S[0,r]. Hence, we find two
elements in <S[0, T] that are not linked by a direct bridge. This means Fi
and F2 are not very "close".
Next, for any 61,62 e L|-@,r; Wi>°°(M;]R")), we define
\\b1-b2Ut)
D.6) 161 (i, 6>; w) - 61 (i, 5; w) - 62 (i, 6>; w) + 62 (i, 5; w) I
= esssup sup — '■.
uieQ 9,9eM P ~ "I
We define \\hi - /i2||o(i) and \\ai - cr2\\oit) similarly. For 91,92 £
L2,^(fi; Wi'°°(IR";]R™)), we define
llffi -ff2||o
D.7) |ffi(a;;w) - gi{x;uj) - 52(a;; w) + 32(x; w) I
= esssup sup ' j —, ^•
wen x,5"eK." F ~ x\

156 Chapter 6. Method of Continuation
Then, for any Ti = {hi,Gi,hi,gi) £ F[0,r] {i = 1,2), set
D.8)
llri - r2||o(i) - ||6i - 62||o(i) + IK - ^2||o(i)
+ \\hi-h2Ut) + \\9i~92\\o-
Note that || • ||o(i) is just a family of semi-norms (parameterized by i £
[0, T]). As a matter of fact, ||ri - r2||o(i) = 0 for all t £ [0, T] if and only if
D.9)
r2 = Tl + 7,
for some 7 £ 'H[0,T].
Theorem 4.2. Let T > 0 and T E H[0,T]. Let $ £ B'{T; [0,T]). Then,
there exists an e > 0, such that for any T' £ H[0, T] with
D.10)
|r-r'||o(i)<e, Vi£[o,r],
we have $ £ ^'(F'; [0,r]).
Proof. Let T = {b,a,h,g) and T' = {b',a',h',g'). Suppose $ £
^'(r;[0,r]). Then, for sorne K,S > 0, A.7)-A.9) and A.8)'-A.9)' hold.
Now, we denote (for any 6,6 E M)
D.11)
' X = X — X, 6 = 6 — 6,
b = b{t,6) - b{t,6), a = ait,6) - a{t,6),
h=h{t,e)-h{t,6), g = g{x)-g{x),
b' = b'{t,6) - b'{t,6), a' = a'{t,6) - a'{t,6),
I h' = h'{t,6) - h'{t,6), g' = g'{x) - g'{x).
Then one has
D.12) \g' -g\ = \g'{x) - g'{x) - g{x) + g{x)\ < \\g' - g\\o\x\
Similarly, we have
D.13)
b'-b\<\\b'-bUt)\6\,
a' -a\<\\cj'-cjUt)\e\,
{\h'-h\<\\h'-hUt)\6\.

§4. Properties of the bridges 157
Hence, it follows that
(*m(|,),(|,)) = (*m(|),(|))
.2<.,r,(!),(.,»_),
D.14) +(*(r) (/_.). (/_.))
>i|iP + 2(B(T)J,r-9) + <C(T)(s' + s),s'-9)
>{S- 2|B(T)|||9' - 91k - |C(T)|||9' + gUg' - 9||«}|i|'
provided \\g' — g\\o is small enough. Similarly, we have the following:
<-#p+2(*(,)(!).(|::|))
() +{*wP'-"VP'-')>
0 y ' V 0
< -E|^|2 + 2{A(i)x+B(i)^j/,6'-6)
+ 2{B{t)x + C{t)y,h' -h)
+ 2(B{tfz,d'-d) + (A{t){d' + d),d'-d)
<[-5 + 2{\A{t)\ + \B{t)\)\\h'-hUt)
+ 2{\B{t)\ + \C{t)\)\\h' -hUt)
+2\Bmw - Hio(i) + \Amw+^iio(i)ik' - c7Ut)]\e\\
Then, our assertion follows. D
The above result tells us that if the equation associated with F is
solvable and F admits a strong bridge, then all the equations "nearby" are
solvable. This is a kind of stability result.
Remark 4.3. We see from D.14) and D.15) that the condition D.10) can

158
Chapter 6. Method of Continuation
D.16)
be replaced by
{2{\BiT)\ + \CiT)\\\9'+9\\o)\\9'~9\\o<S,
sup {2{\A{t)\ + \B{t)\)\\b'-b\\o{t)
+ 2{\Bit)\ + \Cit)\)\\h'-hUt)
+ [2\B{t)\ + \A{t)\\\a' + aUt)]\\a' - a\\o{t)} < S,
where E > 0 is the one appeared in the definition of the bridge (see
Definition 1.3). Actually, D.16) can further be replaced by the following even
weaker conditions:
f 2 {B{T)x,9'-9) + {C{T){g' + g),g' -g)> -S\x\\
Wx,xeWi",
{2{A{t)x + B{tfy,b'-b)
+ 2 {B{t)x + Cit)y,h' -h)+2 {B{t)'^z, a' -a)
+ {A{t){a' + a),a'-a)'j <s\e\\ ve,eeM.
The above means that if the perturbation is made not necessarily small but
in the right direction, the solvability will be kept. This observation will be
useful later.
To conclude this section, we present the following simple proposition.
Proposition 4.4. Let T > 0, T = {b,a,h,g) £ H[0,T] and $ £
^/(r;[0,r]). Let Pen and
D.17)
sup
te[o,T]
D.18)
$(i)=e2'^*$(i), te[0,T],
T=ib-Px,a,h-l3y,g)eH[0,T].
Then, $e^/(r;[0,r]).
The proof is immediate. Clearly, the similar conclusion holds if we
replace Bi{T; [0,T]) by Bii{T; [0,T]), ^(F; [0,r]) or ^'(F; [0,r]).
§5. Construction of Bridges
In this section, we are going to present some more results on the solvability
of FBSDEs by constructing certain bridges.
§5.1. A general consideration
Let us start with the following linear FBSDE:
E.1)
. ^@) = X,
te[o,T],
YiT)=GXiT)+go,

-^■''^%)y
§5. Construction of bridges 159
where A £ ]r("+™>x("+™>, g G R™'^", 7 = (bo,ao,ho,90) £ ?^[0,r] (see
A.3)) and x £ R". We have the following result.
Lemma 5.1. Let T > 0. Then, the two-point boundary value problem
E.1) is uniquely solvable for all 7 £ ?^[0, T] if and only if
E.2) det{(-G,/)e-^*(^5)}>0, t e [0,T].
Proof. Let
as)=(-o5)(n':0-
Then we have the linear FBSDE for (^, rj) as follows:
f<^s;0={(--)<-)as:!)
boit)
E.3) I ' \hoit)Gbo
t e@) = x, r?(r) = go,
Clearly, the solvability of E.3) is equivalent to that of E.1). By Theorem
3.7 of Chapter 2, we obtain that E.3) is solvable for all 7 G ?^[0, T] if and
only if C.16) and C.19) of Chapter 2 hold. In the present case, these two
conditions are the same as E.2). This proves the result. Q
Now, let us relate the above result to the notion of bridge. Prom
Theorem 2.1, we know that if Fi and r2 are linked by a bridge, then
Fj and F2 have the same solvability. On the other hand, for any given F,
Corollary 2.4 tells us that if F admits a bridge, then, the FBSDE associated
with F admits at most one adapted solution. The existence, however, is not
claimed. The following result tells us something concerning the existence.
This result will be useful below.
Proposition 5.2. Let Tq > 0 and F = F,0, h,g) with
^^•^) {h{i])=^{l)' 9{x)=Gx, V{t,e)e[0,To]xM.
Then F G S[0, T] for allTe @, To] if B{T; [0, T]) 7^ (p for all T e @, Tq].
Proof. Since B{T][0,T]) ^ 4>, by Corollary 2.4, E.1) admits at most
one solution. By taking 7 = {bo,(To,ho,9o) = 0 and a; = 0, we see that
the resulting homogeneous equation only admits the zero solution. This is
equivalent to that E.1) with the nonhomogeneous terms being zero only

160 Chapter 6. Method of Continuation
admits the zero solution. On the other hand, in this case, the solution of
E.1) is given by
E.5) (?S)=^"(nO))' ^^[^'^1'
with the condition
E.6) 0 = (-G,/) ( J[j;j) = (-G,/)e^^ (j) Y{0).
We require that E.6) leads to Y{0) = 0. Thus, it is necessary that the left
hand side of E.2) is non-zero for t = T. Since T £ (OjTo] is arbitrary, we
must have E.2). Then, by Lemma 5.1, we have F £ <S[0,r]. D
Let us now look at some class of nonlinear FBSDEs. Recall the semi-
norms II . ||o(i) defined by D.8).
Theorem 5.3. Let Tq > 0, A E ]r("+">x("+™> and T = {b,0,h,g) be
defined by E.4). Suppose E.2) holds for T = To and that B'{T; [0, T]) ^ 4>
for allT e {0,To]. J'henjor any T e {0,T], there exists an e > 0, such that
foran/3enandT= {b,a,h,g) e H[0,T] with
E.7) ||r||o(t)<e, te[o,r],
the following FBSDE:
E.8)
+ ("^'^®f)))dW(t), te[o,n
[ X{0) = X, Y{T) = GX{T) + g{X{T)),
admits a unique adapted solution 0 = {X,Y, Z) £ A^[0,r].
Proof. We note that if
E.9) b{t,6) = ef^^bit, e-^^e), V(t,6) £ [0, T] x M,
then,
E.10) ll&llo(i) = ||6||o(i), vte[o,r].
Similar conclusion holds for ct, h and 'g if we define a, h and g similar to
E.9). On the other hand, if 0(t) = (X(t), Y{t), Z{t)) is an adapted solution
of E.8) with /3 = 0, then 0(t) = e''*0(t) is an adapted solution of E.8).
Thus, we need only consider the case /3 = 0 in E.8). Then, by Theorems
2.1, 4.2 and Proposition 5.2, we obtain our conclusion immediately. D
We note that FBSDEs E.8) is nonlinear and the Lipschitz constants
of the coefficients could be large. Also, E.8) is not necessarily decoupled
nor with monotonicity conditions. Thus, Theorem 5.3 gives the unique

§5. Construction of bridges
161
solvability of a (new) class of nonlinear FBSDEs, which is not covered
by the classes discussed before. On the other hand, by Remark 4.3, we
see that condition E.7) can be replaced by something like D.16), or even
D.17). This further enlarges the class of FBSDEs covered by E.8).
We note that the key assumption of Theorem 5.3 is that T = F,0, h, g)
given by E.4) admits a strong bridge. Thus, the major problem left is
whether we can construct a (strong) bridge for F. In the rest of this section,
we will concentrate on this issue.
We now consider the construction of the strong bridges for F =
{b,0,h,g) given by E.4). Prom the definition of strong bridge, we can
check that $ £ 'S'(F; [0,r]) if it is the solution to the following differential
equation for some constants K,K,S,e > 0,
E.11)
^(t) + A'^^it) + ^t)A= -SI,
te[o,T],
0 -K
satisfying the following additional conditions:
E.12)
G,0)$(i)( J)>0, @,7)$(i)E
<-£/, Vie[o,r],
G,G^)$(r)
Gl^''-
On the other hand, we find that the solution to E.11) is given by
E.13)
m
= e-^''
K
0
q_
-K
,-At
Jo
^-A s^-As^^^
ie[o,r].
Thus, in principle, if we can find constants K, K,S,e > 0, such that E.12)
holds with $(i) given by E.13), then we obtain a strong bridge $(•) for F
and Theorem 5.3 applies.
§5.2. A one dimensional case
In this subsection, we are going to carry out a detailed construction of strong
bridges for a case of n = m = d= 1 based on the general consideration of
the previous subsection. The corresponding class of solvable FBSDEs will
also be determined.
Let F = F,0, h, g) be given by
E.14)
h{t,x,y,z)
h{t,x,y,z)
g{x) = -gx,
= A
-A ti\ ( X
0 0 [y
for all {t,x,y,z) £ [0,oo) x ]R^, with A,/i,3 £ IR being constants satisfying

162 Chapter 6. Method of Continuation
the following:
E.15) X,l^,9>0, ^ + ^-ff'^0-
We point out that conditions E.15) for the constants \,H,g are not
necessarily the best. We prefer not to get into the most generality to avoid some
complicated computation. Let us now carry out some calculations. First
of all
E.16) e-=(-;' 5A^""'), «>«.
Thus, for all t > 0,
e-^^^i'l \\e-^^
E.17)
f(l-e>*) l)\0 -K)\o 1
^e2At iQi(gAt _ g2At) N
/^(gAt_g2At) _;^+/^(i_eAtJJ'
and
E.18)
rt ^ rt / „2As M/"pAs _ „2As\ \
Jo Jo [f{e^^-e^^^) 1 + >^il - e^^r) "'
^(e2A*-l) _^(eA*-lJ
We let iiT > 0 be undetermined and choose
E.19) ^=h ^^'^-
Then, according to E.13), we define
*<"^(^s:! Z)

§5. Construction of bridges
with
163
E.20) <
^(„ = ;,."-^(e--I)=(^-A)e»+^
1
4A
(e^^' + 2),
B(i) = ^(eA*_e2A*) + ^(e^*-lJ
4A2
(e2At+e^*-2),
C{t)
--^
(e^*-l)^-g(e^*-2J
<5(A2+^^) V
A2 '^2A3
i2
A2
f'\(^K-^)e'^'-^^
3E ^
(--x)'"-^(--3
2Ar A2 V" XJ^ yV 2\y
_^_Si>^^,
A2
J^lJlXt
2 1 „2
4A3
(e^^* + 2e^' + 3) - ii'
A2
Prom E.12), we need the following: (e > 0 is undetermined)
E.21)
{A{t)>0, C{t)<-e, Vie[0,r],
\A{T)-2gB{T)+g''C{T)>e.
Let us now look at these requirements separately.
First of all, it is clear true that A{t) > 0 for all i G [0,r]. Next,
C{t) < -£ for all t e [0,r], if and only if
E.22) K>e+^{e''' + 2e''+3)-^^±^t^m, t E [0,T].
Since f"{t) > 0 for alH G [0,oo), the function f{t) is convex. Thus, E.22)
holds if and only if
E.23)
K>f{0)VfiT).

164 Chapter 6. Method of Continuation
Finally, we need
e < A{T) - 2gB{T) + g^C{T)
= ^(e^^^ + 2) + ^(e^^^ + e^^-2)
+ £V(^2AT^2e>T^3)_^.^_£!(ANV)^
E.24) ^Ia^^'^ ^^^ +^;-ff^ ^
- J-{^^ iR\\2XT , 9f^ {-,, £^\ AT _ g^(A^ + /f^)
Ar+ aJ ^ +2A2l^ AT A2
2A A2 ^ 4A3 ^
Thus, we need (note E.23))
E.25) , ± _ ff^ , MV _ ^
_2A A2 ^ 4A3
>9^K>g^mvf{T)).
We now separate two cases (with f{T) and /(O), respectively). First of all,
for /(T), we want
0 < F{T) - ffV(r)
We see that T i-> F{T) is monotone increasing. Thus, to have the above,
it suffices to have
E.27) 0<F@) = ^-e(l+ff2).
Hence, in what follows, we take
E.28)
4A(l + ff2)-
Then, E.26) holds. Next, we claim that under E.15) and E.28), the
following holds.
E.29) F(r) - ffV(O) > 0.
In fact, by the choice of e and by E.27),
E.30) F@) - ffV(O) = ^@) = 0.
On the other hand,
E.31) F'(T) = i(l+f)V-+|f(l + f)e--£!i^.

§5. Construction of bridges 165
Thus, by E.15), it follows that
E.32) F'iO)={\ + ^-l^-9')>0.
Then, by F"{T) > 0, together with E.30) and E.32), we must have E.29).
Hence, we obtain E.25). This shows that a strong bridge $(i) has been
constructed with K, S and e being given by E.19) and E.28), respectively,
and we may take
E.33) F=/@)V/(r).
It is interesting that the $(•) constructed in the above is not in
B{To; [0, T]) for any T > 0 since A{t) > 0. On the other hand, we note that
both A(t) and B{t) are independent of T. However, due to the fact that
K depending on T, C{t) depends on T. But, we claim that there exists a
constant Co > 0, only depending on X,lJ.,g (independent of T), such that
E.34)
Co - f{T) < C{t) < -43^(YT^' * ^ [^'^l'
Co < C{T) < ^
4A(l + ff2)'
where f{t) is defined by E.22). In fact, by E.20), E.22), E.28) and E.33),
we have
E.35) C{t) = fit) - /(O) V f{T) - Ji^^^ •
Clearly, C{t) is convex. Thus,
E.36) C{t)<C{0)vC{T) = -^~^^^, Vie[0,r].
On the other hand, by the fact that f{t) is strictly convex and
limt_>oo f{t) = 00, we see that there exists a unique To > 0, only depending
on A and fi, such that
E.37) C{t)>f{To)~f{0)Vf{T)-j^~-^, te[0,T].
This proves the first relation in E.34). Next, we see easily that there exists
a unique Ti > To, such that f{Ti) = /(O), and
.5 3g. f/(i)</@), Vie[o,ri],
\/(i)>/@), Vie(ri,cx)).
Hence, we obtain
E.39) CiT) > /(To) - /(O) -
4A(l + ff2)-
This proves the second relation in E.34).

166
Chapter 6. Method of Continuation
Now, from Remark 4.3 and Theorem 5.3, we know that the following
FBSDEs is solvable on [0,r].
E.40)
dX{t) = {{p- \)X{t) + fiY{t) +b{t, X{t), Y{t),Zit))}dt
+ a{t,X{t),Y{t),Z{t))clW{t),
dY{t) = {CY{t) + hit,X{t),Y{t),Z{t))}dt+ Z{t)dWit),
X{0) = X, Y{T) = -gX{T) + g{X{T)),
where \,tJ.,g > 0 satisfying E.15), /3 £ R, and T = {b,a,h,g) £ H[0,T]
satisfying
E.41) <
B\B{T)\\\g\\o + \CiT)\\\g\\l<eAl,
sup {2{\A{t)\ + \B{t)\)\\b\\o{t) + 2{\B{t)\ + \C{t)\)\\h\\o{t)
te[o,T] ••
+ 2\Bm\a\\oit) + \Am\aUt?}<eAl,
with A{-), B{-) and C() given by E.20) and e > 0 given by E.28). If we
use D.17), then, E.41) can be relaxed to the following:
' 2B{T)xS + C{T)(§ - 2gx)f > -(eAl)|x|^ Va;,x£]R,
E.42) J '"P {HMt)x + Bitfy)l + 2{Bit)x + Cit)y)t
+ 2B{t)z9 + A{t)d'^} < (eAl)l^p, ^6,6 e M.
If b, a, h and 'g are differentiable, then, we see that E.42) is equivalent to
the following:
f 2B{T)g,{x) + C{T){g,{x) - 2g)g,{x) > -(e A 1), Vx £ R,
/Ait) Bit) 0 \
[Bit) at) 0 (V6(i,6>),V/i(i,6>),Vcr(i,6>))
V 0 0 Bit)J
E.43) i
(Ait) Bit) 0 \ _ ,
+ B(i) at) 0 \ivbit,e),vhit,e),vait,e))\
V 0 0 Bit))
+ Ait)Vait,e){Vait,e)} <eAl, V(i,6>) £ [0,T] x M,
where V6(i,0) = ibxit,6),byit,6),bzit,6))^, and so on. Some direct
computation shows that the first relation in E.43) is equivalent to the following:
E.44)
A / eAl /B(r) \^ BiT) ^ _ , ,
=~\l\cm^^W)~')'Of)^'-'^^''^
<
ej^l , fBiT) y
\C(T) ^)
BiT)
\CiT)\ ' \CiT) ""} ■ CiT)
+
+ ff, Va; £ R.

§5. Construction of bridges 167
By E.34), we know that C{T) is bounded uniformly in T, while, B{T) ->
—00 as T -> 00 (see E.20)). Thus, by some calculation, we see that
E.47)
E.45) _J-£AL>_r(r).;-cx), asT-xx),
and ff need only to satisfy the following:
E.46) -r{T) < g^{x) < 0, Vi £ R.
Clearly, the larger the T, the weaker the restriction of E.46). The second
condition in E.43) is also checkable (although it is a little more complicated
than the first one). It is not hard to see that the choice of functions b and ct
are independent of T as A{t) and B{t) do not depend on T. However, since
C{t) depends on T, by some direct calculation, we see that in order FBSDE
E.40) is solvable for all T > 0, we have to restrict ourselves to the case that
h{t,6) = h{t,y). Clearly, even with such a restriction, E.40) is still a very
big class of FBSDEs, which are not necessarily decoupled, nor monotone.
Also, CT is allowed to be degenerate. We omit the exact statement of the
explicit conditions on 6, ct and h under which E.40) is solvable to avoid some
lengthy computation. Instead, to conclude our discussion, let us finally look
at the following FBSDE:
' dX{t) = {{l3-X)X{t) + iJLY{t) + b{t,X{t),Y{t),Z{t)))dt
+ a{t,X{t),Y{t),Z{t))dW{t),
dY{t) = {f3Y{t) + ho{t)}dt + Z{t)dWit),
tX@) = a;, YiT) = -9XiT) + go,
with \,iJ,,g>0 satisfying E.15) and
sup {2{\A{t)\ + \B{t)\)\\bUt) + 2\Bit)\ ||a||o(i)
E.48) *e["'°°'
+ \Ait)\\\aUt?}<eAl.
This is a special case of E.40) in which h = ho and 'g = go- Then, by the
above analysis, we know that E.47) is uniquely solvable over any finite time
duration [0,r]. Condition E.48) can be carried out explicitly as follows:
{2(e2>*+2) + ?^(e2^*+e^*-2)}||6||o(i)
E.49) +?^B2A*+e^*-2)||a||o(i)
+ (e^^* + 2)\\a\\o{tf < min{4A, j^}, t £ [0, oo).
It is clear that although E.47) is a special case of E.40), it is still very
general and in particular, it is not necessarily decoupled nor monotone.
Also, if we regard E.47) as a nonlinear perturbation of E.1) (with m =

168 Chapter 6. Method of Continuation
n = d= 1 and E.14) holds), then the perturbation is not necessarily small
(for t not large).

Chapter 7
Forward-Backward SDEs with Reflections
In this chapter we study FBSDEs with boundary conditions. In the simplest
case when the FBSDE is decoupled, it is reduced to a combination of a well-
understood (forward) reflected diffusion and a newly developed reflected
backward SDE. However, the extension of such FBSDEs to the general
coupled case is quite delicate. In fact, none of the methods that we have
seen in the previous chapters seems to be applicable, due to the presence of
the reflecting process. Therefore, the route we take in this chapter to reach
the existence and uniqueness of the adapted solution is slightly different
from those we have seen before.
§1. Forward SDEs with Reflections
Let O be a closed convex domain in R". Define for any x £ dO the set of
inward normals to O at a; by
A.1) A4={7:|7| = 1, and {7,a;-y)<0, Vy G O}.
It is clear that if the boundary dO is smooth (say, C^), then for any x £ 90,
the set Mx contains only one vector, that is, the unit inner normal vector
at X. We denote By([0,r]; R") to be the set of all R"-valued functions of
bounded variation; and for t] G By([0,r];R"), we denote |T?|(r) to be the
total variation of t? on [0,r].
A general form of (forward) SDEs with reflection (FSDER, for short)
is the following:
A.2) X(t) = x+ f bis,X{s))ds+ f a{s,X{s))dW{s)+ri{t).
Jo Jo
Here the b and a are functions of {t,x,Lj) G [0,r] x R" x fi (with w being
suppressed, as usual); and t] G BV}r([0,r];R'"), the set of all {!Ft}t>o-
adapted processes r] with paths in BV{[0,T];TR!").
Definition 1.1. A pair of continuous, {:Ft}t>o-adapted processes {X,r)) G
L2,([0,r];R") X BV>([0,r];R") is called a solution to the FSDER A.2) if
1) x{t) eO,vte [o,r],a.s.;
2) nit) = /o l^xis)€ao}lis)d\v\is), where 7E) G Mxis), 0<s<t<T,
d|T?|-a.e.;
3) equation A.2) is satisfied almost surely.
A widely used tool for solving an FSDER is the following
(deterministic) function-theoretic technique known as the Skorohod Problem: Let the
domain O be given.

170 Chapter 7. FBSDEs with Reflections
Problem SP{-;0): Let V £ C([0,r];]R") with V@) £ O be given. Find
a pair (93,t?) G C([0,r];]R") x By([0,r];]R") such that
1) ip{t) = i^{t)+r]{t), Vi e [0,r], and <^@) = V@);
2) V3(i) GO, fori e [0,r];
3) |r?|(i) = /o hv{s)€dO}d\'n\{s)\
4) there exists a measurable function 7 : [0,r] i-> R", such that 7(i) £
■Mp(t) M^l a.s.) and r}{t) = !^-i{s)d\r}\{s).
A pair ((/?, tj) satisfying the above l)-4) is called a solution of the
It is known that under various technical conditions on the domain O
and its boundary, for any tp G C([0, r];]R") there exists a unique
solution to SP{ip;0). In particular, these conditions are satisfied when O
is convex and with smooth boundary, which will be the case considered
throughout this chapter. Therefore we can consider a well-defined
mapping r : C([0,r];]R") ^ C([0,r];]R") such that T{iP){t) = ip{t), t £ [0,r],
where ((/?,??) is the (unique) solution to SP{'tp;0). We will call F the
solution mapping of the SP{-; O).
An elegant feature of the solution mapping F is that it may have a
Lipschitz property: for some constant K > 0 that is independent of T, such
that for ipi e C([0,r],]R"), i = 1,2, it holds that
A.3) |F(V'i)(-) - F(V'2)(-)It < K\M-) - M-)\t,
where \^\*f. denotes the sup-norm on [0,i] for ^ G C([0,r];]R").
Consequently, if ((/5j,??i), 2 = 1,2 are solutions to SP{rpi; O), i = 1,2, respectively,
then for some constant K independent of T,
A.4) |<^i(-) + M-)\t + |r?i(-) - r?2(-)lT < K\M-) - M'Wt-
In what follows we call a (convex) domain O C R" regular if the
solution mapping of the corresponding SP{-;0) satisfies A.3). The
simplest but typical example of a regular domain is the "half space" O =
1R-+={(a;i,-• • ,a;n) £ R" : a;„ > 0}. With a standard localization
technique, one can show that a convex domain with smooth boundary is also
regular. A much deeper result of Dupuis and Ishii [1] shows that a convex
polyhedron is regular, which can be extended to a class of convex domains
with piecewise smooth boundaries. We should note that proving the
regularity of a given domain is in general a formidable problem with independent
interest of its own. To simplify presentation, however, in this chapter we
consider only the case when the domains are regular, although the result we
state below should hold true for a much larger class of (convex) domains,
with proofs more complicated than what we present here.
We shall make use of the following assumptions.
(Al) (i) for fixed x £ IR", b{-,x,-) and a{-,x,-) are {.Ft}t>o-progressively
measurable;

§2. Backward SDEs with Reflections 171
(ii) there exists constant K > 0, such that for all (i,w) £ [0, T] x ft and
a;,a;' e R", it holds that
\b{t,x,u;)-b{t,x',ij)\<K\x-x'\;
\a{t,x,uj) - a{t,x',uj)\ < K\x — x'\.
Theorem 1.2. Suppose that O C R" is a regular, convex domain; and
that (Al) holds. Then the SDER A.2) has a unique strong solution.
Proof. Let T be the solution mapping to SP{-; O). Consider the
following SDE (without reflection);
A.6) X{t) = x+ [ b{s,X{-))ds + [ a{s,X{-))dW{s),
Jo Jo
where for y(-) £ C{[0,T];WC),
b{t,y{-),ij) = b{t,riy)it),ij); a{t,y{-),ij) = a{t,T{y){t),cj).
Note that for any {.Ft}t>o-adapted, continuous process Y, the processes
b{-,Y{■),■) and a{-, ¥{■),■), are all {.Ft}t>o-progressively measurable.
Further, the regularity of the domain O implies that there exists a constant
Kq > 0 depending only on the Lipschitz constant of T and K in (Al), such
that for any {.Ft}t>o-adapted, continuous processes Y and Y', it holds that
\bis,Yi;u;),u;) ~bis,Y'i;u;),u;)\: < Ko\Yis,u;) -Y'is,u;))\:;
\dis,Yi;u;),u;) ~^is,Y'i;u;),u;)\: < Ko\Yis,u;) -Y'is,u;))\l
for all (t, Lj) e [0, T] x O. Therefore, by the standard theory of SDEs (cf.
e.g., Protter [1]), we know that the SDE A.6) has a unique strong solution
X.
Next, we define a process X{t) — r{X){t), t G [0, T]. Then by definition
of the Skorohod problem, we see that there exists a process r] such that
{X,r]) satisfies the conditions l)-3) of Definition 1.1. Consequently, for all
te [0,r], wehave
Xit) = Xit)+r,{t)
= x+ bis,Xi-))ds+ a{s,X{-))dW{s)+r]{t)
Jo Jo
= x+ I b{s,X{s))ds+ I a{s,X{s))dW{s)+r){t).
Jo Jo
In other words, iX,r]) is a solution to the SDER A.5). The uniqueness
follows easily from the construction of the solution and the Lipschitz property
A.3) and A.4). The proof is complete. D
§2. Backward SDEs with Reflections
In this section we study the reflected BSDEs (BSDERs, for short). For
clearer notation we will call the domain in which a BSDE lives by O2,

172 Chapter 7. FBSDEs with Reflections
to distinguish it from those in the previous section. A slight difference is
that we shall allow O2 to "move" when time varies, and even randomly.
Namely, we shall consider a family of closed, convex domains {02(^1^) :
(i,w) e [0,r] X ft} in R™ satisfying certain conditions. Let ^ G 02{T,uj)
be given, we consider the following SDE:
rp rp
B.1) Y{t) = i + j h{s,Y{s),Z{s))ds- j Z{s)dW{s)+C{T)-at)-
Analogous to the FSDER, we define the adapted solution to a BSDER
as follows:
Definition 2.1. A triplet of processes (F,Z,C) e L|-(n;C([0,r];]R'")) x
L2,@,r;]R'"'"*) X BVj:{[0,T\,Br) is called a solution to B.1) if
A) Y{t,u}) e 02(i,w), for all t £ [0,r], P-a.e.w;
B) for any {.Ft}t>o-adapted, RCLL process V{t) such that V{t) £
02it, •), Vi e [0,r], a.s., it holds that {Y{t) - y(i),dC(i)) < 0, as a signed
measure.
We note that Definition 2.1 more or less requires that the domains
{02(', •)} be "measurable" (or even "progressively measurable") in (i, w) in
a certain sense, which we now describe. Let y £ R™ and A C R™ be any
closed set, we define the projection operator Pr with respect to A, denoted
Pr{-; A), hy
B.2) Pr{y;A)=y-^VycF{y,A), y G R";
where d{-,-) is the usual distance function:
B.3) d{y,A)=mf{\y-x\:xeA}.
For each y £ R™, we define C{t,y,Lj) = Pr{y;02{t,Lj)). Throughout this
chapter we shall assume the following technical condition.
(A2) (i) For every fixed y £ R™, the process {t,uj) i-> P{t,y,uj) is {Tt}t>o-
progressively measurable;
(ii) for fixed y £ R", it holds that
B.4) E [ m,y,-)\^dt<cx^.
Jo
Before we go any further, let us look at some examples.
Example 2.2. Let 'Hm be the collection of all compact subsets of R™,
endowed with the Hausdorff metric d*, that is,
B.5) d*{A,B) =ma.x{svipd{x,B), supd{y,A)}, VA,Be'Hm-
It is well-known that CHm, d*) is a complete metric space. Now suppose that
02 ={02{t,uj) : (i,w) £ [0,r] X fi} C {'Hm,d*), then we can view O2 as an

§2. Backward SDEs with Reflections 173
CHm, d*)-va.lned process, and thus assume that it is {^t}t>o-progressively
measurable. Noting that for fixed y £ R™, the mapping A i-> d{y,A) is a
continuous mapping from {'Hm,d*) to R, as
\d{y,A)-d{y,B)\<d*{A,B), Vy £ R", 'iA,B eUm,
the composition function (i,w) i-> dP{y,0{t,u})) is {^t}t>o-progressively
measurable as well, which then renders VyCp(?/,02(-, )) '^n {-^t}t>o-
progressively measurable process, for any fixed y £ R™. Consequently,
O2 satisfies (A2)-(i).
Next, using elementary inequality \d{zi,A) — d{z2,A)\ < \zi — z-i\,
V^i, ^2 £ R", VA C R" one shows that
\^yS{y,02(tM)\ < 2diy,02it,uj)).
Assumption (A2)-(ii) is easily satisfied provided d{y,02{-,-)) £ L'^{[0,T] x
ri), which is always the case if, for example, 0 £ 02(^1^) for all (i, w), or,
more generally, 02{t,uj) has a selection in L^{0,T; fi). D
Example 2.3. As a special case of Example 2.2, the following moving
domains are often seen in applications. Let {0{t,x) : {t,x) £ [0,r] x R"} be
a family of convex, compact domains in R™ such that
(i) the mapping (i, x) i-> 0{t, x) is continuous as a function from [0, T] x
R" to i'Hm,d*).
(ii) for each (t, x), 0 £ 0{t,x); and there exists a constant C > 0 such
that
sup d*iO{t,x),O{t,0)) <C\x\.
te[o,T]
Let X £ L2,(fi;C([0,r];R")), and define 02{t,uj) = 0it,X{t,uj)), (i,w) £
[0,r] X n. We leave it to the readers to check that O2 satisfies (A2). Q
Example 2.4- Continuing from the previous examples, let us assume that
m = 1 and 0{t, x) = [L{t, x),U{t, x)], where -00 < L{t, x) < 0 < U{t, x) <
00 for all {t, x) £ [0, T] x R". Suppose that the functions L and U are both
uniformly Lipschitz in x, uniformly in i £ [0, T]. Then a simple calculation
using the definition of the Hausdorff metric shows that
d*{O{t,0),Oit,x)) =max{\L{t,x) - L{t,0)\, \Uit,x) ~ Uit,0)\} < C\x\.
Thus O2 satisfies (A2), thanks to the previous example. Q
Let us now turn our attention to the well-posedness of the BSDER B.1).
We shall make use of the following standing assumptions on coefficient
h : [0,r] X R" X R"'''* x fi h^ R" and the domain {02{t,uj)}.
(A3) (i) for each {y, z) £ R" xR"'''*, hi-,y, z, •) is an {jc-t}t>o-progressively
measurable process; and for fixed {t,z) £ [0,r] x R"'"* and a.e.w £ n,
h{t,-,z,Lj) is continuous;

174 Chapter 7. FBSDEs with Reflections
(ii) Ef^\h{t,0,0)\'^dt<cx;
(iii) there exist a £ R and k2 > 0, such that for all i £ [0,r], y,y' £
R", and z, z' G R"'"*, it holds P-a.s. that
' {y-y',Kt,y.^) - Ht,y',z)) <a\y-y'p;
< \hit,y,z)-hit,y,z')\<k2\z-z'\;
Jh{t,y,z) - h{t,0,z)\ < k2{l + \y\).
(iv) The domains {02(i,")} is "non-increasing". In other words, it
holds that
0{t,Lj) C 0{s,Lj), Vi>s, a.s.
Our main result of this section is the following theorem.
Theorem 2.5. Suppose that (A2) and (A3) are in force. Then theBSDER
B.1) has a unique (strong) solution. Furthermore, the process Ct is
absolutely continuous with respect to Lebesgue measure, and for any process Vj
such that Vt{uj) G 02{t,uj), Vt £ [0,r], a.s., it holds that
B.6) {^^Yt-Vt)<0, Vte[0,r], a.s.
at
Remark 2.6. Suppose m = 1 and O2 = [L,U], for appropriate processes
L and U. Denote by ^ = C"*""~ C~i Co^ = Co" = Oi the minimal decomposition
of C as a difference of two non-decreasing processes. By replacing V in B.6)
by
respectively, we obtain
B.7) {Yt - LudC+ ) = 0, {Yt-UudQ; ) = 0, Vt £ [0,T], a.s.
Proof of Theorem 2.5. Since the proof is quite lengthy, we shall split it
into several lemmas. To begin with, let us first recall the notion of Yosida
approximation, which is another typical route of attacking the existence and
uniqueness of an SDE with reflection other than using Skorohod problem.
Let ip be any proper, lower semicontinuous (l.s.c, for short), convex
function (by proper we mean that ip is not identically equal to -l-oo). Let
^(v) = {^ '■ v{x) < 00}. We define the subdifferential of ip, denoted by dip,
as
dip{y)={x* eWL"" ■.{x*,y-x)>0, yxeV{ip)}.
ows we denote A — dip. Define, for each e ;
B.8) ^M= inf {f|y-a;|'+<^(a;)}.
In what follows we denote A — dip. Define, for each e > 0, a function

§2. Backward SDEs with Reflections 175
Since R™ is a Hilbert space, and t/s is a l.s.c. proper convex mapping, the
the following result can be found in standard text (cf. Barbu [1, Chapter
II]):
Lemma 2.7. (i) The function ipe is (Frechet) diiferentiable.
(a) The Frechet differential ofipe, denoted by Dip^, satisfies Dip^ = A^,
where A^ is the Yosida approximation of A, define by
B.9) A,iy) = ^iy-My)), where My) = {I + eA)-'{y).
(Hi) \Mx) - My)\ <\x- y\; \Mx) - AMI < 7k - y\,
(iv) AM e d^iMy))-
(v) \AM\ A-.0 I ''^"^''^'' '\l ^ ^' whereA«(y)^Pra^(,)@),
1+00, otherwise,
y e R". D
Let us now specify a l.s.c. proper convex function to fit our discussion.
For any convex, closed subset O C R™, we define its indicator function,
denoted by ip := lo to be
, , A f 0 yeO;
[ +00 y fO,
In this case, 'D{ip) = O. Now by definitions B.8) and B.9), we have
^M= miUy-x\^ = ^dHy.O),
x^O ZE Z£
AM = D^M = ^Vd2(y,0) = \{y - Pr{y,0)\
Consequently, we have
■ JM=Pr{y;0), Ve>0;
B.10) \aM=^. yyeO,We>0;
.A°{y)=0, VyeO.
Further, we replace O by the CHm, d*)-vahied process {O2}, then
Veit,y,uj) = —Io2{t,u)iy), Ve > 0;
B.11) Mt,y,^) = iI + eA{t,;u;))-'iy);
A,{t,y,uj) = -{y - Je{t,y,uj)).
By B.10) we know that Js{t,y,Lj) = Pr{y,02{t,uj)), and by assumption
(A2) we have that for every e > 0, J,(-,y, •) £ L2,@,r;]R'") for all y £ R".

176 Chapter 7. FBSDEs with Reflections
Let us now consider the following approximation of B.1):
Y'{t)=i+ [ his,Y'{s),Z'{s))ds~ f Z'{s)dWis)
B.12) ^;^ •''
MY'is))ds,
I
where A^ is the Yosida approximation of A{t,Lj) = dIo^(^t,u) defined by
B.11). Since A,, is uniform Lipschitz for each fixed e, by Lemma 2.7-(iii)
and by slightly modifying the arguments in Chapter 1, §4 to cope with the
current situation where a in (A3) is allowed to be negative, one shows that
B.12) has a unique strong solution (y^,Z^) satisfying
B.13) e{ sup \Y'{t)\^+l \\Z'{t)fdt\<oo.
^ 0<t<T Jo '
We will first show that as e -> 0, (Y^, Z^) converges in a certain sense, then
show that the limit will give the solution of B.12). To begin with, we need
some elementary estimates.
Lemma 2.8. Suppose that condition (A3) holds, and that ^ £ L^^{Q.).
Then there exists a constant C > 0, independent of e, such that the
following estimates hold
B.14)
e\ sup |y^(t)p + f \Z'{t)\^dt} < C;
•• te[o,T] Jo -■
e[J \A,{t,Y^{t))\^dt]<C.
Proof. The proof of the first inequality is quite similar to those we
have seen many times before, with the help of the properties of Yosida
approximations listed in §2.2, we only prove the second one. First note that
since O2 is convex, so is ipg{t,-,u}) (recall B.11)). We have the following
inequality (suppressing w):
B.15) 'Pe{t,y) + {Dip,{t,y),y-y) <ip,{t,y), V(t,y), a.s.
Now let t = to < ii < • • • < in = r be any partition of [t,T]. Then B.15)
leads to that
B 16) V^iti^Y'it^)) + {Dve{ti,Y'{ti)),Y'{ti+i) - Y^iti))
<Veiti,Y'iti+i))<^eiti+l,Y'{ti+i)), a.S.,
where the last inequality is due to Assumption (A3)-iv). Summing both
sides of B.16) up and letting the mesh size of the partition maxj |ti+i ~ti\ ->
0 we obtain that
B.17) ^,{t,y'{t)) + l {D^,{s,Y'{s)),dY'{s)) <^,{T,0 =0.

§2. Backward SDEs with Reflections 177
Thus, recall the equation for Y^ we have
^,{t,Y'{t)) + - [ \D^,iY^{s))\^ds
B.18) <^eiT,0 + j {D^,is,Y^{s)),h{s,Y^{s),Z'{s)))ds
- j {D^,{Y'{s)),Z'dWs).
By Cauchy-Schwartz inequality and (A3)-(iii),
{Dve{t,y)Mt,y,z)) <Y^\D^e{t,y)\'' +eC{l + \\zf + \y\''), y{t,y,z).
We now recall that ip^ > 0; ^ e 02{T,-) (i.e., ipe{T,0 = 0); and
As{t,y,Lj) = Dips{t,y,Lj). Using the first inequality of this lemma we
obtain that
rp rp
eJ \A,{t,Y'{s))\^ds = E j \D^,{Y'{s))\''ds
<c{l + E sup \Y'{t)\^ + E[ \\Z'{t)fdt\<C,
^ te[o,T] Jo '
where G > 0 is some constant independent of e. Thus, by a slightly abuse
of notations on the constant C, we obtain the desired estimate. Q
Lemma 2.9. Suppose that the assumptions of Lemma 2.8 hold. Then
there exists a constant C > 0, such that for any e,S > 0, it holds that
B.19) e\ sup \Y'{t) - Y^{t)\^ + [ \Z'{t) - Z\t)\^dt} <{e + S)C.
*• te[o,T] Jo '
Proof. Applying Ito's formula we get
\Y'{t)-Y'{t)\' + l^ \\Z'{s)-Z'{s)fds
+ 21 {A,is,Y'is)) ~ Asis,Y\s)),Y'is) ~Y'is))ds
2 j { h{s, Y^{s),Z'is)) ~ h{s, Y'{s),Z\s)),Y'is) - Y\s) )ds
- 2 ^ {Y'{s)~ Y\s), [Z'{s) - Z'{s)]dW{s)).
Since Ae{t,y,Lj) £ dip{Je{y)), we have by definition that
{A,{t,y,uj),J,{t,y,uj)-x) > 0, V a; £ 02(i,w).
B.20)

178 Chapter 7. FBSDEs with Reflections
In particular for any y £ R™, and any S > 0, Js{t,y,uj) £ 02it,uj) and
therefore
{A,{t,y,ij),Mt,y,ij)-Js{t,y,ij))>0, VyGlR", a.e.wGfi.
Similarly,
{As{t,y,Lj),Js{t,y,Lj)- J,{t,y,Lj)) >0, V y £ R", a.e.wGfi.
Consequently, we have (suppressing w)
{A,(i,y)-Arf(i,y),y-y)
= {A,(i,y),[y- J,(i,y)] + [J,(i,y) - Jsit,y)] +Jsit,y) ~y)
B.21) +{Ai(i,y),[y-Ji(i,y)] + [Ji(i,y)-J.(i,y)] + J.(i,y)-y)
>-{A,(i,y),<5Ai(i,y))-{Ai(i,y),eA,(i,y))
= -{e + S){Mt,y),As{t,y}).
Also, some standard arguments using Schwartz inequality lead to that
B.22) 2{h{t,y,z) - h{t,y,7),y-y)) < ^\\z - z\\^ + C\y - y\\
Combining B.20)—B.22) and using the Burkholder and Gronwall
inequalities we obtain, for some constant C > 0,
e{ sup \Y'{t)~Y\t)\'+ [ \\Z^{t)-Z\t)fdt}
"• te[o,T] Jo ^
<{e + 6)E [ {A,{t,Y'{t)),As{t,Y\t))) dt
Jo
<ie + S){E f \Ae{Y'{t))\''dt-E j \A6{Y\t))\''dty <{e + 6)C,
thanks to B.14). This proves the Lemma. D
As a direct consequence of Lemma 2.8, we see that if we send e to
zero along an arbitrary sequence {£«}, then there exist processes Y G
L2,(fi;C([0,r];]R'")), Z G L2,(fi x [0,r];]R'")), independent of the choice
of the sequence {e„} chosen, such that
(y",Z")=(y^",Z^")->(y,Z), asn->oo,
strongly in ^^.(fi; C([0,r];]R'")) x L%{^ x [0,r];]R'").
Furthermore, by Lemma 2.8 and the equation B.12), it follows that
for somer? G L|-@,r;]R'"), C G L|.(fi; C([0,r];]R'")), and possibly along a
subsequence which we still denote by {£«}, it holds that
' A.^iY'-'i-)) -> ~n{-), weakly in L2,@,r;]R'");
e{ sup / A^^{Y^^{s))ds + C,{t)\ j->0, asn->oo.
•• 0<t<T Jo I -"

§2. Backward SDEs with Reflections 179
Here, we use —r] and —( to match the signs in B.1) and B.12). Obviously,
we see that the limiting processes Y, Z, and ^ will satisfy the SDE B.1),
and the proof of Theorem 2.6 will be complete after we prove the following
lemma.
Lemma 2.10. Suppose that the process iY,Z), r], and ( are defined as
before. Then {Y,Z,Q satisfies B.11), such that
(i) ElCliT) = E J^ \rjit)\dt< <x^;
(ii) Y{t) e 02it,-),yt e [0,T], a.s.;
(Hi) for any ROLL, {J-t} t>o-a-dapted process V, {Y{t)-V{t),r]{t)) < 0,
a.s., as a signed measure.
Proof, (i) We first show that C has absolutely continuous paths
almost surely and that C = V- To see this, note that r) is the weak limit of
Ag^{Y^")'s. By Mazur's theorem, there exists an convex combination of
Ag^iY^^Ys, denoted by Ag^iY"^), such that Ag^{Y^^) -> r], strongly in
L^{Q. X [0, T]; R™)). Note that for this sequence of convex combinations of
the sequence As^{Y^"), we also have
e\ sup / Ae„{Y'''{s))ds + C{t)\ j->0, asn->oo.
•• o<t<T Jo I -•
Thus the uniqueness of the limit implies that ((t) = J^ r]{s)ds, Vi £ [0,T].
Furthermore, since L^(ri) C L^(ri), we derive (i) immediately.
(ii) In what follows we denote d{y,t,Lj) = d{y,02{t,Lj)). Since 02{t,Lj)
is convex for fixed (i, w), d{-, 02{t,uj)) is a convex function. Further, since
O2 has smooth boundary, one derives from B.9) that
d{y,t,ij) = \y-Pr{y,02{t,u;))\^\y-My)\=£\My,t,^)\-
for all y e R™, and t £ [0,r], P-a.s.. Hence by part (i), we see that
B.23)
rp rp
e[ d{Y'{t),t,Lj)dt<eE [ \A,{Y'
Jo Jo
{t))\dt
< eVTE[ I \A,{Y'{t))\'^dty -> 0.
Next, define for each (i, w) £ [0,r] x Vt the conjugate function of d(-,t,u))
by
B.24) G{z,t,u>)=mi{d{y,t,u>) - {z,y)],
y
and define the effective domain of G by
B.25) V^{t,Lj) = {ze'K: G{z,t,uS) > ~oo}.
Since d(-,t,u)) is convex and continuous everywhere, it must be
identical to its biconjugate function, or equivalently, its closed convex hull (see

180 Chapter 7. FBSDEs with Reflections
Hiriart-Urruty-Lemarechal [1]). Consequently, the following conjugate
relation holds:
B.26) d{y,t,ij)= sup {G{z,t,ij) + {z,y)};
and both the infimum of B.24) and the supremum of B.26) are achieved
for every fixed (i, w). Now for fixed (i, w), and any zq G V'^{t,Lj), we let
yo = yo(i)W) be the minimizer in B.24). Then
d{yo,t,u)) ~ {yo,zo) = G{zo,t,u)) < d{y,t,u)) ~ {y,zo), Vy £ R",
and hence
{y~yo,zo)< d{y,t,Lj) ~ d{yo,t,Lj), Vy £ R".
Since it is easily checked that d{-, t,Lj) is uniformly Lipschitz with Lipschitz
constant 1, we deduce from above that |2;o| < 1. Namely 'D'^{t, w) C [—1,1].
Now let Y be the limit process of F^", we apply a measurable
selection theorem to obtain a (bounded) {.Ft}t>o-adapted process R, such that
R{t,uj) e V^{t,u)) C [-1,1], Vi, a.s.; and
B 27) I d{Y{t,Lj),t,Lj) = G{R{t,Lj),t,Lj) + {R{t,Lj),Y{t,Lj)),
^' ' \d{Y'-{t,uj),t,uj)>G{R{t,uj),t,uj) + {R{t,uj),Y'-{t,uj)),
Therefore, recall that F^" -> F, we have
E f d{Y{t),t,-)dt = E I ^G{Y{t),t,-) + (R{t),Y{t))dt
= lim E [ \G{Y{t),t,-) + {R{t),Y"{t)) dt
n->oo Jq I
< lim E [ d{Y^-{t),t,-)dt = 0,
n->oo Jq
thanks to B.23). That is, Ejjd{Y{t),t,-)dt = 0, which implies that
Y{t,Lj) e 02{t,Lj), dt X dP-a.e. Thus the conclusion follows from the
continuity of the paths of Y.
(iii) Let V{t) be any {.Ft}t>o-adapted process such that V{t,Lj) £
02it,uj), Vi e [0,r], P-a.s. For every e > 0, and i G [0,r], consider
B.28) A,{t) = E [ {MY%s))~V{s),A{Y'{s)))ds.
Jo
Since V{t) e 02{t,-), for all t, and A,{Y'{t)) £ a/o,(t,.)(J,(y^(i)) (see
Lemma 2.7-(iv)), we have
{MY'{t))-V{t),Mt,Y'{t)))>0.

§3. Reflected FBSDEs 181
Namely, Aj(i) > 0, Ve > 0 and te [0,T]. On the other hand, since
KS) = E j {{UY'{s)) - Y^{s), Ms, Y'{s)))
B.29) + {Y'{s) ~ V{s),A,{s, Y'{s))) ]ds
^EJ [~e\A,{s,Y'{s))\^ + {Y'{s)~V{s),A,{s,Y'{s)))]ds
Now using the uniform boundedness B.14) and the weak convergence
of {A5„(-,y^")(•))}, and the fact that F^" converges to Y strongly in
L|.(fi;C([0,r];]R'")), one derives easily by sending n -> oo in B.29) that
0<E [ {Y{s) ~ Vis), ~T]{s)) ds, Vi e [0, T].
Jo
Or equivalently,
{Y{t)-Vit),r,it)) = {Y{t)-V{t),^{t))<0, Vie[0,r], a.s.
as a (random) signed measure. Thus completes the proof of Lemma 2.10.
D
§3. Reflected Forward-Backward SDEs
We are now ready to formulate forward-backward SDEs with reflection
(FBSDER, for short). Let Oi be a closed, convex domain in R", and
O2 = {02{t,uj) : (i,w) e [0,r] X R" X fi} be a family of closed, convex
domains in R™. Let x £ Oi, and g : R" x fi h^ R" be a given Tt-
measurable random field satisfying
C.1) g{x,ij)e02{T,ij), y{x,uj).
Consider the following FBSDER:
C.2)
Xt = x+ / b{s,Xs,Ys,Zs)ds+ / a{s,Xs,Ys,Zs)dWs+T]t;
Jo Jo
Yt = g{XT) + j his,Xs,Ys,Zs)ds~ j Z^dWs + Ct ~ Q;
Definition 3.1. A quintuple of processes {X, Y, Z, r], Q is called an adapted
solution of the FBSDER C.2) if
1) iX,Y) e LJ,{n,CiO,T;WC x R")), Z £ L'^y,{0,T;Wi"''"^), (t?,C) G
BVriO,T;WC x R");
2) XteOi,Yte02it,-),Wte [0,r], a.s.;
3) \v\t = lo Mx.edo^}d\v\s; Vt = Io1sd\ri\s, Vi £ [0,r], a.s., for some
progressively measurable process 7 such that 7s G ^/^^(Oi), d|T?|-a.e.;
4) for all ROLL and progressively measurable processes U such that Ut G
02(i,-), Vie [0,r], a.s., one has {Yt-Ut,d(:t) < 0, Vie [0,r], a.s.;

182 Chapter 7. FBSDEs with Reflections
5) {X,Y,Z,r),C) satisfies the SDE C.2) almost surely.
In light of assumptions (Al)-(A3), we will assume the following
(A4) (i) Oi has smooth boundary;
(ii) 02{t,uj) C 02{s,uj), Vi > s, a.s.; and for fixed y £ R™, the
mapping {t,Lj) ^ p{t,y,ij) = Pr{y;02{t,ij) belongs to L2,([0,r];]R'").
(iii) The coefficients b, h, a, and g are random fields defined on
[0,r] X R" X R" X R"""* such that for fixed ix,y,z), the
processes b{-,x,y,z,-), h{-,x,y,z,-), and a{-,x,y,z,-) are {Tt}t>o-
progressively measurable, and g{x,-) is .Fy-measurable.
(iv) For fixed (i, x, z) and a.e. w, h{t, x, -, z,lj) is continuous, and there
exists a constant K > 0 such that \h{t,x,y,z,Lj)\ < iir(l-l-|a;|-l-|y|),
for all {t,x,y,z,Lj). Moreover,
rp rp
e[ \b{t,0,0,0)\'^dt + E [ \a{t,0,0,0)fdt + E\g{0)\^ <oo.
Jo Jo
(v) There exist constants h > 0,i = 1,2 and 7 £ R such that for all
(i,w) e [0,r] X fi and x={x,y,z),Xi={xi,yi,Zi) e R" x R" x
j^mxd^ i = l_2, andx''=(a;,y) for x = ix,y,z).
• \b{t,xi,Lj) ~ b{t,X2,uj)\ < K\xi -X2I;
• {h{t,x,yi,z,u;) ~ h{t,x,y2,z,u)),yi -2/2) < 7lyi -Vi?;
• \h{t,xi,y,zi,u}) ~ h{t,X2,y,Z2,Lj)\ < K{\xi ~ X2\ + \\zi ~ Z2\\);
. Mt,xi,cj) ~ a{t,X2,ij)W' < K^\x° ~ x«|2 + kUzi ~ Z2f;
• \g{xi,Lj) ~ g{x2,uj)\ < k2\xi - X2\.
We should note that if fci = ^2 = 0, then a and g are independent
of z, just as the many cases we considered before. Therefore, the FBSDE
considered in this chapter is more general. We note also that the method
presented here should also work when there is no reflection involved (e.g.,
Oi = R", O2 = R").
§3.1. A priori estimates
We first establish a new type of a priori estimates that is different from
what we have seen in the previous chapters. To simplify notations we shall
denote, for t G [0, T), H(i, T) = L%-{t, T; R), and let H<=(i, T) be the subset
of H(i, T) consisting of all continuous processes. For any A £ R, define an
equivalent norm on H(i, T) by:
T' ]
Then Ux{t,T)={^ G H(i,r) : ||^||t,A < 00} = U{t,T). We shall also use
the following norm on H'^(i, T):

§3. Reflected FBSDEs 183
and denote Hx^p{t, T) to be the completion of H"^(i, T) under norm | • | txp-
Then for any A and C, Ilx^p{t,T) is a Banach space. Further, if i = 0, we
simply denote IHU ^ IMIo.a; l-ll^ =1-l^,;,,^; H = H@,r); H<= = H<=@,r);
Ha = Hx{t,T), and H^.^ = Ux^T).
Moreover, the following functions will be frequently used in this section:
for A e Randie [0,r],
C.3)
A{\, t) = e-(^'^«)*; B{\, t) = L_^ = t f e-^'^dB.
A 70
It is easy to see that, for all A G R", B(A, •) is a nonnegative, increasing
function, A{\,t) > 1; and B(A,0) = 0, A{X,0) = 1.
Lemma 3.2. Let (A4) hold. Let {X,Y,Z,r],0 and {X',Y',Z',ri',C)
be two solutions to the FBSDER C.2), and let ^=^ - ^', where (, =
X, Y, Z, T], (, respectively.
(i) Let A e R, Ci, C2 > 0, and let Xi=\~K{2 + C^^ + C^^) - K^.
Then, for all A' £ R,
C.4)
e-^*£;|Xt|2 + (Ai-A') / e-^^e-^'(i--^E\Xr\^dT
Jo
< [ e-^^e-^'^'-^^{K{Ci + K)E\Yr\'' + {KC2 + kl)E\Zr\''}dT.
Jo
(ii) Let A e R and C3, C4 > 0, and let A2 = -A - 27 - KiC^^ + C^^).
Then, for all A' £ R,
e-At£;|y> |2 ^ (^^ _ y) f e'^^e"^'(-''^E\%\''dT
C.5) +{i^KCi)l e-^^e--^'(--*)£;|Z,|2dr
Consequently, if KC4 = 1 — a for some a £ @,1), then
C.6) e-'^E\XT\' + Xi\\X\\l < K{C, + K)\\Y\\l + {KC-, + k\)\\Z\\l.
C.7) ||X||^ < B{\,T)[K{C, + K)\\Y\\l + {KC-, + k\)\\Z\\l].
C.8) \\Y\\l < B{M,T)[kle-^''E\XT\^+KC^\\X\\l],
C.9) \\Z\\l < ^ii^[fc|e-^^£|lT|' + KC^\\X\\l]■
Proof. We first show C.4). Let t £ @,r], A, A' be arbitrarily given, and
consider the function Ft(s,a;) =e-^'e-^'(*-')|a;p, for {s,x) £ [0,i] x R".

184 Chapter 7. FBSDEs with Reflections
Applying Ito's formula to Ft{s, Xg) from 0 to t, and then taking expectation
we have
Jo
+ \\a{T,Xr,Yr,Zr)-a{T,X',X,Z'r)f}dr
+ 2E [ e-^^e-^'^'-^^{Xr,dfjr).
Jo
Since Xt,XI. e Oi, Wt e [0,T], a.s., we derive from Definition 3.1-C)
that e^^'^e"^'^'^'"'') {Xt,df]r) < 0 (as a signed measure), Vs G [0,r], a.s..
Therefore, repeatedly applying the Schwartz inequality and the inequality
2ab < co^ + c""^6^, Vc > 0, using the definition of Ai, together with some
elementary computation with the help of (A4), we derive C.4).
To prove C.5), we let Ft{s,x) — e"''^'e~'^'('~*'|a;p, and apply Ito's
formula to Ft{s, Yg) from i to T to get
-At
E\Yt\'' + (A' + A)£; f e-^^e-^'("-*'|y,|2dr
-XT-X'(T-t)pi„(y^ _ „/y/ m2
e e
E\giXT) - 9{X!r)\'
rT
+ 2 J e-^'^^-'^e-^- {Yr,hiT,Xr,Yr,Zr) ~hiT,X',,Y;,Z'r))dT
rT
+ 2E e-^'(^-*'e-^^{y^,dCr).
Again, since Y{t,-),Y'{t,-) £ 02{t,-), P-as., by Definition 3.1-D) we have
{Ft (w), d^t (w)) < 0, dt X dP-a.s.. Thus, by using the similar argument as
before, and using the definition of A2, we obtain C.5).
Now, letting A' = 0 and i = T in C.4) yields C.6); letting A' = Ai in
C.4) and then integrating both sides from 0 to T yields C.7), since B{Xi, •)
is increasing; letting A' = A2 in C.5) and integrating from 0 to T yields
C.8). Finally, note that if A2 < 0, then letting A' = A2 and i = 0 in C.5)
one has (remember KC4 = 1 — a)
Jo

§3. Reflected FBSDEs 185
while if A2 > 0, then let A' = 0 in C.4) one has
\\Z\\l<l{kle-'^E\Xr\'+KCs\\X\\l}-
Combining the above we obtain C.9). D
We now present another set of useful a priori estimates for the adapted
solution to FBSDER C.2). Denote a°{t,uj) = a{s,0,0,0,uj), f°it,uj) =
/(s,0,0,0,w), h°{t,uj) = h{t,0,0,0,uj), and g°{uj) = ff@,w).
Lemma 3.3. Assume (A4). Let {X,Y,Z,t],Q be an adapted solution to
the FBSDER C.2). For any A,_A' £ R.e > 0,_Ci,C2,_C3,C4 > 0, we deGne
A[ = Ai — A + K'^)e and A^ = A2 — e, where Ai and A2 are those deBned in
Lemma 3.2. Then
ft
e-^'E\Xt\^ + {\\ - A') / e-^'^'-^h-^'E\Xr\^dT < e-^''\x\^
Jo
C-10) +£e-^'(*--)e-^-{i£;|/(r,0,0,0)P+('l + ^)Kr,0,0,0)|2
+ K{Ci + K{l+e))E\Yr\'' + {KC2 + k\{l + e))E\Z,\''^dT.
and
e-^'E\Yt\^ + m - A') j e-'^^^-''^e-^^E\Y,\'dT
+ {I-kid) I e-^'("-*'e-^"£;|Z,pdr
C.11) "'*
</t2'(l + e)e-^'(^-*'e-^^£;|XT|2 + {l + %-^'^'^-''^e-^'^ E\g{<S)\''
nn
Consequently, ifC^ = i^, for some a £ @,1), we have
e-^'^E\XT\'+\{\\X\\l<
C.12)
\x\'' + K{C,+K{l+e))\\Y\
+ (Kd + klil + e))\\Z\\l + \\\ni +il + \] Ik' lu
\\X\\l<B{\\,T)
C.13)
\x\^+K{C,+K{l + e))\\Y\
+ (Kd +k\{l + e))\\Z\\l + \\\ni + A + J ) \W IIA

186
Chapter 7. FBSDEs with Reflections
C.14)
\Y\\l<B{\l,T)
kl{l+e)e-^'^E\XT\^+KCz\\X\\l
+ (l+Me-^^£;|ff«|^ + iMP,
C.15)
2 / A{\l,T)
\z\\i <
kl{l+e)e-^'^E\XT\^ + KC^\\X\\l
+ (l+lV-^^£;|ff«|2+i||;i«||2
§3.2. Existence and uniqueness of the adapted solutions
We are now ready to study the well-posedness of the FBSDER C.2). To
begin with we introduce a mapping T : H"^ i-> H"^ defined as follows: for
fixed X e R", let X = r(X) be the solution to the FSDER:
C.16) 'Xt = x+ [ b{s,Ys,Ys,Zs)ds+ [ a{s,Ys,Ys,Zs)dWs+TJt,
Jo Jo
where the processes Y and Z are the solution to the following BSDER:
rp rp
C.17) Yt=g{XT) + j h{s,Xs,Ys,Zs)ds~ j ZsdWs + QT-Qt-
Clearly, the assumption (A4) enables us to apply Theorem 2.5 to
conclude that the BSDER C.17) has a unique solution (F, Z, Q), which in turn
guarantees the existence and uniqueness of the adapted solution X to the
FSDER C.16), thanks to Theorem 1.2. Furthermore, by definition of \\
(Lemma 3.3) we see that if A is chosen so that Ai > 0, then it is always
possible to choose e > 0 small enough so that A[ > 0 as well; and C.12) will
lead to X e H;^ j^^ (since Ai > 0 and \\ > 0). Let us try to find a suitable
Ai > 0 so that F is a contraction on 'H.x^x^, which will lead to the existence
and uniqueness of the adapted solution to the FBSDER C.2) immediately.
To this end, let X\X^ e H<=; and let {Y\ Z\ Q) and (T,7y0,« = 1,2,
be the corresponding solutions to C.17) and C.16), respectively. Denote
^i = e - e, for i = X,Y,Z,'X. Applying C.6)-C.9). (with d = i^)
we easily deduce that
e-^^£;|AXTp + Ai||AX|
C.18)
where
< nia,T){kie-^' E\AXt\' + KCsWAXWi).
n{a,T)^KiCi+K)BCX2,T) + ^^^^iKC2 + kly,
a
C.19)
and (recall Lemma 3.2)
C.20) Ai = X~K{2 + Ci;^+C2^)~K^; A2 = ~X~2'y-K{C3^+ C^^).

§3. Reflected FBSDEs 187
Clearly, the function /*(-,•) depends on the constants K,ki,k2,'y, the
duration T > 0, and the choice of C1-C4 as well as A, a. To compensate the
generality of the coefficients, we shall impose the following compatibility
conditions.
(C-1) 0<kik2<l;
(C-2) k2 = 0; 3a e @,1) such that ii{a,T)KCz < Ai,
(C-3) k2 > 0; 3ao £ (^1^2,1), such that i^{al,T)k^ < 1 and Ai = ^.
We remark here that the compatibility condition (C-1) is not a surprise.
We already saw it in Chapter 1 (Theorem 1.5.1). In fact, in Example 1.5.2
we showed that such a condition is almost necessary for the solvability of
an FBSDE with general coefficients, even in non-reflected cases with small
duration. The first existence and uniqueness result for FBSDER C.2) is
the following.
Theorem 3.4. Assume (A4) and fix C4 = "j^". Assume that the
compatibility conditions (C-1), and either (C-2) or (C-3) hold for some choices of
constants A, a, and C1-C3. Then the FBSDER C.2) has a unique adapted
solution over [0,T].
Proof. Fix d = ^~^. First assume that (C-1) and (C-2) hold. Since
k2 = 0, C.18) leads to that
IIAXIP, < ^^^^^IIAXIP,
Since we can find Ci—C3 and a £ @,1) so that n{a,T)KCz < 1, F is a
contraction mapping on (H, \\ -Ha)- The theorem follows.
Similarly, if (C-1) and (C-3) hold, then we can solve A from C.20) and
Ai = KCzlk2, and then derive from C.18) that
\^X\lo-^^<,x{al,T)kl\l^X\%-^^,
Let Ci, i = 1,2,3 and ao £ (fcife,l) be such that iJ,{aQ,T)k2 < 1, the
mapping F is again a contraction, but on the space H;^j^j, proving the
theorem again. D
A direct consequence of Theorem 3.4 is the following.
Corollary 3.5. Assume (A4) and the compatibility condition (C-1). Then
there exists To > 0 such that for all T G @,ro], the FBSDER C.2) has a
unique adapted solution.
In particular, if either ki — 0 or k2 = 0, then the FBSDER C.2) is
always uniquely solvable on [0, T] for T small.
Proof. First assume ^2 = 0. In light of Theorem 3.4 we need only show
that there exists To = To(Ci,C2,C3,X,a) such that (C-2) holds for some
choices of C1-C3 and A, a, for all T £ @,ro].

188 Chapter 7. FBSDEs with Reflections
For fixed Ci, C2, C3, A, and a £ @,1) we have from C.19) that
a
Therefore, let C1-C3 and a be fixed we can choose A large enough so that
/i(a, 0)KC3 < Ai holds. Then, by the continuity of the functions A{a, •)
and B{a, •), for this fixed A we can find To > 0 such that /i(a, T)KCz < ^1
for all T e @,ro]. Thus (C-2) holds for all T G @,ro] and the conclusion
follows from Theorem 3.4.
Now assume that ^2 > 0. In this case we pick an qq G (^1^2) 1), and
define
C.21) s^-j--^>0.
Now let C2 = j^, Ci = —^, and choose A so that Ai = {k3C3)/k2 > 0.
Since in this case we have
, 2 r.^ -^'^2 + kj 1 kj 1
Ma^,0) = -^5-^ = ^ + ^<^,
thanks to C.21). Using the continuity of/i(aQ, •) again, for any Ci,C3 > 0
we can find ToiCi,C3) > 0 such that n{al,T)k^ < 1 for all T G @,ro].
In other words, the compatibility condition (C-3) holds for all T £ @, To],
proving our assertion again.
Finally if ki = 0, then (C-1) becomes trivial, thus the corollary always
holds. D
From the proofs above we see that there is actually room for one to play
with constant C1-C3 to improve the "maximum existence interval" [0, To).
A natural question is then is there any possibility that To = 00 so that the
FBSDER C.2) is solvable over arbitrary duration [0,r]? Unfortunately,
so far we have not seen an affirmative answer for such a question, even
in the non-reflecting case, under this general setting. Furthermore, in the
reflecting case, even if we assume all the coefficients are deterministic and
smooth, it is still far from clear that we can successfully apply the method
of optimal control or Four Step Scheme (Chapters 3 and 4) to solve an
FBSDER, because the corresponding PDE will become a quasilinear
variational inequality, thus seeking its classical solution becomes a very difficult
problem in general.
We nevertheless have the following result that more or less covers a
class of FBSDERs that are solvable over arbitrary durations.
Theorem 3.6. Assume (A4) and the compatibiUty condition (C-1). Then
there exists a constant A > 0, depending only on the constants K, ki,k2,
such that whenever 7 < -A, the FBSDER C.2) has a unique adapted
solution for allT > 0.
Proof. We shall prove that either (C-2) or (C-3) will hold for all T > 0
provided 7 is negative enough, and we shall determine the constant A in
each case, separately.

§3. Reflected FBSDEs 189
First assume ^2 = 0. In this case let us consider the following
minimization problem with constraints:
C.22) min F(C7i,C72, C3, Ai,a),
Cj>0, i = l,2,3;Ai>0,0<a<l,
Xl-2K[KC2+kj)C3>0
where
i^(Gi,G2,C3,Ai,a) = -= ^/^^^ , ,.2M^^ + ^
\i
C 23) ^ ^' - - - ^ \,-2{KC2 + kl)KC3
Let A be the value of the problem C.22) and C.23). We show that if
7 < -A/2, then (C-2) holds for all T > 0.
Indeed, if 7 < -A/2, then we can find Ci, C2, C3, Ai > 0 and a £ @,1),
such that Ai - 2(ii:C2 + k^KCs > 0, and
C 24) Xi~2{KC2 + kl)KC, '
+ Ki2 + Cr'+C2'+C^')+K'[^).
On the other hand, eliminating A in the expressions of Ai and A2 in C.20),
and letting C4 = ^^'^ we have
A2 = - (Ai + K{2 + C^^ + C2^ + C3-I) + j^ + K^) - 27.
Thus C.24) is equivalent to
C.25) J-|^a±^ + i^^±ii)W3<l,
Ai I- A2 a i
and A2 > 0. Consequently, A{X2,T) = 1 and B{\2,T) < AJ^ (recall C.3));
and C.25) implies that n{a,T)KCz < Ai, i.e., (C-2) holds for all T > 0.
Now assume ^2 > 0. Following the arguments in Corollary 3.5 we
choose Ai = =^ > 0, C4 = ^~^, and ao G (^1^2, !)• Let E > 0 be that
defined by C.21), and consider the minimization problem:
C.26) min ^(^1,^2,^3),
Ci>0, i = l,2,3; ^
Sa2-KC2>0
where
nC, C2, Cs) ^ "f2^!'jXf ^ + K{2 + Cf^ + C2' + ^3-^)
C.27) "V ^
+^+^.(a^).

190 Chapter 7. FBSDEs with Reflections
Let A be the value of the problem C.26) and C.27), one can show as in
the previous case that if 7 < -A/2, then A2 > 0 (hence A(\2,T) = 1 and
B{X2,T) < Aji), and /i(a2,r)fc| < 1. Namely (C-3) holds for all T > 0.
Combining the above we proved the theorem. D
§3.3. A continuous dependence result
In many applications one would like to study the dependence of the adapted
solution of an FBSDE on the initial data. For example, suppose that there
exists a constant T > 0 such that the FBSDER C.2) is uniquely
solvable over any duration [t,T] C [0, T], and denote its adapted solution by
(X*''^,y*''^,Z*''^,?7*''^,C*''^). Then an interesting question would be how the
random field {t,x) ^ (X*>^,y*'^, Z*'^,??*'*, C*'"^) behaves. Such a
behavior is particularly useful when one wants to relate an FBSDE to a partial
differential equation, as we shall see in the next chapter.
In what follows we consider only the case when m = 1, namely, the
BSDER is one dimensional. We shall also make use of the following
assumption:
(A5) (i) The coefficients b, h, a, g are deterministic;
(ii) The domains {02i-,-)} are of the form 0E,0;) = 02{s,X*''^{s,uj)),
(s,w) e [t,T] X R", where 02it,x) = {L{t,x),U{t,x)), where L(-,-) and
[/(•, •) are smooth deterministic functions of (i, x).
We note that the part (ii) of assumption (A5) does not cover, and is
not covered by, the assumption (A4) with m = 1. This is because when
m = 1 the domain O2 is simply an interval, and can be handled differently
from the way we presented in §2 (see, e.g., Cvitanic & Karatzas [1]). Note
also that if we can bypass §2 to derive the solvability of BSDERs, then
the method we presented in the current section should always work for the
solvability for FBSDERs. Therefore in what follows we shall discuss the
continuous dependence in an a priori manner, without going into the details
of existence and uniqueness again. Next, observe that under (A5) FBSDER
C.2) becomes "Markovian", we can apply the standard technique of "time
shifting" to show that the process {5^*'^(s)}s>t is .F*-adapted, where Tl =
a{Wr,t <r<s}. Consequently an application of the Blumenthal 0-1 law
leads to that the function u(t,x) = F/'^ is always deterministic!
In what follows we use the convention that X*'^(s) = x, F*'^(s) =
y*'^(i), and Z*'^(s) = 0, for s e [0,i]. Our main result of this subsection is
the following.
Theorem 3.7. Assume (A5) as well as (A4)-(m)-(v). Assume also
that the compatibility conditions (C-1) and either (C-2) or (C-3) hold- Let
u{t,x)=Yt''', {t,x) e [0,r] X Oi. Then u is continuous on [0,T] x O and
there exists C > 0 depending only on T, b, h, g, and a, such that the
following estimate holds:
C.28) \u{ti,Xi)-u{t2,X2)\'^ <C(\xi-X2\'^ + {l + \xi\'^V\x2\'^)\ti-t2\y

§3. Reflected FBSDEs 191
Proof. The proof is quite similar to that of Theorem 3.4, so we only
sketch it.
Let (ii,a;i) and (^2,3:2) be given, and let X = X'^'^i -X*^'^^ Assume
first ti > t2, and recall the norms || • \\t^x and |-|t,A,/3 at the beginning of
§3.1. Repeating the arguments of Theorem 3.4 over the interval [^2)^], we
see that C.8) and C.9) will look the same, with || • ||;^ being replaced by
II • \\t2,x; but C.6) and C.7) become
C.6)'
C.7)'
<K{C, + K)\\Y\\l,x + {KC2 + k\)\\Z\\l^x + E\X{t,)\\
\\X\\l^^<B{\i,T)[K{C,+K)\\Y\\l,x
+ {KC,+ki)\\Z\\l^x + E\X{t,)\%
where B{\T) ~ ^—^f^ . Now similar to C.18), one shows that
C.18)' ' ii'^ii*=>^
<,ji{a,T){kle-^'^E\XT\^ + KC,\\X\\l^^) + E\X{h)\\
Arguing as in the proof of Theorem 3.4 and using compatibility conditions
(C-l)-(C-3), we can find a constant C > 0 depending only on T > 0 and
K, k\, ^2 such that
C.29) \X\l^^^0<CE\X{h)\^=CE\x2-X''''''{t2)\\
where /3 = Ai - ii{a, T)KCz if ^2 = 0; and /3 = /i(a, T)kl if fe > 0.
Prom now on by slightly abuse of notations we let C > 0 be a generic
constant depending only on T,K, ki and ^2, and be allowed to vary from
line to line. Applying standard arguments using Burkholder-Davis-Gundy
inequality we obtain that
C.30) E sup \X\s)f + E sup \Y^{s)f < CE\X{t2)f,
To estimate E\X{t2)\^ let us recall the parameters Af and Al defined
in Lemma 3.3. For each e > 0 define
Since Af -> Ai, Al -> A2, and n^{a,T) -> ij,{a,T), as e -> 0, if the
compatibility condition (C-1) and either (C-2) or (C-3) hold, then we can choose
e > 0 such that n^{a,T)kl{l -I- e) < 1 when k2=0 and if{a,T)KC^ < \\
when k2 ^ 0. For this fixed e > 0 we can then repeat the argument of
Theorem 3.4 by using C.12)—C.15) to derive that
(l-"^^"f^^Oll^-|P.<^(^)k|-+(l + ^)
AI
^2 = 0;

192
Chapter 7. FBSDEs with Reflections
or
(l - ^^%a,T)kl)\X'\l0 < C{e) L ^ + (l + ^)
^2 7^0,
where C(e) is some constant depending on T, K, k\, ^2, and e. Since e > 0
is now fixed, in either case we have, for a generic constant C > 0,
\\X'\\l<C{l + \x,\''),
which in turn shows that, in light of C.12)-C.15) \\Y^\\\ < C(l + |a;ip),
and \\Z\W < C(l + \x\\'^). Again, applying the Burkholder and Holder
inequalities we can then derive
C.31) e\ sup \X\t)\''\ + E{ sup |F'(i)|'j<C(l + |a;in.
•• ti<s<T J ^ti<s<T -'
Now, note that on the interval [^1,^2] the process {X, Y, Z) satisfies the
following SDE:
C.32) <
nS nS
X{s) = (xi -X2)+ b\r)dr + / a\r)dW{r),
Y{s) = y(i2) + f ' h\r)dr + f ' Z\r)dW{r),
s e [ti,t2].
where b\r) = b{r,X^{r),Y\r),Z\r)), a^{r) = a{r,X\r),Y\r),Z\r)),
and h^ (r) = h{r, X^ (r), F^ (r), Z^ (r)). Now from the first equation of C.32)
we derive easily that
E{ sup \Xis)\-'}<C{\xi~-X2\^ + il + \xi\')\h-t2\}.
tl<S<t2
Combining this with C.30), C.31), as well as the assumption (A4-iv), we
derive from the second equation of C.32) that
E\Y{h)\^ < E\Y{t2)\^ + 0A + \xi\^ V \x2\^)\h - t2\
< C{\xi - a;2p + A + Ixip V |a;2p)|ii - ^2!}.
Since Y{ti) = u{ti,xi) —u{t2,X2) is deterministic, C.28) follows. The case
when ii < ^2 can be proved by symmetry, the proof is complete. D

Chapter 8
Applications of FBSDEs
In this chapter we collect some interesting applications of FBSDEs. These
applications appear in various fields of both theoretical and applied
probability problems, but our main interest will be those that related to the
truly coupled FBSDEs and their applications in mathematical finance. Let
us first recall the FBSDE in its general form: denote 0 = {X,Y, Z),
A-1) <^
X{t) = a; +
/ b{s,Q{s))ds+ I a{s,Q{s))dWis),
Jo Jo
Y{t)=g{X{T)) + j b{s,Q{s))ds-I Z{s)dW{s), t £ [0,T],
In different applications we will make assumptions that are variations of
what we have seen before, in order to suit the situation.
§1. An Integral Representation Formula
In this section we consider a special case: 6 = 0, and a is independent of z.
Thus A.1) takes the form:
rt rt
A.2)
X{t) = x+ [ b{s,e{s))ds+ [ a{s,Xis),Yis))dW{s),
Jo Jo
Y{t) = 9{X{T)) - [ Z{s)dW{s), t e [0,r],
L Jt
Prom the Four Step Scheme (see Chapter 4), we know that if we define
z{t,x,y,p) = pa{t,x,y), and let 6{t,x) be the classical solution of the
following system of PDEs:
A.3) <^
1
e!+-tT[e^,Mt,x,e)a{t,x,ey]+{b{t,x,e,z{t,x,e,e,)),e^^) = o,
L, ,/lC,,
[e{T,x) = g{x),
then the (unique) adapted solution of A.2) is given by
A.4)
where
A.5)
X{t)=x+ [ b{s,X{s))ds+ [ a{s,X{s))dWis),
Jo Jo
Y{t) = e{t,x{t)y,
^ zit) = e,it,xit))a{t,xit),e{t,xit))).
b{t,x) = b{t,x,e{t,x),e^{t,x)a{t,x,e{t,x)));
a{t,x) = a{t,x,6{t,x)),

194 Chapter 8. Applications of FBSDEs
Now from the second (backward) equation in A.2), and noting that Yq is
non-random by Blumenthal 0-1 law, we have Yq = EYq = Eg{XT)', and
setting i = 0 in A.2) we then have
A.6) g{X{T))^Eg{X{T))+f e,{s,X{s))u{s,X{s),e{s,X{s)))dW{s).
Jo
Let us compare A.6) with the Clark-Haussmann-Ocone formula in this
special setting. For simplicity, we assume m = n = 1. Recall that the
general form of the Clark-Haussmann-Ocone formula in this case is:
A.7) g{X{T))=Eg{X{T))+ f E{Dsg{X{T))\Ts)dWs,
Jo
where D is the so-called "Malliavin derivative" operator. Note that by
Malliavian calculus we have, for each s £ [0,r], that Dsg{X{T)) =
g'{X{T))DsX{T), and
DsX{t) = d{s,X{s)) + [ h^{r,X{r))DsX{r)dr
J s
+ f a^{r,X{r))DsX{r)dW{r), t £ [s,T].
Denote
Z{t)= f h{r,X{r))dr+ [ a,{r,X{r))dW{r),
J s J s
and let £{Z)t be the Doleans-Dade stochastic exponential of Z, that is,
£{Z)t=exp{Z{t)~-hz,Z]{t)}
A 8)
= exp U a,{r,X{r))dW{r) + l^ [kir,X{r)) - ^al{r,X{r))]dr
Then the process u{t)=^DsX{t), t £ [s,T] can be written as u{t) —
£{Z)td{s,X{s)). Therefore,
E{Dsg{X{T))\Ts] = E{g'{X{T))DsX{T)\Ts]
^ ■ ^ = E{g'{X{T))£{Z)T\Ts}d{s,X{s)).
Putting this back into A.7) and comparing it to A.6) we obtain immediately
that
j [E{Dsg{X{T))\T,} ~a{s,X{s))e,{s,X{s))]dW{s) =0,
and consequently,
{E{Dsg{X{T))\Ts]^d{s,X{s))e,{s,X{s));
^'■''^ [E[g'iXiT))EiX)r\T.) = e.is,Xis)), "^ ^''-'■'■

§1. An integral representation formula 195
Since the expressions on the right sides of A-10) depend neither on the
Malliavin derivatives, nor on the conditional expectations, they are more
amenable in general. Also, since forward SDE in A.4) depends actually on
Y and Z, we thus obtained an integral representation formula A.6) that is
more general than the "classical" Clark-Haussmann-Ocone's formula, when
the Brownian functional is of the form g{X{T)).
It is interesting to notice that the second equation in A.10) does not
contain the Malliavin derivative, and it leads to Haussmann's version of
integral representation formula. Let us now prove it directly without using
Malliavin calculus. To do this, we define a the process pt = 6x{t,X(t))
(such a process is often of independent interest in, e.g., stochastic control
theory). For simplicity we assume m = n = 1 again and that the FBSDE
is decoupled. That is
A.11)
X{t)^x+ [ bis,X{s))ds+ [ a{s,X{s))dW{s),
Jo Jo
Y{t) = 9{X{T)) - f Z{s)dW{s), t e [0,r],
Jt
and the PDE A.3) becomes
\e{T,x)=9{x),
We should note that the following arguments are all valid for the coupled
FBSDEs with 6 = 0, in which case we should simply replace A.11) by A.4).
Proposition 1.1 Tiiere exists an adapted process {K{t) : t > 0} sucii tiiat
(p, K) is the unique adapted solution of tiie following backward SDE:
Pt = 9'{X{T))+ I %{s,X{s)fps+(Tx{s,X{s))K{s)]ds
A.13) ^*
- / K{s)dW{s).
In particular, if the function 6 is C^, thenK{t) — dxx{t,X{t))a{t,X{t)) for
t >0.
Proof. We first assume that 6 is C^. Taking one more derivative in the
X variable to the equation A.12) and denote u — 9x v/e have
n 141 1 "* + ■^Uxxcr'^ityx) + [b{t,x) + {aax){t,x)]ux +b{t,x)u = 0,
[u{T,x) = Qxix).
On the other hand, if we apply Ito's formula to u from t to r @ < t < r).

196 Chapter 8. Applications of FBSDEs
then we have
w(r,X(r))=w(i,X(i))+ f {utis,X{s))+u^{s,Xis))b{s,Xis))
A-15) + ^uUs,X{s))a\s,X{s))}ds
+ f u^{s,X{s))a{s,X{s))dW{s).
Using A.14) and denoting K{t) = Uxit,X{t))a{t,X{t)), we obtain from
A.15) that
u{t,X{t)) =u{t,X{t)) - / [ub:,+U:,{aa:,)]{s,X{s))ds
+ f Ux{s,X{s))a{s,X{s))dWis)
A-16) ^*
= u{t,X{t))- / [ubx{s,X{s))+K{s)ax{s,X{s))]ds
+ f K{s)dWis),
Now setting pt = u{t,X{t)) and t =:T, we obtain A.13) immediately.
In the general case where 6 is not necessarily C^ we argue as follows.
Let {p, K) be the adapted solution to the backward SDE A.13), and we are
to show that pt = ^^(i, X(i)), that is, V/i £ R,
A.17) e{t,X{t) + h)~-e{t,X{t))=pth+o{h), Vi, a.s.
To this end, fix t G [0, T] and consider the SDE
A.18) X''{t) = X{t) + h+ f bis,X''is))ds + r ais,X''{s))dW{s),
for t <T <T. Define 0 = X''{t) - X{t), t G [t,T]. Then it is easy to
verify that ('^ satisfies
A.19) dC\T) = b,iT,X{T))C\T)+a,{T,XiT)X\T)dW{T)+e''{T),
where
eHr) = f {J [bAs,X{s) + f3eis)) - b,is,X{s))W}c''{s)ds
= f {J WAs,Xis)+f3(:Hs)) - a,is,Xis))W]<:''is)dW{s).
Thus by the standard results in SDE we have -E{sup«^<y |e''(r)| J^t} =
o{h).
On the other hand, using Four Step Scheme one shows that
e{t,X{t)) = E{g{X{T)) ^a, eit,X{t)+h) = E{g{X{T)) Tt},

A.20)
§2. A nonlinear Feynman-Kac formula 197
thus
e{t,x{t) + h)-e{t,x{t))
=E{g{X'^{T))-g{X{T))\Tt}
=E{g'iXiT))(:^\Tt} + E{j [g'iXr + fiC^) - 9'{XT)W<:^\Tt}
=E{g'iXiT))(:!},\Tt} + oih).
Now applying Ito's formula to PrCr from r = i to r = T we have
{g'iXiTX^iT) = pth + o{h) + m{T) - m{t),
where m stands for some {^t}t>o-martingale. Taking conditional
expectation we obtain from A.20) that
e{t,X{t) + h)-e{t,X{t))=pth + oih), P-a.s., Wt e [0,T].
Using the continuity of both X and p we have 6x{t,X{t)) = pt, Vi, P-a.s.,
proving the proposition. D
§2. A Nonlinear Feynman-Kac Formula
In this section we establish a stochastic representation theorem for a class
of quasilinear PDEs, via th route of FBSDEs. We note that following
presentation will include the BSDEs as a special case. To begin with, let
us rewrite A.1) again, on an arbitrary time interval [t,T], t £ [0,T): for
t<s<T,
B.1)
Xis) = x+ f b{r,Qir))dr+ f air,X{r),Y{r))dW{r),
Y{s) = g{X{T))+ f h{r,Q{r))dr- f Z(r)dW(r)
J s J s
We would like to show that if the FBSDE B.1) has unique adapted solutions
on all subintervals [t,T] C [0,r], denoted by (X*'^,y*'^, Z*'^), then the
function w(i,a;) =F*'^(i) would give a viscosity solution to a quasilinear
PDE. Thus if we can prove the uniqueness of such viscosity solution (see
Chapter 3, §3), then clearly we obtain a certain "probabilistic solution"
to the corresponding PDE, in the spirit of the celebrated Feynman-Kac
formula. For this purpose, in what follows we shall always assume the
solvability of the the FBSDE B.1), under the following assumptions:
(Al) (i) m = 1; and the coefficients b, h, cr, g are deterministic,
(ii) The functions b and h are differentiable in z.
Note that (Al)-(i) amounts to saying that coefficients of A.2) are
"Markovian". Thus the standard technique of "time shifting" can be used
to show that the process {5^/'^}s>t is .Ff-adapted, where T^ = a{Wr,t <

198 Chapter 8. Applications of FBSDEs
r < s}. Consequently the function u{t,x) = F/'^ is deterministic, thanks
again to the Blumenthal 0-1 law.
In order to describe the quasilinear PDE that an FBSDE is
corresponding to, let us denote S{n) to be the set of n x n symmetric non-negative
matrices, and for p £ R", Q G S{n), define
B 2) H{t,x,u,p,Q) = -ti {aa'^{t,x,u)Q + {b{t,x,u,a{t,x,u)p),p)
+ h{t,x,u,a{t,x,u)p),
and denote Du = Vw = {dxj^u, ■ ■ ■, dx„u)'^, D'^u = (cJj.j. .w)i,j (the Hessian
of w), and ut = dtu. The quasilinear PDE that we are interested in is of
the following form:
{ut+H{t,x,u,Du,D'^u) = 0,
\u{T,x)=g{x),
We have the following theorem.
Theorem 2.1. Assume (Al). Suppose that for a given time duration
[t,T], the FBSDE B.1) has an adapted solution (X*'^,y*'^,Z*'^). Then
the function w(i, x) = F^ '^, (t, x) E [0,T] x R" is a viscosity solution of the
quasilinear PDE B.3).
Proof. We shall prove only that w is a viscosity subsolution to B.3).
The proof of the "supersolution" is left as an exercise. First note that
u{t,x) = y*'^(i) is continuous on [0,r] x R", locally Lipschitz-continuous
in X, and locally H6lder-i in t.
Let (t, x) e [0, T) X R" be given; and let v? e C^'^{[0, T] x R") be such
that (i, x) is a global maximum point of u— ip such that w(i, x) = ip(t, x).
We are to check that the inequality C.27) of Chapter 3 holds.
To simplify notations, in what follows we suppress the superscript " *'^ "
for the processes X, F, and Z. First note that by modifying ip slightly at
"infinite" if necessary we assume without loss of generality that and Dip
is uniformly bounded, thanks to the uniform Lipschitz property of u in x.
Next note that the pathwise uniqueness of the FBSDE leads to that for
any 0 < r < r < T one has u{t,X{t)) = Y{t), hence we can rewrite the
backward SDE in B.1) as
u{t,x)=u{T,X{T))+ f h{s,X{s),Y{s),Z{s))ds
B.8) ^'
- / Z{s)dW{s).

§2. A nonlinear Feynman-Kcic formula
Now applying Ito's formula to ip{-,X{-)) from i to r we have
^iT,X{T)) =ip{t,x)+ ipt{s,X{s))ds
+1 {D^{s,X{s)),b{s,X{s)Ms,X{s)),Z{s)))ds
r 1
+ / ■-tv{<j<j'^{s,X{s),u{s,X{s)))D'^ip{s,X{s))}ds
+ j\D^{s,X{s)),ais,X{s),u{s,Xis)))dW{s)).
Write
199
B.9)
B.10)
where
h{s,X{s),Y{s),Zis)) = his,X{s),Yis),[a''D^]{s,X{s),Y{s)))
+ {a{s),Z{s) - [a''D^]{s,Xis),Y{s)));
b{s,X{s),Y{s),Z{s)) = b{s,X{s),Y{s),[a''D^]{s,X{s),Yis)))
+ m{Z{s) ~ [a''D^]{s,Xis),Yis)))},
a{s) = J —{s,X{s),Y{s),Zr{s))de;
^^•^^^ 1^^") = / f^is,X{s),Y{s),Ze{s))de;
. Zeis) = eZ{s) + A - e)cj'^{s,X{s),Y{s))D^{s,X{s)).
By assumption (Al), we see that a and C are bounded, adapted
processes. Therefore, subtracting B.9) from B.8), using B.10) and B.11),
and noting the facts that u{t, x) = ip{t,x) and u{t,X{t)) < ip{t,X{t)), we
obtain
0>u{t,X{t))-^{t,X{t))
= f{- ^(^'^(^)) -F{s,X{s).Y{s)M''D^]{s,X{s),Y{s))
B 12) •''^
-{Z{s)-[c7'^D^]{s,X{s),Y{s)),a{s)-D^{s,X{s))(i{s))]ds
+ j^ {Z{s) - [cj'^D^]{s,X{s),Y{s)),dW{s)).
Since d{s) = a{s) + Dip{s,X{s))^{s), s £ [t,T] is uniformly bounded, the
following process is a P-martingale on [t,T]:
0* ^exp{ - I' {e{r),dW{r))-\j^ |%)Nr}, s £ %T].
By Girsanov's Theorem, we can define a new probability measure P via
% = 6^, so that W\s) = W{s) - W{t) - J'e{r)dr is a P-Brownian

200 Chapter 8. Applications of FBSDEs
motion on [t,T]. Furthermore, since the processes {X,Y,Z)) satisfies
rT
e{ sup \X{s)\'']+e{ sup \Y{s)\'']+E |Z(s)pds<
•• t<s<T ' •• t<s<T ' Jt
00,
the boundedness of Dip and the uniform Lipschitz property of a imply that,
for some constant C > 0,
e[J \Z{s)-[cj'^D^]{s,X{s),Y{s))fdsy
< CE{Qir{j\l + \Z{s)\' + \Xis)\' + |y(s)|>) '}
< C{EiQirf}'{E f [1 + |Z(s)|2 + |X(s)|2 + |y(s)p]ds}' < oo.
In other words, the integral
M\u)^j^{Zis)-[a''D^]is,Xis),Yis)),dW{s)), ue[t,T]
is a P-local martingale on [t,T] satisfying E{M*')j. < oo, the by
Burkholder-Davis-Gundy's inequality, one shows that it is a P-martingale
on [t,T]. Hence, by taking expectation E{-} on both sides of B.12) we
obtain that
B.13)
0> EJ\-^is,Xis))-H{s,Xis),Yis),[a''D^]is,X{s),Y{s)))}ds.
Dividing both sides by r and then sending r -> 0 we obtain C.27) of
Chapter 3 immediately. D
Remark 2.2. For a more complete theory, one should also prove that the
viscosity solution to the quasilinear PDE B.3) is unique. This is indeed the
case when the coefficient a is independent of y as well (i.e., a = a{t,x));
and when the solution class is restricted to, for example, bounded,
continuous functions that are uniform Lipschitz in x and Holder -^ in t. We note
that due to the special quasilinearity, the function B.2) is neither
monotone, nor even one-sided uniform Lipschitz in the variable x, therefore H
is not "proper" in the sense of Crandall-Ishii-Lions [1], or convertible to a
proper function using the standard technique of "exponentiating" (see, e.g.,
Fleming-Soner [1]). Consequently, the uniqueness of the viscosity solution
is by no means trivial. However, since this issue is more or less beyond the
scope of this book, we will not include the proof here. We refer the
interested readers to the works of Barles-Buckdahn-Pardoux [1], Pardoux-Tang
[1], or Cvitanic-Ma [2].

§3. Black's Consol Rate Conjecture 201
§3. Black's Consol Rate Conjecture
One of the early applications of FBSDE is to confirm and explore a
conjecture by Fischer Black regarding consol rate models for the term structure
of interest rates. A consol is by definition a perpetual annuity, that is, a
security that pays dividends continually and in perpetuity. A consol rate
model is one in which the stochastic behavior of the short rate, taken as a
non-negative progressively measurable process below, is influenced by the
consol rate process. The relation between the two rate processes then yields
a special term structure of interest rates.
In order to set up a mathematical model, let us consider the following
simplest situation in which the short rate is a constant r > 0, then there
should be no difference between the short rate and long term (consol) rate.
In this case the consol price Y can be calculated as the simple actuarial
present value of a perpetual annuity. Assuming, for instance, that the
annuity is in a form of annuity-immediate in terms of actuarial mathematics,
that is , it pays, say $1, at the end of each year, then the price Y can be
calculated easily as
oo
1 1
^A-Fr)*^ (l + r) 1-
(l+r)
In other words, the price for the (unit) consol is the reciprocal of the interest
(consol) rate. In general, let us define the consol rate to be the reciprocal
of the consol price, then instead of studying the original term structure
of interest rates, it would be equivalent to study the relation between the
consol price and the short rate.
Now let us generalize the above idea. For a given short rate process
r = {rt : t > 0},we use the standard expected discounted value formula (an
extension of the aforementioned actuarial present value formula) to evaluate
the consol price process Y = [Yt : i > 0} f:
-ir
C.2) Y{t) = El I e~r'"f"''^"ds
Tt>, t>0.
(One can check that if r{t) = r, then Y{t) = ^, as C.1) shows!). The
Consol rate problem can be formulated as follows. Assume that the short
rate process depends on the consol price (whence consol rate) in a non-
anticipating manner, via the following SDE:
C.3) dr{t) =fi{r{t),Y{t))dt + a{r{t),Y{t))dW{t).
where W is a standard Brownian motion in R^, and n, a are some
appropriate functions. Then is there actually a pair of adapted processes (r, Y)
f Without getting into the associated definitions and related notions of
arbitrage, it is not unusual in applications to work from the beginning with the
so-called "equivalent martingale meeisure," in the sense of Harrison and Kreps
[1], and we do so.

202 Chapter 8. Applications of FBSDEs
that satisfies both C.2) and C.3)? If so, can Y also he described by an
SDE? In an earlier work Breunan and Schwartz [1] proposed a model of
term structure of interest rates in which both short rate and long rate are
characterized by SDEs. However, it was shown later by Hogan [1] by
examples that such a model may not be meaningful in practice. Sensing that
the controversy might be caused by the inappropriate specification of the
coefficients, together with a simple observations by using Ito's formula and
C.1), the late economist/mathematician Fisher Black made the following
conjecture:
(Black's Conjecture). Under at most technical conditions, for any (n, a)
there is always a function A : @, oo) x @, oo) -> @, oo) depending on fi and
a, such that
dY{t) = {r{t)Y{t) - I)dt + A{r{t),Y{t))dW{t).
Black's conjecture essentially re-confirms the SDE model of Brennan
and Schwartz, but it was not clear at the time for how to determine the
function A, and how it should related to the coefficients n and a in C.3).
We now show how to confirm Black's conjecture by using the theory of
FBSDEs. To this end, let us assume first that the short rate r process is
"hidden Markovian". That is, there is a (Markovian) "state process" X in
R" such that the short rate is given by rt = h{Xt), for some well behaved
function h. To be more specific, we will assume that X satisfies an SDE:
Ux{t) = b{X{t),Y{t))dt + a{X{t\Y{t))dW{t),
\x{o) = x, ie[o,r],
where 6, a are some appropriate functions defined on R" x R. Since the
coefficients b and u can be computed explicitly in terms of ju, a, and h using
Ito's formula, we can recast the consol rate problem as follows.
Infinite Horizon Consol Rate Problem (IHCR). Find a pair of
adapted, locally square-integrable processes {X,Y), such that
C.5)
' dX{t) = b{X{t),Y{t))dt + a{X{t),Y{t))dW{t), te [0,oo),
Y{t) = eI Te-1'''(^f"»"""ds \Tt\,
Xo = X, t e [0, oo).
Any adapted process {X, Y) satisfying C.5) is called an adapted solution of
Problem IHCR. Moreover, an adapted solution (X, Y) of Problem IHCR is
called a nodal solution with representing function 9 if there exists a bounded
C^ function 6 with 6^ being bounded, such that
C.6) Y{t)=e{X{t)), te[0,oo).
Recall that the term "nodal solution" was first introduced in Chapter 4,
§3, where we studied the FBSDE in a infinite horizon [0, oo) of the following

§3. Black's Consol Rate Conjecture 203
type:
C.7)
( dX{t) = b{X{t),Y{t)) dt + a{X{t),Y{t))dW{t),
dY{t) = {h{X{t))Y{t) - l)dt-{Z{t),dW{t)), t e [0,oo),
X{0)=x,
, Y{t) is bounded a.s., uniformly in i £ [0, oo).
The following theorem shows that C.7) is exactly the system of SDEs that
can characterize the process X and Y simultaneously, which will be the
first step towards the resolution of Black's conjecture. First let us recall
the technical assumptions C.4)-C.6) of Chapter 4:
(A2) The functions a, b, h are C^ with bounded partial derivatives and
there exist constants A,/i > 0, and some continuous increasing function
1/: [0, oo) -> [0, oo), such that
' XI < a{x, y)a{x, yY < /i/, {x, y) £ K." x R,
\b{x,y)\<pi\y\), (a;,y)elR"xlR,
inf h{x) = S > 0, sup h{x) = 7 < oo.
Theorem 3.1. Assume (A2). If {X,Y,Z) is an adapted solution to C.7),
then {X,Y) is an adapted solution to Problem (IHCR).
Conversely, if {X, Y) is an adapted solution to Problem IHCR, then
there exists an adapted, IR -valued, locally square-integrable process Z,
such that {X, Y, Z) is an adapted solution of C.7).
Proof. To see the first assertion, let {X, Y, Z) he an adapted solution to
C.7). Let r(t) = e~ Jo ''(-^(«))'i«^ i g [g^r]. Then using integration by parts
(or Ito's formula) one shows easily that
r(r)y(r) = T{t)Y(t) + f r{s)dY{s) + f Y{s)dr{s)
or
Y{t) = e- r M^(^))d. ^ r g- /; HMu))du^^ ^ ^^^^ _ ^^^y
where m denotes some {.Ff}f>o-martingale, as usual. Taking conditional
expectation E{ ■ \Tt) on both sides and letting T —^ 00, we prove the first
assertion.
Conversely, suppose that {X,Y) is an adapted solution to Problem
IHCR. Define
C.8) U{t) = I
00
e'
t
J'h{X{u))du^^

204 Chapter 8. Applications of FBSDEs
Clearly, U{t) is well-defined for each t > 0, thanks to (A2). We claim
that U is the unique bounded solution of the following ordinary differential
equation with random coefficients:
C.9) ^ = h{X{t))U{t) - 1, te [0,00).
Indeed, by a direct verification one shows that the function U defined by
C.8) is a bounded solution of C.9). On the other hand, let U be any
bounded solution to C.9) defined on [0,oo). Then for any 0 < i < T, we
can apply the variation of constants formula to get
C.10) U{t) = e- r M^(«))d«f^(j.) + / e"" ^' '''^'"'''^"ds.
Since [/(T) is bounded for all T > 0, and by (A2), h{X{u)) > E > 0
Vu e [0,r], P-a.s., sending T -^ oo on both sides of C.10) we obtain C.8),
proving claim.
Next, define Y{t) = E{U{t)\Tt}- Note that since the filtration {J^t}t>o
is Brownian, the process Y is continuous and is indistinguishable from the
optional (as well as predictable) projection of U. Hence, for any bounded,
{.7^t}t>o-adapted process H, it holds thatf
H{s)U{s)dsM =En H{s)Y{s)ds\Tty
Now for 0 < i < T < 00 we have from C.9) and C.11) that
Y{t) = E{U{t)\J^t) = e{u{T) - / [h{X{s))U{s) - l]dsk)
C.12) r^
= e[y{T)-J [h{X{s))Y{s)-l]ds\Tt].
Thus, by using the martingale representation theorem one shows that there
exists an adapted, square-integrable process Z^'^^ defined on [0,T], such
that for alH £ [0,r],
C.13) Y{t)=Y{T)- j [h{X{s))Y{s)-l]ds + j (^'^'(s).
dWs
Since C.13) holds for any T > 0, let 0 < Ti < r2 < oo, we have for
t e [0,Ti] that
Y{t)=Y{Ti) -| \h{X{s))Y{s) - l]ds+l \z^''^\s)dW{s))
= Y{T2)-j \h{X{s))Y{s)-l]ds+j \z('^-\s),dW{s)).
f See, for example, Dellacherie and Meyer [1, Chapter VI].

§3. Black's Consol Rate Conjecture 205
Prom this one derives easily that
C.14) f \z^^^\s) - Z^'^'\s),dWis)) = 0, for all
This leads to that E [jo'[Z^^^^{s) - Z'-'^^^s)f ds\ = 0. In other words,
^(^i) = ^(^2), di (gi dP-almost surely on [0,ri] x 0. Consequently, modulo
a, dt^ dP-null set, we can define a process Z hy Zt = Z^'^^t), if t e [0,N],
where A'' = 1,2, • • •. Clearly Z is locally square-integrable, and C.13) can
now be rewritten as
C.15) Y{t)=Y{T)- j [h{X{s))Y{s)-l]ds + l {Z{s),dW{s)),
for all T > 0, or equivalently, {X,Y) satisfies the SDE C.7). Finally, the
boundedness of Y follows easily from the definition of Y and the fact that
Ut < j,yt>0, P-a.s., proving the proposition. D
We remark here that Theorem 3.1 shows that the Black's conjecture
can be partially solved if the FBSDE C.7) is solvable. However, in order to
confirm Black's conjecture completely, we have to show that the process Z
can actually be written as Z{t) = (p{X{t),Y{t)) for some function (f, which
in turn will give Z{t) = A{r{t), Y{t)) for some function^, as the conjecture
states. But this is exactly where the nodal solution comes into play, and
the Chapter 4, Theorem 3.3 essentially solves the problem. We recast that
theorem here in the new context.
Theorem 3.2. Assume (A2). Then there exists at least one nodal solution
{X, Y) of Problem IHCR. Moreover, the representing function 6 satisfies
(i) 7-1 < e{x) < S-\ for all x eU.
(ii) 6 satisfies the following differential equation for x £ IR":
C.16) itr (e^Mx,o)a'^{x,e)') + {b{t,e),e^) -h{x)e + 1 = 0,
Consequently, The Black's conjecture is solved (in terms of Problem IHCR)
with A{x,y) = a'^{x,yN^{x).
Proof. This is the direct consequence of Chapter 4, Theorem 3.3; and
the last statement if due to the fact that Z{t) = A{X{t),Y{t)) whenever
the nodal solution exists. D
Remsirk 3.3. We should point out here that although the bounded solution
U of the random ODE C.9) with infinite-horizon is unique, the uniqueness
of the adapted solution to the FBSDE C.7) over an infinite duration is
still unknown. In fact, as we saw in Chapter 4 (§3), the uniqueness of the
adapted solution, as well as that of the nodal solution, to FBSDE C.7),
is a more delicate issue, especially in the higher dimensional case.
However, since Black's conjecture concerns only the existence of the function

206 Chapter 8. Applications of FBSDEs
A, Theorem 3.2 provides a sufficient answer. Interested readers could of
course revisit Chapter 4, §3 for more details on various issues regarding
uniqueness.
Finite-Horizon Valuation Problem and its limit.
In the standard theory of term structure of interest rates the time
duration is often set to be finite. Namely, we content ourselves only in
a finite time interval [0,T]. Let us now view the process y as a long
term interest rate (or the price of a long term bond to be comparable to
the consol price), and view X as the state process for the short rate r,
with r{t) = h{X{t)), and h satisfies (A2). In order to study the explicit
relation between X and Y, let us assume that they have an explicit relation
at terminal time T: Y{T) = g{X{T)). We consider the following Finite-
Horizon- Valuation Problem. Note that Such a problem is a generalization of
the well-known finite horizon annuity valuation problem, which corresponds
to the case when g = Q below, by allowing the annuity price to influence
the short rate.
Problem FHV. Find an adapted process {X, Y) such that for
rt rt
C.17)
X{t)=x-\- f b{X{s),Y{s))ds-\- f a{X{s),Y{s))dW{s),
Jo Jo
Y{t) = ElrJg{X{T)) +1^ r'ds\Tt\, te [o,T],
where rf^e-i'>'^<"»'^".
Any adapted process {X,Y) satisfying C.17) is called an adapted
solution of Problem FHV. Further, an adapted solution {X, Y) of Problem
FHV is called a nodal solution of Problem FHV if there exists a function
e : [0, T] X K." -^ R, which is C^ in t and C^ in x, such that
C.18) Y{t)=e{t,X{t)), te[0,T].
Conceivably the Problem FHV will associate to an FBSDE as well, as
was seen in the IHCR case. In fact, some similar arguments as those in
Theorem 3.1 shows that if {X,Y) is an adapted solution to the Problem
FHV, then there exist a progressively measurable, square integrable process
Z such that (X, Y, Z) is an adapted solution to the following FBSDE:
' dX{t) = b{X{t),Y{t))dt-\-a{X{t),Y{t))dW{t),
C.19) idY{t) = {h{X{t))Y{t)-l)dt-{Z{t),dW{t)), te[0,T],
^X{0)=x, Y{T)=g{X{T)).
Conversely, if (X, Y, Z) is an adapted solution to C.19), then a variation
of constant formula applied to the backward SDE in C.19) would lead
immediately to that Y satisfies C.17). Furthermore, using the results in

§4. Hedging options for large investors 207
Chapter 4 (Four Step Scheme) we see that if g is regular enough, then any
adapted solution of C.19) must be a nodal solution. These facts, together
with Chapter 4, Theorem 3.10, give us the following theorem, which slightly
goes beyond the Black Conjecture.
Theorem 3.4. In addition to (A2), assume further that the function g
belongs boundedly to C2+"(]R") for some a e @,1). Then, Problem FHV
admits a unique adapted solution {X, Y). Moreover, this solution is in fact
a nodal solution.
Furthermore, if the Problem IHCR has a unique nodal solution, denoted
by {X,Y), where Y = 6{X) and 6 satisfies the differential equation C.16);
and if we denote {X^,Y^) to be the nodal solution of Problem FHV on
the interval [0,K], then it holds that
C.20) lim E\Y/^ - Yt\'^ + ElXf - Xt\'^ = 0,
jff—>00
uniformly in t £ [0, oo) on compacts.
§4. Hedging Options for a Large Investor
In this section we apply the theory of FBSDEs to another problem in
finance: hedging contingent claims for a large investor. We recall that the
problem of hedging a contingent claim was discussed briefly in Chapter 1,
§1.3. In this section we shall remove one of the fundamental assumptions
on which the Black-Scholes theory is built, that is, the "small investor"
assumption. Roughly speaking, the "small investor" assumption says that no
individual investor is influential enough so that his/her investment
strategy, or wealth, once exposed, could affect the market prices.
Mathematically, under such an assumption the coefficients of the stochastic differential
equation that characterizes the price of underlying security should be
independent of the portfolio of any investor. Although such an assumption has
long been deemed as common sense, it has been also noted recently that
the investors that are "not-so-small" could really make disastrous effect to
a financial market. A probably indisputable evidence, for example, is the
"Hedge Fund" crisis of 1998 in the global financial market, in which the
"large investors" obviously played some important roles. In this section, we
try to attack the problem of hedging a contingent claim involving "large
investors" . We should point out here that the model that we will be studying
is still quite "ad hoc", and we shall only concentrate on the mathematical
side of the problem.
Recall from Chapter 1, §1.3 the mathematical model of a continuous-
time financial market. There are d-l-1 assets traded continuously: a money
market account and d stocks, whose prices at each time t are denoted by
Po{t), Pi{t), i = l,--,d, respectively. An investor is allowed to trade
continuously and frictionlessly. The "wealth" of the investor at time t is
denoted by X{t); and the amount of money that the investor puts into the
i-th stock at time t is denoted by 7ri(i), 1 = 1, • • •, d (thus the amount of
money that the investor puts into the money market at time t is X{t) —

208
Chapter 8. Applications of FBSDEs
12i=i '^i{t))- We assume that the investor is "large" in the sense that his
wealth and strategy, once exposed, might influence the prices of the financial
instruments. More precisely, let us assume that the prices (Pq, Pi,- ' y Pd)
evolves according to the following (stochastic) differential equations on a
given finite time horizon [0,T] (comparing to Chapter 1, A,26)):
' dPQ{t) = Po{t)r{t,X{t),7r{t))dt, 0<t<T;
dPi{t) = Pi{t){bi{t,P{t),X{t),7r{t))dt
D.1)
+ J2^vit,Pit),X{t),7r{t))dWj{t)},
tPo@) = l, Pi{0)=pi>0, i = l,---,d,
where W = {Wi,- ■ ■ ,Wd) is a d-dimensional standard Brownian motion
defined on a complete probability space (Q, T, P), and we assume as usual
that {!Ft}t>o is the P-augmentation of the natural filtration generated by
W. To be consistent with the classical model, we call b the appreciation
rate and a the volatility matrix of the stock market.
Further, we assume that the investor is provided an initial endowment
a; > 0, and is allowed to consume, and denote C{t) to be the cumulative
consumption time t. It is not hard to argue that the change of the wealth
"dX(i)" should now follow the dynamics:
D.2)
dm = t ?S^^^(^) + (^(^)-^f=^"'(^»dPo@ - dCit)
^ X@) = a; > 0.
To simplify presentation, from now on we assume that d = 1 and that
the interest rate r is independent of tt and X, i.e., r = r{t), t>0. Denote
D.3)
b{t,p,x,Tr)={x - Tr)r{t) + Trb{t,p,x,Tr);
A
a{t,p,x,Tr) =Tra{t,p,x,Tr),
for {t,p, x,Tr) e [0,r] X K.^ We can rewrite D.1) and D.2) as
Po(i)=exp{ f r(s)ds},
P{t)=p+ [ P{s){b{s,P{s),X{s),7r{s))ds
Jo
+ a{s,P{s),X{s),7r{s))dW{s)},
D.4)
te[o,n
X{t)=x +
P{s),X{s),Tr{s))ds
D.5)
+
[bis,
Jo
f a{s,P{s,X{s),7r{s))dW{s) - C{t);
Jo

§4. Hedging options for large investors 209
Before we proceed, we need to make some technical observations: first,
we say a pair of {^f}f>o-adapted processes Gr,C) is a hedging strategy
(or simply strategy) if C() has nondecreasing and RCLL paths, such that
T I
C@) = 0 and C{T) < oo, a.s.-P; and E J^ |7r(s)pds < oo. Clearly, under
suitable conditions, for a given strategy (tt, C) and the initial values p > 0
and a; > 0 the SDEs D.4) and D.5) have unique strong solutions, which will
be denoted by P = pp-^-'^.C' g^.^^ ^ _ j^p,x,7r,c^ whenever the dependence
of the solution on p, x, tt, C needs to be specified.
Next, for a given a; > 0, we say that a hedging strategy (tt, C) is
admissible w.r.t. x, if for any p > 0, it holds that pP'^''^'C'(^t) > 0 and
j^p,x,7r,C(^) > 0, Vi e [0,r], a.s.P. We denote the set of strategies that
are admissible w.r.t. x by A{x). It is not hard to show that A{x) ^ 0
for all X. Indeed, for any a; > 0, and p > 0, consider the pair tt = 0 and
C = 0. Therefore, under very mild conditions on the coefficients (e.g., the
standing assumptions below) we see that both P and X can be written as
"exponential" functions:
D.6)
P{t) = pexp { j [bis] - \\ais)\']ds + j a{s)dWis)} > 0,
X{t) = xexp{ / r{s)ds} > 0,
Jo
where 6(s) = b{s,P{s),0,0) and ct(s) = a{s,P{s), 0,0). Thus @,0) £ ^(a;).
Recall from Chapter 1, §1.3 that an option is an .Fr-measurable random
variable B = g{P{T)), where 5 is a real function; and that the hedging price
of the option is
D.7) h{B) = M{x e E : 3Gr, C) e ^(a;), s.t. X'^'^'^iT) > B a.s. }.
In light of the discussion in Chapter 1, §1.3, we will be interested in the
forward-backward version of the SDEs D.4) and D.5):
P{t)=p+ f P{s){b{s,P{s),X{s),7r{s))ds
Jo
+ a{s,P{s),X{s),7r{s))dW{s)},
X{t)=g{P{T))- I 'b{s,P{s),X{s),7r{s))ds
D.8)
r
is
It
'd{s,P{s,X{s),-K{s))dW{s);
H
We first observe that under the standard assumptions on the coefficients
and that 5 > 0, if (P, X, tt) is a solution to FBSDE D.8), then the pair (tt, 0)
must be admissible w.r.t. X@) (a deterministic quantity by Blumenthal
0—1 law). Indeed, let [P,X,-k) be an adapted solution to D.8). Then a
similar representation as that in D.6) shows that P{t) > 0, Vi, a.s. Further,
define a (random) function
j(t,x,z) = r{t)x ^ ZG-\t,P(t),X(t),i:{t)){b(t,P(t),X{t),'K(t)) - r(t)\.

210 Chapter 8. Applications of FBSDEs
thena;(i) =X{t),z{t) = a{t, P{t),X{t),Tr{t))Tr{t) solves the following
backward SDE
x{t)=g{P{T)) + l f{s,x{s),z{s))ds + l z{s)dW{s).
Applying the Comparison theorem (Chapter 1, Theorem 6.1), we conclude
that X{t) = x{t) > 0, Vi, P-a.s., since g{P{T)) > 0, P-a.s. The assertion
follows.
§4.1. Hedging without constraint
We first seek the solution to the hedging problem D.7) under the following
assumptions.
(H3) The functions b, a : [0,T] x IR^ i-> IR are twice continuously difFer-
entiable, with bounded first order partial derivatives in p, x and tt being
uniformly bounded. Further, we assume that there exists a isT > 0, such
that for all (i,p, a;,7r).
96
dp
+
da_
dp
+
da_
dx
+
da_
<K.
(H4) There exist constants K > Q and /i > 0, such that for all (i,p, a;,7r)
with p > 0, it holds that
/i < a'^{t,p,x,Tr) < K.
(H5) g e Cj+"AR) for some a e @,1); and g>0.
Remark 4.1. Assumption (H4) amounts to saying that the market is
complete. Assumption (H5) is inherited from Chapter 4, for the purpose
of applying the Four step scheme. However, since the boundedness of g
excludes the simplest, say, European call option case, it is desirable to
remove the boundedness of g. One alternative is to replace (H5) by the
following condition.
(H5)' lim|p|^oo5(p) = oo; but g £ C^(IR) and g' £ C^{K). Further, there
exists K > 0 such that for all p > 0,
D.9) \pg'ip)\ < K{1 + g(p)); \p''g"{p)\ < K.
The point will be revisited after the proof of our main theorem. Finally, all
the technical conditions in (H3)-(H5) are verified by the classical models.
An example of a non-trivial function a that satisfies (H3) and (H4) could
be a{t,p,x,Tr) = a{t) + arctan(a;^ -I- |7rp).
We shall follow the "Four Step Scheme" developed in Chapter 4 to solve
the problem. Assuming C = 0 and consider the FBSDE D.8). Since we
have seen that the solution to D.8), whenever exists, will satisfy P{t) > 0,
we shall restrict ourselves to the region {t,p,x,Tr) £ [0,T] x @, oo) x IR^

§4. Hedging options for large investors 211
without further specification. The Four Step Scheme in the current case is
the following:
Step 1: Find z : [0,r] x @,oo) x IR^ -^ ]R such that
D.10) qpa{t,p,x,z{t,p,x,q)) - z{t,p,x,q)a{t,p,x,z{t,p,x,q)) =0,
In other words, z{t,p,x,q) = pq since ct > 0 by (H4).
Step 2: Using the definition of b and a in D.3), we deduce the following
extension of Black-Scholes PDE:
.4 ^^. (o = Ot + \a\t,p,e,pej,)pHj,j,+pej, - r{t)e,
\e{T,p)=9(j>), p>0.
Step 3: Let 6 be the (classical) solution of D.11), set
D ^2) {^^*'^) = b{t,p,e{t,p),pep{t,p))
and solve the following SDE:
D.13) P{t)=p+ f P{s)b{s,P{s))ds+ f P{s)a{s,P{s))dW{s).
Jo Jo
Step 4'- Setting
{x{t) = e{t,p{t))
\T:{t) = P{t)ep{t,P{t)),
show that {P,X,7r) is an adapted solution to D.8) with C = 0.
The resolution of the Four Step Scheme depends heavily on the
existence of the classical solution to the quasilinear PDE D.11). Note that
in this case the PDE is "degenerate" near p — 0, the result of Chapter 4
does not apply directly. We nevertheless have the following result that is
of interest in its own right:
Theorem 4.2. Assume (H3)-(H5). There exists a unique classical solution
e{-,-) to the PDE D.11), defined on (t,p) e [0,T] x @,oo), which enjoys
the following properties:
(i) 6 - g is uniformly bounded for {t,p) £ [0, T] x @,00);
(ii) The partial derivatives of 6 satisfy: for some constant K > 0,
D.15) \pep{t,p)\ < K{1 + \p\); \p'ej,j,{t,p)\ < K.

212
Chapter 8. Applications of FBSDEs
Proof. First consider the function 6 — 6 — g. It is obvious that 6t = Of,
6p = Op — Qp and Opp = Opp — gpp-, and 6 satisfies the following PDE:
D.16)
1
o = Ot + ^<j'{t,p,e + g{p),p{ep + g'{p)))p\epp + g")
+ r{t)\p{ep + g')-{e + g)],
[^(r,p)-o, p>o.
To simplify notations, let us set a{t,p,x,Tr) = a{t,p,x + gip),Tr + pg'ip)),
then we can rewrite D.16) as
D.17)
o = dt + ^^^{t,P,o,pep)p%p + r{t)p§p + r{t,p,e,pdp),
eiT,p) = o, p>o,
where
D.18) r{t,p,x,7r) = ^a\t,p,x,7r)p^g"{p) + r{t)pg'{p) - r{t){x + gip)).
Next, we apply the standard Euler transformation: p = e^, and
denote e{t,O=0{t,e^)- Since et{t,0 = Ot{t,e^), 9^A,0 = e%{t,e^), and
9ii{t,0 = e'^^9pp{t, e^) + e^6p{t, e^), we we derive from D.17) a quasilinear
parabolic PDE for 6:
D.19) <
0 = et + -a2(i,e«,^,^c)(% - e%) + r{t)e^+f{t,e^,e,e^),
where
D.20)
ao{t,£,,x,7r) = a{t,e^,x,Tr);
boit,^,x,Tr) =r{t) - -[ao{t,£,,x,Tr)];
bo{t,^,x,Tr) =r{t,e^,x,Tr).
Now by (H3) and (H4) we see that ao{t,^,x,Tr) > /i > 0, for all
{t,^,x,Tr) e [0,r] X IR^ and for all {t,^,x,Tr), it holds (suppressing the
variables) that
Thus, either (H5) or (H5)', together with (H3), will imply the boundedness

§4. Hedging options for large investors
of %^. Similarly, we have
-{t,e^,x + g{e^),-K + e^g'{e^))e^ < oo;
-(i,a; + 5(e«),7r + eV(e«)M'(e^)e^
213
sup
(t,{,x,7r)
sup
(t,{,x,7r)
dp
da_
dx
< K sup
da
da_
dx
{t,x + g{e^),Tr + e^g'{e^)) [1 + {x + g{e^))] < oo;
sup
(t,{,x,7r)
dn
{t,x + g{e^),7r + eig'{ei))g'{ei)ei
< K sup
(t,{,x,7r)
da_
d-K
{t,X + g{e^),7r + eV(e^)) [1 + (a; + gie^))] < oo,
Consequently, we conclude that the function ctq has bounded first order
partial (thus uniform Lipschitz) in the variables ^, x and tt, and thus so is
6o- Moreover, note that for any
ia2(i,e«,rr,7rM"(e«) = ^aUt,C,x,7r)e'^2g"{e^)
is uniformly bounded and Lipschitz in ^, x and tt by either (H5) or (H5)', we
see that bo is also uniform bounded and uniform Lipschitz in (a;,^, tt). Now
we can apply Chapter 4, Theorem 2.1 to conclude that the PDE D.11) has
a unique classical solution 6 in C^+t'2+" (for any a £ @,1)). Furthermore,
6, together with its first and second partial derivatives in ^, is uniformly
bounded throughout [0, T] x IR. If we go back to the original variable,
then we obtain that the function 6 is uniformly bounded and its partial
derivatives satisfy:
sup\p9p{t,p)\ < oo;
(t,p)
sup \p'^6pp{t,p)\ < 00.
(t,p)
This, together with the definition oi 6 and condition (H5) (or (H5)'), leads
to the estimates D.15), proving the proposition. D
A direct consequence of Theorem 4.2 is the following
Theorem 4.3. Assume (H3), (H4), and either (H5) or (H5)'. Then for
any given p> 0, the FBSDE D.8) admits an adapted solution (P,X, tt).
Proof. We follow the Four Step Scheme. Step 1 is obvious. Step 2 is the
consequence of Theorem 4.2. For step 3, we note that since Op and 6pp may
blow up when p4-0, a little bit more careful consideration is needed here.
However, observe that b and a are locally Lipschitz in [0, T] x @, oo) x IR ,
thus one can show that for ant p > 0, the SDE D.13) always has a "local
solution" for t sufficiently small. It is then standard to show (or simply
note the exponential form D.6)) that the solution, whenever exists, will
neither go across the boundary p = 0 nor explode before T. Hence step 3
is complete. Since step 4 is trivial, we proved the theorem. D

214
Chapter 8. Applications of FBSDEs
Our next goal is to show that the adapted solution of FBSDE D.8)
does give us the optimal strategy. Also, we would like to study the
uniqueness of the adapted solution to the FBSDE D.8), which cannot be easily
deduced from Chapter 4, since in this case the function a depends on tt (see
Chapter 4, Remark 1.2). It turns out, however, under the special setting of
this section, we can in fact establish some comparison theorems which will
resolve all these issues simultaneously. We should note that given the
counterexample in Chapter 1, §6 (Example 6.2 of Chapter 1), these comparison
theorems should be interesting in their own rights.
Theorem 4.4. (Comparison Theorem): Suppose that the assumptions
of the Theorem 4.3 are in force. For given p £ 1R_,_, let (tt, C) be any
admissible pair such that the corresponding price/wealth process {P, X)
satisfies X{T) > g{P{T)), a.s. Then X{-) > e{-,P{-)), where 6 is the
solution to D.11).
Consequently, if {P',X') is an adapted solution to FBSDE D.8)
starting from p G 1R_,_, constructed by the Four-Step scheme. Then it holds that
X@)>^@,p)=X'@).
Proof. We only consider the case when condition (H5)' holds, since
the other case is much easier. Let (P, X, 7r,C) be given such that Gr,C) £
A{Y{0)) and X{T) > 9{P{T)), a.s. We first define a change of probability
measure as follows: let
' Ooit) = [a-'[b-rl]{t,P{t),X{t)Mt))-
Zo{t) = exp { - I eo{s)dW{s) - i I \eo{s)\^ds};
D.21)
SO that the process Wo{t) = W{t) -\- Jq 9o{s)ds is a Brownian motion on the
new probability space @, J^, Pq). Then, the price/wealth FBSDE D.4) and
D.5) become
P{t)=P+ [ P{s){r{s,X{s),7r{s))ds
Jo
+ / a{s,P{s),X{s),7r{s))dWo{s)},
Jo
X{t) = g{P{T)) - I r{s,X{t),7r{s))X{s)ds
- f 7r{s)a{s,P{s),X{s),7r{s))dWo{s) -^ C{T) - C{t),
Since in the present case the PDE D.11) is degenerate, and the function
g is not bounded, the solution 6 to D.11) and its partial derivatives could
blow up as p approaches to 91R^ and infinity. Therefore some modification
of the method in Chapter 4 are needed here. First, we apply Ito's formula
D.22) <

§4. Hedging options for large investors 215
to the process g{P{-)) from i to T to get
g{P{t)) = 9{P{T)) - 1^ {g,{P)r{s,X,7r)P-'^a\s,P,X,7r)gpp{P)}ds
-J gp{P)a{s,P,X,7r)dWo{s),
here and in what follows we write (P,X,tt) instead of (P(s),X(s),7r(s)) in
all the integrals for notational convenience.
Next, we define a process X = X - g{P), then X satisfies the following
(backward) SDE:
X{t) = X{T) - I {r{s,X,7r)[X-gp{P)P] - ^a\s,P,X,7r)gj,j,{P)}ds
- I Gr(s) - Pgj,{P))a{s, P, X, 7r)dWo {s) + C{T) - C{t)
We now use the notation 6 = 6 — g as that in the proof of Theorem 4.2;
then it suffices to show that X{t) > 9{t, P{t)) for all t e [0,T], a.s. Pq. To
this end, let us denote X(i) =e{t,P{t)),Tr{t) = P{t)[ep{t,P{t))+gp{P{t))];
and Ax{t) = X{t) - X{t), A^{t) = 7r(i) - jr{t). Applying Ito's formula to
the process Ax{t), we obtain
Ax{t) = X{T) - ^ {r(s,X,7r)[y - (^^(P) + ?p(s,P))P]
It
1
Os{s,P) - -a\s,P,X,Tr)[epp{s,P) + gpp{P)]}ds
2
^^•^^^ -J {tt - P[gp{P) +ej,{s,P)]a{s,P,X,7r)dWo{s)
=X{T)- I A{s)ds- I A^c7(s,P,X,7r)dWo(s)
+ C{T) - C{t),
where the process A{-) in the last term above is defined in the obvious way.
Recall that the function ?satisfies PDE D.16), that e{t, P{t)) + g{P{t)) =
X{t) — Ax{t), and the definition of tt, we can easily rewrite A{-) as follows:
A{s) = r{s,X,7r)X{s) - r{s,X - Ax,^)[X{s) - Ax{s)]
- r{s, X, 7rOr(s) - r{s, 6{s, P), 7rOr(s)
+ ^{Aa{s,P,X,7r,n,e{s,P))Q{s,P) = his) + his) + his),
where
Qit,p) =p^iOppit,p) + gppip));
Aait,p,x,Tr,Tr,q) = a'^it,p,q + gip),n) - a'^it,p,x,Tr)),

216 Chapter 8. Applications of FBSDEs
and li's are defined in the obvious way. Now noticing that
his) = [r{s,X,7r)X{s) -r{s,X - Ax,7r){X - Ax)]
+ [ris,X - Ax,7r) - r(s,X - Ax,^)][X(s) - Ax(s)]
= f/ -^{r{s,x,7r)x} dXlAxis)
+ J -^is,X-Ax,7r + XA^)[X-Ax]dXA„is)
= ai{s)Ax{s)+p,{s)A^{s)),
we have from condition (A3) that both cci and P2 are adapted processes
and are uniformly bounded in {t,uj). Similarly, by conditions (H1)-(H3)
and (H5'), we see that the process 0(-,P()) is uniformly bounded and
that there exist uniformly bounded, adapted processes 0:2, 0:3 and P2, Ps
such that
his) = r(s,X,7rOr(s) -r(s,^(s,P),7rOr(s)
+ [r{s,e{s,P),7rOris)~r{s,e{s,P),^)^s)]
= a2{s)Ax{s) + p2{s)A^{s);
his) = aais) Axis) + Pais) A^{s).
Therefore, letting a = Y^^-i at, P = Y^i-i Pi, we obtain that
Ait) = ait)Axit) + Pit)A„it),
where a and P are both adapted, uniformly bounded processes. In other
words, we have from D.23) that
D.24)
Ax it) = XiT) - I {ais)Axis) + Pis)A^is)}ds
-J A^is)ais,P,X,7r)dWois)) +CiT) - Cit).
Now following the same argument as that in Chapter 1, Theorem 6.1
for BSDE's, one shows that D.24) leads to that
D.25)
exp(^- j ais)ds)Axit) = E^exp(- j ais)ds)AxiT)
+ / expT- j aiu)du)dCis)\Tt\.
Therefore AxiT) = XiT) - giPiT)) > 0 implies that Axit) > 0, Vi £
[0,T], P-a.s. We leave the details to the reader.
Finally, note that if (P',X') is an adapted solution of D.8) starting
from p and constructed by Four Step Scheme, then it must satisfy that
X'@) = 61@,p), hence X@) > X'@) by the first part, completing the
proof. n

§4. Hedging options for large investors 217
Note that if {P, X, tt) is any adapted solution of FBSDE D.8) starting
from p, then D.25) leads to that X{t) = e{t,P{t)), 'it e [0,T], P-a.s.,
since C = 0 and A(r) = X{T) - g{PiT)) = 0. We derived the following
uniqueness result of the FBSDE D.8).
Corollary 4.5. Suppose that assumptions of Theorem 4.4 are in force.
Let {P,X,7r) be an adapted solution to FBSDE D.8), then it must be the
same as the one constructed from the Four Step Scheme. In other words,
the FBSDE D.8) has a unique adapted solution and it can be constructed
via D.13) and D.14).
Reinterpreting Theorem 4.4 and Corollary 4.5 in the option pricing
terms we derive the following optimality result.
CoroUsiry 4.6. Under the assumptions of Theorem 4.4, it holds that
h{g{P{T))) = X{0), where P, X are the first two components of the adapted
solution to the FBSDE D.8). Furthermore, the optimal hedging strategy is
given by Gr,0), where tt is the third component of the adapted solution to
FBSDE D.8). Furthermore, the optimal hedging prince for D.7) is given
by X{0), and the optimal hedging strategy is given by (tt, 0).
Proof. We need only show that Gr,0) is the optimal Strategy. Let
(tt', C) e H{B). Denote P' and X' be the corresponding price/wealth pair,
then it holds that X'{T) > g{P'{T)) by definition. Theorem 4.4 then tells
us that X'@) > X@), where X is the backward component of the solution
to the FBSDE D.8), namely the initial endowment with respect to the
strategy Gr,0). This shows that h(g(P(T))) = X@), and therefore Gr,0) is
the optimal strategy. D
To conclude this section, we present another comparison result that
compares the adapted solutions of FBSDE D.8) with difFerent terminal
condition. Again, such a comparison result takes advantage of the special
form of the FBSDE considered in this section, which may not be true for
general FBSDEs.
Theorem 4.7. (Monotonicity in terminal condition) Suppose that the
conditions of Theorem 4.3 are in force. Let (P%X',7r'), i = 1,2 be the
unique adapted solutions to D.8), with the same initial prices p > 0 but
different terminal conditions X^(T) = g^(P^(T)), i = 1,2 respectively. If
g^, g^ all satisfy the condition (H5) or (H5)', and g^(p) > g^ip) for all
p>0, then it holds that X^@) > X'^@).
Proof. By Corollary 4.5 we know that X^ and X^ must have the form
x^t) = e\t,p\t)); x\t)=e\t,p\t)),
where 6^ and 6'^ are the classical solutions to the PDE D.11) with terminal
conditions g^ and g'^, respectively. We claim that the inequality 9^(t,p) >
e'^(t,p) must hold for all (t,p) £ [0,r] x K.^.
To see this, let us use the Euler transformation p = e^ again, and define
u'(t,0 = e\T - t,e^). It follows from the proof of Theorem 4.2 that v}

218 Chapter 8. Applications of FBSDEs
and u^ satisfy the following PDE:
respectively, where
a{t,^,x,Tr) = e~^a{T — t,e^,x,Tr);
bo{t,^,x,7r) = r{T - t,x,7r) - ^a^T - t,^,x,7r)-
f{t,x,Tr) = r{T — t,x,Tr).
Recall from Chapter 4 that u"s are in fact the (local) uniform limits of the
solutions of following initial-boundary value problems:
D.27)
0 = ut- -CTi(i,^,u,U5)u{{ -bo{t,(,,u,u^)u^ +ur{t,u,u{),
u\i,j,^{U)=9\e^), \i\=R;
I «@,0=5'(e«), ieBn,
i = 1,2, respectively, where -Bj^ ={^;|^| < R]- Therefore, we need only
show that u\{t,0 > "le(^0 for all {1,0 e [0,T] x Br and R>0.
For any e > 0, consider the PDE:
D.27e)
ut = --a'^{t,£,,u,u^)u^^ + bo{t,£,,u,u^)u^ -ur{t,u,u{) + e.
\dBR
{t,0=9\e^)+e, \i\^R;
. «@,0 = 5'(e«)+e, i^Bn,
and denote its solution by u)j ^. It is not hard to check, using a standard
technique of PDEs (see, e.g., Friedman [1]), that u)j^ converges to v}^,
uniformly in [0, T] x K. . Next, We define a function
Fit,^,x,q,q) = -CT^ {t, C, X, q)q + bo {t, C, q, q)q - xf{t, x, q).
Clearly F is continuously differentiable in all variables, and u)j ^ and u|j
satisfies
dt
a«2
>^(i,'e,<e,«e)c,«,J«);
dt
f^ = F{t,i,ul,{ul)^,{ul)^{)-
. <.(^0 > <{t,0, it,0 e [0,T] X Bt[J{0} X OBr,
Therefore by Theorem 11.16 of Friedman [1], we have u]^^ > uj^ in Br.
By sending e —^ 0 and then i? —^ oo, we obtain that u^{t,^) > u'^{t,^)

§4. Hedging options for large investors 219
for all (i,0 e [0,r] X W^, whence e^{-,-) > e'^{-,-). In particular, we have
X^{0) = e^{0,p) > 6l2@,p) = X2@), proving the theorem. D
Remark 4.8. We should note that from 6^(t,p) > 6'^{t,p) we cannot
conclude that X^{t)> X^ (t) for all t, since in general there is no comparison
between 6^{t,P^{t)) and 9'^{t,P'^{t)), as was shown in Chapter 1, Example
6.2!
§4.2. Hedging with constraint
In this section we try to solve the hedging problem D.7) with an extra
condition that the portfolio of an investor is subject to a certain constraint,
namely, we assume that
(Portfolio Constraint) There exists a constant Cq > 0 such that |7r(i)| <
Co, for allte [0,r], a.s.
Recall that Tr{t) denotes the amount of money the investor puts in the
stock, an equivalent condition is that the total number of shares of the stock
available to the investor is limited, which is quite natural in the practice.
In what follows we shall consider the log-price/wealth pair instead of
price/wealth pair like we did in the last subsection. We note that these two
formulations are not always equivalent, we do this for the simplicity of the
presentation. Let P be the price process that evolves according to the SDE
D.1). We assume the following
(H6) b and a are independent of tt and are time-homogeneous; g > 0 and
belongs boundedly to C^"*"" for some a £ @,1); and r is uniformly bounded.
Define x{t) = In P{t)- Then by Ito's formula we see that x satisfies the
SDE:
X{t) =Xo + l [b{e^^'\x{s)) - ^a\e^^^\X{s))]ds
D.21) +a{e^'^'\X{s))dW{s)
= Xo+ [ b{x{s),X{s))ds+ [ a{x{s),X{s))dW{s),
Jo Jo
where xo = Inp; b{x,x) = b{e^,x) — ^a'^{e^,x); and a{x,x) = a{e^,x).
Next, we rewrite the wealth equation D.5) as follows.
X{t) =x+ f [r{s)X{s) + 7r{s){b{P{s),X{s)) - r{s))]ds
Jo
D.22) + f Tr{s)a{P{s),X{s))dW{s)-C{t)
Jo
= x- [ f{s,x{s),X{s),7r{s))ds+ [ 7r{s)dx{s) ~ C{t).
Jo Jo
where
D.23) fit, X, X, tt) = -r{s)x - 7r[^a^ (x, x) - r{s)].

220
Chapter 8. Applications of FBSDEs
In light of the discussion in the previous subsection, we see that in order
to solve a hedging problem D.7) with portfolio constraint, one has to solve
the following FBSDE
x{t)=Xo+ [ b{x{s),X{s))ds+ [ a{x{s),X{s))dW{s),
Jo Jo
D.24) lx{t)=g{x{T)) + I /(s,x(s),X(s),7r(s))ds - ^ 7r{s)dx{s)
+ C{T) - C{t),
. k(-)| < Co, dt X dP-a.e. {t,Lo) £ [0,r] x 0.
In the sequel we call the set of all adapted solutions (x, X, 7r,C) to
the FBSDE D.24) the set of admissible solutions. We will be interested
in the nonemptyness of this set and the existence of the minimal solution,
which will give us the solution to the hedging problem D.7). To simplify
discussion let us make the following assumption:
(H7) b and a are uniformly bounded in (x, a;) and both have bounded first
order partial derivatives in x ^nd x.
We shall apply a Penalization procedure similar to the one used in
Chapter 7 to prove the existence of the admissible solution. Namely, we let
(^ be a smooth function defined on IR such that
D.25)
'0
^p{x) = lx-{Co + l)
^-x-{Co + l)
\ip'{x)\ < 1, Va; e R;
\x\ < Co;
x>Co+2;
X < -Co - 2;
and consider the penalized FBSDEs corresponding to D.24) with C = 0:
for each n > 0, and 0 <t < s <T,
D.26)
' X"(s) = Xo + y(x"(r),X"(r))dr + j\{x''{r),X^{r))dW{r),
^"(s) = 5(X"(T)) + r[/(r,x"(r),X"(r),7r"(r)) +n^Gr"(r))]dr
J s
^r
7r"(r)dx"(r).
Applying the Four Step Scheme in Chapter 4 (in the case m = 1), we see
that D.26) has a unique adapted solution that can be written explicitly as
D.27)
X"(s)=^"(s,x"(s));
7r"(s)=^;j(s,x"(s)),
se[t,T\,

§4. Hedging options for large investors 221
where ^" is the classical solution to the following parabolic PDE:
l^"(r,x) = 5(x).
Further, the solution ^", along with its partial derivatives ^", 6^ and 6^^
are all bounded (with the bound depending possibly on n). The following
lemma shows that the bound for 6"' and ^" can actually be made
independent of n.
Lemma 4.9. Assume (H6) and (H7). Then there exists and constant
C > 0 such that
0<^"(x,a;)<C; \e^{x,x)\<C, V(x,a;) £ E^
Proof. By (H6) and (H7), definitions D.23) and D.25), we see that there
exist adapted process a" and P" such that |q;"(s)| < L, \P"'{s)\ < Ln,
Vs e [t, T], Vn > 0, P-a.s., for some L > 0, L„ > 0; and that
f{s, x(s), X"(s), 7r"(s)) + n^Gr"(s)) = a"(s)X"(s) + ^"(sOr"(s).
Define R"{s) = exp{/j* a^dr}, s £ [t, T]. Then by Ito's formula one has
i?"(s)X"(s) = i?"(rM(x"(r)) + f R"{r)p"{r)Tr"{r)dr
J s
D.29) - I i?"(sOr"(r)dW^(r)
J s
= R''{T)gix"iT)) - f i?"(rOr"(r)dW^"(r).
J s
where W"{s) = Wis) - W(t) - // /3"{r)dr. Since P" is bounded for each n,
there exists probability measure Q" <C P such that W"' is a Q" Brownian
motion on [t,T], thanks to Girsanov's Theorem. We derive from D.29) and
(H6) that
0 < R''is)X''{s) = E'3"{i?"(rM(x"(r))|J^,} < C, Q"-a.s.
where the constant C > 0 is independent of n. Consequently X", is
uniformly bounded, uniformly in n, almost surely. In particular, there exists
C > 0 such that 0 < Xf = 0"{t,x) < C, proving the first part of the
lemma.
To see the second part, denote Z"{s) = d^x^^^ x(*))^(*^ x(*))- Since we
can always assume that 9" is actually C^ by the smoothness assumptions
in (H6) and (H7), we can use the similar argument as that in Proposition
1.1 to show that the pair (tt", Z") is an adapted solution to the BSDE:
7r"(s) = g'ixiT)) + f [A"(r)Z"(r) + B"(rOr"(r) + C''ir)]dr
J s
Z''{r)dW{r),
I

222
where
D.30) <
Chapter 8. Applications of FBSDEs
+ Uis,x,x,Tr) \\
B"{s) = U{s,x,x,7r)\
C^{s) = Us,x,x,t:)
l(x,a:,vr)=(x''(s),e''(s,X"(s)),e;(s,X''(s)))
(X,^,^) = (x^(s),0'^(s,x''(s)),0^(s,x^(s)))
Since B" and C" are uniformly bounded, uniformly in n, by (H6) and (H7),
and A" is bounded for each n, a similar argument as that of part 1 will
lead to the uniform boundedness of tt", with the bounded independent of
n. The proof of the lemma is now complete. D
Next, we prove a comparison theorem that is not covered by those in
Chapter 1, §6.
Lemma 4,10. Assume (H6) and (H7). For any n > I it holds that
e^+\t,x) > o^{t,x), v(i,x) e [o,r] x r.
Proof. For each n, let (x",X",7r") be the adapted solution to
D.26), defined on [t,T]. Define X"(s) = 6l"(s,x"+Hs)) and t:1{s) =
^J(s,x""'"^(s)). Applying Ito's formula and using the definition of X", tt",
and 6^ one shows that
dX"(s) = { - /(s,x"+^(s),X"+Hs),7r"+i(s))
- (n + l)^Gr"+i(s)) + 7r"(sN(x"+'(s),X"+i(s))}ds
+ 7r"(s)<T(x"+' (s), X"+i (s))dW^(s).
On the other hand, by definition we have
dX^+\s) = { - f{s,x"^\s),X"+\sU"+\s))
~{n+ l)^Gr"+i(s)) + 7r"(sN(x"+ns),X"+i(s))}ds
+ 7r"(s)a(x"+^(s),X"+i(s))dW^(s).
Now denote X" = X"+^ - X" and tt" = 7r"+i - tt", and note that ip is
uniform Lipschitz with Lipschitz constant 1, b and ^" are uniformly bound,
we see that for some some bounded processes a" and P"' it holds that
dX"(s) = { - a"(s)X"(s) - ;5"(sOr"(s) - ^Gr"+i(s))
+ 7r"{s)a{x"+\s),X"+'{s))dW{s).
Since X"(T) = 0 and (^ > 0, the same technique of Theorem 4.4 than shows
that under some probability measure Q which is equivalent to P one has
X"(s) =E'5| f i?"(r)(^Gr"+i(r.))dr|jf^,} > 0,

§4. Hedging options for large investors 223
where r(s) = exp{J^ a"^{r)dr}. Setting s = t we derive that 0"'~^^{t,x) >
Combining Lemmas 4.9 and 4.10 we see that there exists function 6{t, x)
such that 0"^{t,x) Z' S{t,x)y as n —^ oo. Clearly 6 is jointly measurable,
uniformly bounded, and uniform Lipschitz in x, thanks to Lemma 4.9. Thus
the following SDE is well-posed:
D.31) xis) = f b{x{r),e{r,x{r))dr + l\{x{r),e{r,x{r))dW{r);
Now define X{s) = 6{s,xis)). It is easy to show, using the uniform
Lipschitz property of 6 (in a;) and some standard argument for the stability of
SDEs, that
D.32) lim e\ sup |;^"(s) - ;^(s)|| = 0,
n->oo L t<s<T '
and, together with a simple application of Dominated Convergence
Theorem, that
D-33) ^ _3
E{|X"E) - X{s)\} - E{rE,x"(s)) - e{s,x{s))\)
< 2C|E{|x"(s) - x{s)\} + 2E{\e^{s,x{s)) - 0{s,x{s))\} ^ 0,
as n —^ 00. We should note that at this point we do not have any
information about the regularity of the paths of process X, and neither do we
know that it is even a semimartingale. Let us now take a closer look.
First notice that Lemma 4.9 and the boundedness of r() and a
eJ K{s)\'ds<C- eJ \f{s,x"{s),X"{s),7r"{s))\'ds<C.
Therefore for some processes tt,/" G L'jr{t,T;WC) such that, possibly along
a subsequence, one has
D.34) Gr",/(s,x"(s),X"(s),7r"(s)))^Gr,/0),in {L'^{t,T;U)r.
Next, let us define
D.35)
Since
D.36)
A"{s) = f nip{Tr"{r))dr, 0<t<s<T,
Jt
Ais) = e{t, x) - X{s) - /' f{j)dr - r 7r{r)dx{r).
v Jt Jt
A"{s) = e"{t,x) - X"{s) - |'[/(r,x"(r),X"(r),7r"(r))dr
-yr"(r)dx"(r).

224 Chapter 8. Applications of FBSDEs
Combining D.32)-D.34), one shows easily that, A" converges weakly in
1/^@, T;1R) to A{s). Therefore it is not hard to see that for any fixed
t<si < S2 <T it holds that
D.37) P{^, <^,} = 0,
since A"'s are all continuous, monotone increasing processes. Thus one
shows that both A{s—) and A{s+) exist for all s £ [t,T]. Denote A{s) =
A{s+), then A is cadlag, and for fixed s, A{s) < A{s), P-a.s.. We claim
that the equality actually holds. Indeed, from D.35) we see that X{-) + A{-)
is continuous. Let Q be the rationals in IR, then for each s £ [t, T], it holds
almost surely that
D.38) lim X(r) = lim [X(r) + A(r) - A(r)] =X(s) + A(s) - A{s).
r-€Q r-eQ
On the other hand, since for each r £ [t,T] one has X{r) = d{r,x{''')) >
^"(r, x(r)), using the continuity of the functions ^"'s and the process x(-)
we have
limX(r) > lim^"(r,x(r)) = e"{s,x{s)).
r i s r l s
Letting n —^ oo and using D.38) we derive
X{s) + A{s) = lim X{r) + A{s) > lim e"{s, x{s)) + A{s)
rls n—>-oo
= e{s,x{s)) + A{s) = X{s) + A{s).
Consequently, A{s) > A{s), P-a.s., whence A{s) = A{s), P-a.s.. In other
words, A{s) is a cadlag version of A.
Prom now on we replace A by its cadlag version in D.35) without further
specification. Namely the process X {=0{-,x{'))) is a semimartingale with
the decomposition:
D.39) X{s)=e{t,x)-{l f{r)dr+A{s))-j 7r{r)dx{r), t<s<T,
and is cadlag as well. We have the following theorem.
Theorem 4.11. Assume (H6) and (H7). Let x, /") ^^ be defined by
D.31) and D.34), respectively; and let X(s) — 0(s,xis)), where 9 is the
(monotone) limit of the solutions of PDEs D.28), {6"}. Define
D.40) Cis)^l\fis)-f{x{r),Xir),7r(r))}dr + A(s), t < s < T.
Then (x, X, tt, C) is an adapted solution to the FBSDE with constraint
D.24).
Furthermore, if (x, X,7r,C') is any adapted solution to D.24) on [t,T],
then it must hold that X{t) < X{t). Consequently, x*=X{0) is the

§4. Hedging options for large investors 225
minimum hedging price to the problem D.7) with the portfolio constraint
\At)\ < Co.
Proof. We first show that f{r) - f {x{r), X(r), Z(r))} >0,dtx dP-a.e.
In fact, using the convexity of / in the variable z and that f{t,x, 0,0) = 0
we have, for each n,
/(s,x"(s),X"(s),7r"(s)) - f{s,x{s),X{s)Ms))
>-L(|x"E)-xWI + |^"(s)-X(s)|)
+ [/(s,x(s),X(s),7r"(s)) - fis,xis),X{s),7ris))]
>-L{\x"{s)-x{s)\ + \X^{s)-Xis)\)
+ Gr"(s)-7r(s))/.(x(s),X(s),7r(r)).
Using the boundedness of f-^, we see that for any t] G L'jr{0, T; IR) such that
T] >0, dt X cfP-a.e., it holds that, as n —^ oo,
eJ [fir] - /(r,x(r),X(r),7r(r))Kdr
= \\m E [ [/(r,x"(r),X"(r),7r"(r))-/(r,x(r),X(r),7r(r))Kdr
n—>oo J.
> -LeJ [\x"ir)-x{r)\ + |X"(r) - X(r)|]r,(r)dr
+ E j {7r"{r)-7r{r))U{x{s),X{s)Mr))v{r)dr^0.
Therefore /°(s) - /(s,x(s),X(s),7r(s)) >0,dtx dP-a.e., namely C(-) is a
cadlag, nondecreasing process. Now rewriting D.39) as
X{s)=g{x{T))+l f{r,x{r),X{r),7r{r))dr+l 7r{r)dx{r)+C{T)~C{s),
for t < s < T, we see that {x,X,Tr,C) solves the FBSDE in D.24). It
remains to check that 7r(t) e T, dt x dP-a.e. But since <^r@) = 0 and
\'Pr\ ^ 1) we have
E [ |<^rGr"(s))|2ds <E [ |7r"(s)|2ds < C.
Jo Jo
Thus, possibly along a subsequence, we have ¥'rGr"(-)) —^ 'Pr ^'^^ some (^p £
1/^@, T;1R). Since (po is convex and C^ by construction, we can repeat the
argument as before to conclude that <^p(s) > (^rGr(s)) > 0, dt x dP-a.e..
But on the other hand,
E f <fr{Tr{r))dr <E f <f^{r)dr = lim E f <fr{Tr"{r))dr
Jt Jt "-^°° Jt
= lim -EA^'iT) = 0,

226 Chapter 8. Applications of FBSDEs
we have that (^rGr(s)) = 0, dt x cfP-a.e.
To prove the last statement of the theorem let (x, X, tt, C) be any other
solutions of the FBSDE D.24). Denote for each n, X"(s) = e"'{s,x{s)) and
7f"(s) = 6^{s,x{s))- Applying Ito's formula and Using D.28) one can show
that X" is a solution to the BSDE
X"{s) = gixiT)) + f [f{r,x{r),X"{r),n"{r)) + nv{n^{r))
D.41) ^^ ^
- ^{aHx{r),X{r))-a\x{r),X"{r))]dr - J ^^r)dx{r).
It then follows, with X = X - X", tt = ? - S=", that
X{s)= f [a"(r)X(r)+^"(rOr(r)]dr- /" 7r(r)dx(r) + C(r) - C(s),
J s J s
where a" and ^5" are some bounded, adapted processes, thanks to the
assumptions on the coefficients. Thus some similar arguments as those in
Lemma 4.10 shows that X(s) > 0, Vs £ [t,T], P-a.s. In particular, one
has X{t) > X"{t) — 0^{t,x), for all n. Letting n -^ oo we obtain that
X{t) > 0{t,x) = X{t). Thus (x,X,7r,C) is the minimum solution of D.24)
on [t, T]. Finally, if i = 0, then we conclude that a;* = X@) is the minimum
hedging price to D.7) with portfolio constraint, proving the theorem. D
§5. A Stochastic Black-Scholes Formula
In this section we present another application of the theory established in
the previous chapters to the theory of option pricing. First recall that in the
last section we essentially assumed that the market is "Markovian", that is,
we assumed that all the coefficients in the price equation are deterministic
so that the Four Step Scheme could be applied. We now try to explore
the possibility of considering more general market models in which the
market parameters can be random. To compensate this relaxation, we
return to a standard "small investor" world. Namely, we assume that the
price equations are (compared to D.4)):
E 1) (dP\t) = r{t)P°{t)dt; (bond)
^'' \ dP{t) = P{t)[b{t)dt + a{t)dW{t)], (stock)
where r, b, and a are now assumed to be bounded, progressively
measurable stochastic processes. We also assume that a is bounded away from
zero. To simplify discussion, we shall assume that both P and W are one
dimensional. Thus the wealth equation D.5) now becomes (replacing X by
Y in this section)
E.2) dY{t) = [Y{t)r{t) + 7r{t){b{t) - r{t))]dt + 7r{t)a{t)dW{t) - dC{t).
In the case where r(-) = r, b{-) = b, and ct(-) = a are all constants,
the standard Black-Scholes theory tells us that the fair price of an option

§5. A stochastic Black-Scholes formula 227
of the form g{P{T)) at any time t £ [0,r] is given by
E.3) Yit)=E{e-^^^-'^g{P{T))\Tt},
Here E is the expectation with respect to some risk-neutral probability
measure (or "equivalent martingale measure"). Furthermore, if we denote
u{t,x) to be the (classical) solution to the backward PDE:
rr A\ \ ut +-a'^x'^u^x+rxu^-ru = 0, (i,a;) £ [0,T) x @, oo);
E.4) < 2
[u{T,x) =g{x),
then it holds that Y{t) = u{t,P{t)), Vi £ [0,T], a.s.. Further, using the
theory of BSDE, it is not hard to show that if (Y, Z) is the unique adapted
solution of the backward SDE:
Y{t)=g{P{T))- f [rY{s)+a-\b-r)Z{s)]ds- f Z{s)dW{s),
then Y coincides with that in E.3); and the optimal hedging strategy is
given by 7r(i) = a-'Z{t) = v^{t,P{t)).
In light of the result of §4, we see that the valuation formula E.3) is not
hard to prove even in the general cases when r, 6, cr, and g{-) are allowed to
be random. But a more subtle problem is to find a proper replacement, if
possible, of the "Black-Scholes PDE" E.4). We note that since the
coefficients are now random, a "PDE" would no longer be appropriate. It turns
out that the BSPDE established in Chapter 5 will serve for this purpose.
§5.1. Stochastic Black-Scholes formula
Let us consider the price equation E.1) with random coefficients r, 6, ct;
and we consider the general terminal value g as described at the beginning
of the section. We allow further that r and b may depend on the stock
price in a nonanticipating way. In other words, we assume that r{t,uj) =
r{t,P{t,uj),uj); b{t,uj) = b{t,P{t,uj),uj), and a{t,uj) = a{t,P{t,uj),Lo)
where for each fixed p £ IR, r(-,p, •), 6(-,p, •), and <t(-,p, •) are predictable
processes. Thus we can write E.1) and E.5) as an (decoupled) FBSDE:
P{t)=p+ f P{s)b{s,P{s))ds+ [ P{s)a{s)dWs,
Jo Jo
E.6) \ Y{t) = g{P{T)) - I [Y{s)r{s, P{s)) + Z{s)e{s,P{s))]ds
-J Z{s)dW{s),
where 6 is the so-called risk premium process defined by
e{t,P{t)) = a-'{t,P{t))[b{t,P{t)) - r{t,Pm, Vi £ [0,T];

228 Chapter 8. Applications of FBSDEs
and Z{t) = Tr{t)a{t). We shall again make use of the Euler transformation
X = logp introduced in the last section. By Ito's formula we see that the
log-price process X = log P and the wealth process Y will satisfy
X(i) = ^+ [ 6(s,X(s))ds+ / a{s,X{s))dW{s),
Jo Jo
Y{t) = g{X{T)) - I [Yis)fis,Xis)) + Z{s)e{s,Xis))]ds
-J Z{s)dW^s),
E.7)
where
E.8)
b{t,x,uj) = b{t,e^,uj) — -a'^{t,e^uj);
r{t,x,Lo) = r{t,e^ ,Lo)\ a{t,x,uj) = a{t,e^,uj);
e{t,x,uj) = e{t,e'',uj), g{p,Lj) = 5(e^,w).
We have the following result.
Theorem 5.1. {Stochastic Black-Scholes Formula) Suppose that the
random fields b, f, 6 and g defined in E.8) are progressively measurable in
{t,bj), and are m-th continuously differentiable in the variable x, with all
partial derivatives being uniformly bounded, for some m > 2. Let Let
the unique adapted solution of E.7) be {X,Y,Z). Then the hedging price
against the contingent claim g{P{T), •) at any t £ [0, T] is given by
y(i) = ^|e"i"«^''''^''"'*'5(X(T),-)bt|
= EJe" X"'"<^'^<^"%(P(T), ^l^t},
where E{-\Tt} is the conditional expectation with respect to the equivalent
martingale measure P defined by
^ = exp { - 1^ e{t,X{t))dW{t) -\j^ m,X{t))\'dt].
Furthermore, the backward SPDE
E.10)
(t, x) = g{x) + / I -a'^Uxx + {b- (jO)ux
— fu-\- aqx — qO Ids — / q{s, x)dWs
has a unique adapted solution {u,q), such that the log-price X and the
wealth process Y are related by
E.11) Y{t)=u{t,X{t),-), Vie[0,T], a.s.

§5. A stochastic Black-Scholes formula 229
Finally, the optimal hedging strategy tt is given by, for all t £ [0, T],
E 12) At)=&-\t,Xit))Zit)
= Vu{t,X{t),-) + a{t,X{t))-\{t,X{t),-), a..s.
Proof. First, since the FBSDE E.7) is decoupled, it must have unique
adapted solution. Next, under the assumption, the backward SPDE E.10)
admits a (classical) adapted solution, thanks to Chapter 5, Theorems 2.1-
2.3. Applying the generalized Ito's formula, and the following the Four Step
Scheme one shows that the adapted solution {X,Y,Z) to E.7) satisfies
E.13) Y{t) = u{t,X{t)), Z{t) = q{t,X{t))+a{t,X{t))Vu{t,X{t)).
On the other hand, using the comparison theorem for BSDE (Chapter
1, Theorem 6.1), and following the same argument of Corollary 4.6, one
shows that the hedging price at any time t is Y{t), and the hedging
strategy is given by E.12). Finally, since the Y satisfies a BSDE in E.7), an
argument as that in Theorem 4.4 gives the expression E.9). D
Remsirk 5.2. In the case when all the coefficients are constants, by
uniqueness we see that the adapted solution to the BSPDE E.10) is simply (w,0),
where u is the classical solution to a backward PDE which, after a change of
variable x = log x' and by setting v{t, x') = u{t, log a;'), becomes exactly the
Black-Scholes PDE (8.4). Thus Theorem 5.1 recovers the classical Black-
Scholes formula.
§5.2. Convexity of the European contingent claims
In this and the following subsection we apply the comparison theorems for
backward SPDEs derived in Chapter 5 to obtain some interesting
consequences in the option pricing theory, in a general setting that allows random
coefficients in the market models. Our discussion follows the lines of those
of El Karoui-Jeanblanc-Picque-Shreve [1].
The first result concerns the convexity of the European contingent
claims. In the Markovian case such a property was discussed by Bergman-
Grundy-Wiener [1] and El Karoui-Jeanblanc-Picque-Shreve [1]. Let us now
assume that r and a are stochastic processes, independent of the current
stock price. From Theorem 5.1 we know that the option price at time t
with stock price x is given by u{t, x)=u{t,\ogx), where u is the adapted
solution to the BSPDE E.10). (Note, here we slightly abuse the notations
X and p!). The convexity of the European option states that the function
u{t, •) is a convex function, provided g is convex. To prove this we first note
that by using the inverse Euler transformation one can show that u is the
(classical) adapted solution to the BSPDE:
E.14)
du = { — -p^cF^Uxx — xrux + ru — xaq^ — 6q}dt — qdW{t),
[0,T)x@,oo)
u{T,x)=g{x), x>0.

230 Chapter 8. Applications of FBSDEs
Differentiating E.14) with respect to x twice and denote v = u^x, P = Qxx,
then we see that {v,p) satisfies the following (linear) BSPDE:
dv = i — -x^a'^Vxx — Ba;CT^ + xr)vx — {cr^ + r)v
- xapx-{2a - e)p\dt -pdW{t); {t,y) £ [0,T) x R;
^v{T,x)=g"{x).
E.15)
Here again the well-posedness of E.15) can be obtained by considering its
equivalent form after the Euler transformation (since r and a are
independent of a;!). Now we can applying Chapter 5, Corollary 6.3 to conclude that
V >Q, whenever g" > 0, and hence u is convex provided g is.
We can discuss more complicated situation by using the comparison
theorems in Chapter 5. For example, let us assume that both r and a are
deterministic functions of {t,x), and we assume that they are both C^ for
simplicity. Then E.10) coincides with E.4). Now differentiating E.4) twice
and denoting v = Uxx, we see that v satisfies the following PDE:
. . . 0 = vt + -x^a'^Vxx + axvx +hv + rxx{xUx - u),
where
v{T,x)=g"{x), x>0,
a = 2c7^ + 2x00^, + r;
b = a'^ + Axaax + {xaxY + x^aaxx + 2xrx + r.
Now let us denote V = xux — u, then some computation shows that V
satisfies the equation:
.g^^. io = Vt+^x^a^Vxx+cxVx + {xrx-r)V, on [0,T) x @,oo),
\v{T,x)=xg'{x)-g{x), x>0,
for some function c depending on a and b (whence r and a). Therefore
applying the comparison theorems of Chapter 5 (use Euler transformation
if necessary) we can derive the following results: assume that g is convex,
then
(i) if r is convex and xg'{x) — g{x) > 0, then u is convex.
(ii) if r is concave and xg'{x) — g{x) < 0, then u is convex.
(iii) if r is independent of a;, then u is convex.
Indeed, if xg'{x) — g{x) > 0, then y > 0 by Chapter 5, Corollary 6.3.
This, together with the convexity of r and g, in turn shows that the solution
V of E.16) is non-negative, proving (i). Part (ii) can be argued similarly.
To see (iii), note that when r is independent of a;, E.16) is homogeneous,
thus the convexity of h implies that of u, thanks to Chapter 5, Corollary
6.3 again.

§5. A stochastic Black-Scholes formula 231
§5.3. Robustness of Black-Scholes formula
The robustness of the Black-Scholes formula concerns the following
problem: suppose a practitioner's information leads him to a misspecified value
of, say, volatility a, and he calculates the option price according to this
misspecified parameter and equation E.4), and then tries to hedge the
contingent claim, what will be the consequence?
Let us first assume that the only misspecified parameter is the
volatility, and denote it by ct = a{t,x), which is C^ in x; and assume that the
interest rate is deterministic and independent of the stock price. By the
conclusion (iii) in the previous part we know that u is convex in x. Now let
us assume that the true volatility is an {.Ff}f>o-adapted process, denoted
by a, satisfying
E.18) ^{t)>cr{t,x), V{t,x),a..s.
Since in this case we have proved that u is convex, it is easy to check that
in this case F.16) of Chapter 5 reads
E.19) (£ - £)« + {M-M)q + f-f= ~x''[a^ - a^]u,, > 0,
where {jC,M) is the differential operator corresponding to the misspecified
coefficients (r, ct). Thus we conclude firom Chapter 5, Theorem 6.2 that
u{t,x) > u{t,x), V{t,x), a.s. Namely the misspecified price dominates the
true price.
Now let us assume that the inequality in E.18) is reversed. Since
both E.4) and E.14) are linear and homogeneous, {—u,—q) and (—u,0)
are both solutions to E.14) and E.4) as well, with the terminal condition
being replaced by —g{x). But in this case E.19) becomes
(£ - C){-u) = ^x^[a^ - a^]{-u^^) > 0,
because u is convex, and ct^ < c^. Thus —u > —u, namely u <u.
Using the similar technique we can again discuss some more
complicated situations. For example, let us allow the interest rate r to be
misspecified as well, but in the form that it is convex in x, say. Assume that
the payoff function h satisfies xh'{x) — h{x) > 0, and that f and a are
true interest rate and volatility such that they are {.Ff}f>o-adapted
random fields satisfying f{t,x) > r{t,x), and a{t,x) > a{t,x), V(i,a;). Then,
using the notation as before, one shows that
(£ - C)u = ■^x'^i^'^ - ct^]mxx + {r-r)[xu^ -u]>0,
because u is convex, and xu^ —u = V>0, thanks to the arguments in the
previous part. Consequently one has u{t,x) > u{t,x), V(i,a;), a.s. Namely,
we also derive a one-sided domination of the true values and misspecified
values.

232 Chapter 8. Applications of FBSDEs
We remark that if the misspecified volatility is not the deterministic
function of the stock price, the comparison may fail. We refer the
interested readers to El Karoui-Jeanblanc-Picque-Shreve [1] for an interesting
counterexample.
§6. An American Game Option
In this section we apply the result of Chapter 7 to derive an ad hoc option
pricing problem which we call the American Game Option.
To begin with let us consider the following FBSDE with reflections
(compare to Chapter 7, C.2))
F.1) <^
f bis,Xs,Y„Z,)ds+ [ a{s,X„Ys,Z,)dWs;
Jo Jo
Yt = g{XT) + j h{s,Xs,Ys,Z,)ds- j ZsdWs + (t - Ct
Note that the forward equation does not have reflection; and we assume
that m = 1 and 02{t,x,uj) = {L{t,x,uj),U{t,x,uj)), where L and U are
two random fields such that L{t,x,uj) < U{t,x,uj), for all {t,x,uj) £ [0,T] x
IR" X 0. We assume further that both L and U are continuous functions
in X for all (i,w), and are {.Ff}f>o-progressively measurable, continuous
processes for all x.
In light of the result of the previous section, we can think of X in
F.1) as a price process of financial assets, and of y as a wealth process
of an (large) investor in the market. However, we should use the latter
interpretation only up until the first time we have dC < 0. In other words,
no external funds are allowed to be added to the investor's wealth, although
he is allowed to consume.
The American game option can be described as follows. Unlike the
usual American option where only the buyer has the right to choose the
exercise time, in a game option we allow the seller to have the same right
as well, namely, the seller can force the exercise time if he wishes. However,
in order to get a nontrivial option (i.e., to avoid immediate exercise to be
optimal), it is required that the payoff be higher if the seller opts to force
the exercise. Of course the seller may choose not to do anything, then the
game option becomes the usual American option.
To be more precise, let us denote by Mt,T the set of {.Ff }f>o-stopping
times taking values in [t,T], and t e [0,T) be the time when the "game"
starts. Let r £ Mt^r be the time the buyer chooses to exercise the option;
and a £ Mt^r be that of the seller. If r < ct, then the seller pays L{T,Xr);
if CT < T, then the seller pays C/(ct, X^). If neither exercises the option
by the maturity date T, then the seller pays B = g{XT)- We define the
minimal hedging price of this contract to be the infimum of initial wealth
amounts Yq, such that the seller can deliver the payoff, a.s., without having
to use additional outside funds. In other words, his wealth process has to
follow the dynamics of Y (with dC > 0)i up to the exercise time a /\t /\T,

§6. An American game option 233
and at the exercise time we have to have
F.2) yo-ArAT > 9i^T)'i-{aAT=T}+L{T,Xr)l{T<T,T<a} + U{cr,X^)l^„^^y.
Our purpose is to determine the minimal hedging price, as well as the
corresponding minimal hedging process.
To solve this option pricing problem, let us first study the following
stochastic game (Dynkin game) is useful: there are two players, each can
choose a (stopping) time to stop the game over an given horizon [t,T]. Let
a e Mt,T be the time that player / chooses, and r £ Mt,T be that of player
IPw. If £7 < T, the player / pays U{a){= C/(ct,X^)) to player //; whereas if
T < a <T, player / pays L{t){= L{t, Xr)) (yes, in both cases the player /
pays!). If no one stops by time T, player / pays B. There is also a running
cost h{t){= h{t,Xt,Yt,Zt)). In other words the payoff player / has to pay
is given by
F.3)
/(tAt
h{u)du + Bl{<,Ar=T}
+ -£'Wl{r<T, r<<7} + C^(«^)l{<7<r},
where B £ 1/^@) is a given .Ft—measurable random variable satisfying
L{T) < B < U{T). Suppose that player // is trying to maximize the
payoff, while player / attempts to minimize it. Define the upper and lower
value s of the game by
V{t)= essinf esssup E{i?f (c7,T)|.Ff},
,^ ,s 'T€Mt,T TeM,,T
' -' A
y(i) = esssup essinf E{Rf{a,T)\Tt}
respectively; and we say that the game has a value if V{t) = V_{t) = V{t).
The solution to the Dynkin game is given by the following theorem,
which can be obtained by a line by line analogue of Theorem 4.1 in Cvitanic
and Karatzas [2]. Here we give only the statement.
Theorem 6.1. Suppose that there exists a solution {X,Y,Z,() to FB-
SDER F.1) (with 02it,x) = {L{t,x),U{t,x)). Then the game F.3)
with B = giXr), h{t) = h{t,Xt,Yt,Zt), and L{t,u) = L{t,Xt{u)),
U{t,bj) = U{t,Xt{uj)) has value V{t), given by the backward component
Y of the solution to the FBSDER, i.e. V{t) = V{t) = V_{t) = Yt, a.s., for
alio <t <T. Moreover, there exists a saddle-point {at,%) £ Mt,T x Mt,T,
given by
at = mi{s e [t,T) : Y^ = C/(s,X,)} AT,
ft = inf{se[i,T): n = L(s,X,)}AT,

234 Chapter 8. Applications of FBSDEs
namely, we have
E{Rf''-\at,T)\j^t} < E{Rf''^\aun)\:Ft}
=Yt<E{Rf''-\a,ft)\Tt}, a.s.
for every (a, r) £ Mt^r x Mt,T- D
In what follows when we mention FBSDER, we mean F.1) specified as
that in Theorem 6.1.
Theorem 6.2. The minimal hedging price of the American Game Option
is greater or equal to V{0), the upper value of the game (at t = 0) of
Theorem 6.1. If the corresponding FBSDER has a solution {X,Y,Z,(),
then the minimal hedging price is equal to Yq.
Proof: Fix the exercise times a, r of the seller and the buyer,
respectively. If y is the seller's hedging process, it satisfies the following dynamics
for i < T A CT A T:
Yt-^ \ h(s,Xs,Ys,Zs)ds= I z^stivvs - L,t,
f h{s,Xs,Ys,Zs)ds= f ZsdWs-Ct
Jo Jo
with C non-decreasing. Hence, the left-hand side is a supermartingale.
Prom this and the requirement that y be a hedging process, we get Yt >
E{Rf^'^\a,T)\Tt}, Vi, a.s. in the notation of Theorem 4.1. Since the
buyer is trying to maximize the payoff, and the seller to minimize it, we
get Yt>Vf Vi, a.s.. Consequently, the minimal hedging price is no less
than y@).
Conversely, if the FBSDER has a solution with Y as the backward
component, then by Theorem 6.1, process Y is equal to the value process
of the game, and by D.4) (with i = 0) and B.10), up until the optimal
exercise time a := ctq for the seller, it obeys the dynamics of a wealth
process, since Ct is nondecreasing for t < ao- So, the seller can start with
Yq, follow the dynamics of Y until t = a and then exercise, if the buyer
has not exercised first. In general, from the saddle-point property we know
that, for any r £ Mx),t,
YaAT > 9{XT)l{aAT=T} +-£'(t, ^T)l{T<T,r<a-} + U{a, X^)l^^^^-^.
This implies that that the seller can deliver the required payoff if he uses
a as his exercise time, no matter what the buyer's exercise time r is.
Consequently, Yo = V{0) is no less than the minimal hedging price. Q

Chapter 9
Numerical Methods for FBSDEs
In the previous chapter we have seen various applications of FBSDEs in
theoretical and applied fields. In many cases a satisfactory numerical
simulation is highly desirable. In this chapter we present a complete numerical
algorithm for a fairly large class of FBSDEs, and analyze its consistency as
well as its rate of convergence. We note that in the standard forward SDEs
case two types of approximations are often considered: a strong scheme
which typically converges pathwisely at a rate 0{-y^), and a weak scheme
which approximates only approximates E{f{X{T))}, with a possible faster
rate of convergence. However, as we shall see later, in our case the weak
convergence is a simple consequence of the pathwise convergence, and the
rate of convergence of our scheme is the same as the strong scheme for pure
forward SDEs, which is a little surprising because a FBSDE is much more
complicated than a forward SDE in nature.
§1. Formulation of the Problem
In this chapter we consider the following FBSDE: for t £ [0,T],
A.1)
X{t)=x+ f b{s,e{s))ds+ f a{s,X{s),Y{s))dW{s);
Jo Jo
Y{t)=g{X{T))+ f b{s,e{s))ds- [ Z{s)dW{s),
where 0 = {X,Y,Z). We note that in some applications (e.g., in Chapter
8, §3, Black's Consol Rate Conjecture), the FBSDE A.1) takes a slightly
simpler form:
A.2)
X{t)=x+ f b{s,X{s),Y{s))ds+ [ a{s,X{s),Y{s))dWs;
Jo Jo
Y{t)=g{X{T)) + I b{s,X{s),Y{s))ds- I Z{s)dW{s).
That is, the coefficients b and b do not depend on Z explicitly, and often in
these cases only the components {X, Y) are of significant interest. In what
follows we shall call A.2) the "special case" when only the approximation
of {X, Y) are considered; and we call A.1) the "general case" if the
approximation of {X, Y, Z) is required. We note that in what follows we restrict
ourselves to the case where all processes involved are one dimensional. The
higher dimensional case can be discussed under the same idea, but
technically much more complicated. Furthermore, we shall impose the following
standing assumptions:

236 Chapter 9. Numerical Methods of FBSDEs
(Al) The functions b, b and a are continuously differentiable in t and twice
continuously differentiable in x,y,z. Moreover, if we denote any one of
these functions generically by %!), then there exists a constant a £ @,1),
such that for fixed y and z, 'il){-, •, y, z) £ C'^+t>2+". Furthermore, for some
L> 0,
m;;y,z)\\i,2,a<L, Viy,z)ewe.
(A2) The function a satisfies
A.3) t^<(T{t,x,y)<C, V{t,x,y)e[0,T]xWe,
where 0 < /i < C are two constants.
(A3) The function g belongs boundedly to C^+" for some a £ @,1) (one
may assume that a is the same as that in (Al)).
It is clear that the assumptions (A1)-(A3) are stronger that those in
Chapter 4, therefore applying Theorem 2.2 of Chapter 4, we see that the
FBSDE A.1) has a unique adapted solution which can be constructed via
the Four Step Scheme. That is, the adapted solution (X, Y, Z) of A.1) can
be obtained in the following way:
ft rt
X(t) =T-\- I
A.4)
X{t)=x+ f b{s,X{s))ds+ f a{s,X{s))dW{s),
Jo Jo
Y{t) = e{t,x{t)), z{t) = a{t,x{t),e{t,x{t))e,{t,x{t)),
where
b{t,x) = b{t,x,6{t,x),a{t,x,0{t,x)Nx{t,x))),
a{t,x) = a{t,x,6{t,x));
and 6 £ C^+t'2+" for some 0 < a < 1 is the unique classical solution to
the quasilinear parabolic PDE:
A.5)
Ot + ~a{t, X, efe^^ + b{t, X, e, a{t, x, 6N^N^
+^{t,x,e,a{t,x,e)e^) = o, {t,x) e (o,t) x r,
\,e{T,x)= g{x), xeU.
We should point out that, by using standard techniques for gradient
estimates, that is, applying parabolic Schauder interior estimates to the
difference quotients repeatedly (cf. Gilbarg & Trudinger [1]), it can be shown
that under the assumptions (A1)-(A3) the solution 6 to the quasilinear
PDE A.5) actually belongs to the space C^+t'^+". Consequently, there
exists a constant K > 0 such that
A.6) ll^lloo + ll^tlloo + ll^ttlloo + ll^xlloo + ll^xxlloo + ll^xxxlloo + ll^xxxxlloo < K.
Our line of attack is now clear: we shall first find a numerical scheme
for the quasilinear PDE A.5), and then find a numerical scheme for the

§2. Numerical approximation of the quasilinear PDE 237
(forward) SDE A.4). We should point out that although the numerical
analysis for the quasilinear PDE is not new, but the special form of A.5)
has not been covered by existing results. In the next Section 2 we shall study
the numerical scheme of the quasilinear PDE A.5) in full details, and then
in Section 3 we study the (strong) numerical scheme for the forward SDE
in A.4).
§2. Numerical Approximations of the Quasilinear PDE
In this section we study the numerical approximation scheme and its
convergence analysis for the quasilinear parabolic PDE A.5). We will first
carry out the discussion for the special case completely, upon which the
study of the general case will be built.
§2.1. A special case
In this case the coefficients b and b are independent of Z, we only
approximate {X,Y). Note that in this case the PDE A.5), although still
quasilinear, takes a much simpler form:
B 1) i^t + \T{t,x,e)^e,, + b{t,x,e)e, + b{t,x,e) = o, ie(o,T),
[e{T,x)=g{x), xeK.
Let us first standardize the PDE C.1). Define u{t,x) = 9{T ~ t,x),
and for (f = a, b, and b, respectively, we define
if{t,x,y)=ip{T -t,x,y), y{t,x,y).
Then u satisfies the PDE
\ ut~ ■-a'^{t,x,u)u^^ -b{t,x,u)u^ -b{t,x,u) =0;
[u@,a;) = g{x).
To simplify notation we replace a, b and bhy a, b and b themselves in
the rest of this section. We first determine the characteristics of the first
order nonlinear PDE
B.3) Ut — b{t,x,u)ux = 0-
Elementary theory of PDEs (see, e.g., John [1]) tells us that the
characteristic equation of B.3) is
det\aijt'{s) — Sijx'{s)\ = 0, s > 0,
where s is the parameter of the characteristic and (aij) is the matrix
0
0
0
0 0
—b{t,x,u) 0
-1 0

238 Chapter 9. Numerical Methods of FBSDEs
In other words, if we let parameter s = t, then the characteristic curve C is
given by the ODE:
B.4) x'{t) = b{t,x{t),u{t,x{t)).
Further, if we let r be the arclength of C, then along C we have
dT= [l + b^{t,x,u{t,x))]^dt,
and
Ot xpXdt dx J
where ip{t,x) = [l + b'^{t,x,u{t,x))]^. Thus, along C, equation B.2) is
simplified to
{du 1 2/ \ T/ \
ip— = -a {t, X, u)u^^ + b{t, X, u);
u{0,x) = g{x).
We shall design our numerical scheme based on B.5).
§2.1.1. Numerical scheme
Let h > 0 and Ai > 0 be fixed numbers. Let Xi = ih, i = 0, ±1,---,
and t'' = kAt, k ^ 0,1,---,N, where i^ = T. For a function f{t,x), let
!''{■) = /(**^r); and let f^ = f{t'',Xi) denote the grid value of the function
/. Define for each k the approximate solution w'' by the following recursive
steps:
Step 0: Set w° = g{xi), i = • ■ •, —1,0,1,-••; use linear interpolation to
obtain a function w" (a;) defined on a; £ IR.
Suppose that w*^~^(a;) is defined for a; £ IR, let w^~^ = 'w''~^{xi) and
'b^ = bit'',Xi,wt'); a^=a{t',Xi,wt'y, b^ = 6(i^a;,,w^l);
B.6) i x^ = Xi - b^At, w^-^ = w''-^ix^);
Step k: Obtain the grid values for the fc-th step approximate solution,
denoted by {■w^ }, via the following difference equation:
B.7) '• ^/ = -{affSliw)'^ + F)f; -oo < i < oo.
Since by our assumption a is bounded below positively and b and g are
bounded, there exists a unique bounded solution of B.7) as soon as an
evaluation is specified for w*^~^(a;).
Finally, we use linear interpolation to extend the grid values of
{w^}g_Qo to all a; e IR to obtain the fc-th step approximate solution w*^(-).

§2. Numerical approximation of the quasilinear PDE 239
Before we do the convergence analysis for this numerical scheme, let
us point out a standard localization idea which is essential in our future
discussion, both theoretically and computationally. We first recall from
Chapter 4 that the (unique) classical solution of the Cauchy problem B.2)
(therefore B.5)) is in fact the uniform limit of the solutions {u^} {R —^ oo)
to the initial-boundary problems:
B.2)fl
ut - -a{t,x,ufux^ - b{t,x,u)u^ - b{t,x,u) = 0,
u{0,x) — g{x), a; e IR;
L u{t, x) = g{x), \x\=R, 0<t<T.
It is conceivable that we can also restrict the corresponding difference
equation B.7) so that —io <i<io, for some io < oo. Indeed, if we denote w*°'*^
to be the following localized difference equation
B.7),„
"""' ' =U<^frSl{w)'! + (b)^; -io<i<io,
At 2
Wi = g{xi), ~io <i < io;
w'^i^=g{x±i,), k = 0,1,2
then by (Al) and (A2), one can show that wf is the uniform limit of {w]°' },
as io —^ 00, uniformly in i and k. In particular, if we fix the mesh size h > 0,
and let R = i^h, then the quantities
B.8) max|u(i^a;i)-w*^| and max \u"-{t'',Xi)-w^^''\
i —io<i<io
differ only by a error that is uniform in k, and can be taken to be arbitrarily
small as io (or ioh = R) is sufficiently large. Consequently, as we shall see
later, if for fixed h and At we choose R (or io) so large that the error
between the two quantities in B.8) differ by 0{h + \At\), then we can
replace B.2) by B.2)^1, and B.7) by B.7)jo without changing the desired
results on the rate of convergence. But on the other hand, since for the
localized solutions the error \u^{t'',x±ia) ~ ^So I = 0 for all fc = 0,1,2, • • •,
the maximum absolute valueoftheerror |u-'*(i*^,a;i)—w]°' \,i = —ior""i*o,
will always occur in an "interior" point of {—R,R). Such an observation
will be particularly useful when a maximum-principle argument is applies
(see, e.g.. Theorem 2.3 below). Based on the discussion above, from now on
we will use the localized version of the solutions to B.2) and B.7) whenever
necessary, without further specifications.
To conclude this subsection we note that the approximate solutions
{w*^(-) are defined only on the times t = t'', k = 0,1,- ■ ■ ,N. An
approximate solution defined on [0,T] x IR is defined as follows: for given h > 0

240 Chapter 9. Numerical Methods of FBSDEs
and Ai > 0,
N
h,At
B.9) w''^'^\t,x) = {
Y,w^{x)\t^-\t<']{t), <e@,T];
k=\
w°(a;), i = 0.
Clearly, for each k and i, ■w'^'^^{t'',Xi) = wf, where {wf} is the solution to
B.7).
§2.1.2. Error analysis
We first analyze the approximate solution {w*^(-)}. To begin with, let us
introduce some notations: for each k and i, let
B.10) 5t^=Xi-b{t'',Xi,u^-')At, «f-i =«(*'=-!, xt).
Let {x{t) : t''~^ < t < t''} be the characteristic such that x{t'') — xt. That
is, by B.4),
x{t)=Xi- b{s,x{s),u{s,x{s)))ds, t''~'^ < t < t''.
Denote x — x{t''~^). It is then easily seen that
sup \x{t) -Xi\ < ||6||ooAi;
t''-^<t<ti'
\^i-x\< [ \b{t'',Xi,u^-')-b{t,x{t),u{t,x{t)))\dt
< {\\bt\\oo + l|6xl|oo||6||oo + ll&ulloodKlloo + |K||oo||6||oo)}Ai2
To simplify notations from now on we let C > 0 to be a generic constant
depending only on b, b, a,T, and the constant isT in A.6), which may vary
from line to line. Thus the above becomes
B.11) sup \x{t) - Xi\ < CAt; |xf -x\< CAt^.
t''-^<t<t''
We now derive an equation for the approximation error. To this end,
recall x and xf defined by B.10); and note that along the characteristic
curve C,
,du n(t'',x) -u(t''~'^,x) ,, ,u(t'',x) -u(t''~^,x)
lb—- K, lb K, w[x)— ^ f
9t ^ At ^^ ^[(a;-fJ + (AiJ]l
_ u{t'',x)-u{t''-\x)
At
The solution of B.5) thus satisfies a difference equation of the following
form: for —00 < i" < 00 and k = I,-- ■ ,N,
B.12) "-• ~^~' = li<r{u)^rSl{u)^ +b{u)^+et

§2. Numerical approximation of the quasilinear PDE
241
where u^~^ = u*^~^(xf) and b{u)^ and a(u)f correspond to bf and af
defined in B.6), except that the values {wf~^} are replaced by {u^~^}; e^
is the error term to be estimated. We have the following lemma.
Lemma 2.1. There exists a constant C > 0, depending only on b, b,
a, T, and the constant K in A.6), such that for all k — 0,---,N and
— 00 < i < 00,
\e'^\<C{h + At).
Proof. First observe that at each grid point {t'',Xi)
tl^{t'',Xi)
du
1
;t^x^) 2
= o^ (* >^iy'^i)'U-x
{t'',Xi)
+ 6(i^a;i,w^).
Therefore, for —oo < i < oo, fc = 1, • • •, A'^,
e? =
At
^{t\Xi)^
)Ul
(t\xi)
I 1
I(t\x0 2
iaiu)frS'iu)
+ {bit\xuu^) - 6(«)f} = ii'"+1^'" + /^^
)f}
We estimate ij''', if''' and I^''' separately. Recall that C will denote
a generic constant that might vary from line to line. Using the uniform
boundedness of by and u^ we have
B.13)
Similarly,
r3,fc
ir\ = \bit'',Xi,ul)~b{t\xi,ur')\ < CAt
lli'^l < \{W\t'',Xi,u^)-a'{t\xi,ut')\K.it\xi)\
('•14) + \a'{t'',x.,ut')\\u..{t',x^) - <L_^^1^||
< C{\\ut\\ooAt + ||«„^||oo/i) < C{h + At).
To estimate I^' we note from B.11) that
u{t''-\x)-u{t''-\5t'l)
B.15)
At
< " '^" ' '- < CAt,
- At -
On the other hand, integrating along the characteristic from (i ,x) to

242
{t'',Xi), we have
Chapter 9. Numerical Methods of FBSDEs
uit'',xi)-u{t''-\x) 1 /■* d
A^ = A^ L. d^"(*'^W)*
1 /■*'=
B.16) "^y^-
d'^u
Since along the characteristics ■—^ depends on uu, utx and u^^ and b,
which are all bounded, one can easily deduce that
.k
B.17) |i^/^_^{[V'|^](*>a:(i))-[v|^](i',a:i)}cii|<C(/i + Ai),
Combining B.11)^B.17), we have
I}M <
u{t'',Xi) - u{t''~'^,x)
At
<C{h + At),
'[^'^]it',^^)
+
l{t'',Xi)
At
proving the lemma.
We are now ready to analyze the error between the approximate
solution w'*'^*(i, a;) and the true solution u{t, x). To do this we define the error
function C{t,x) = u{t,x) - w'^'^*{t,x) for {t,x) £ [0,T] x IR; as before, let
Cf = ({t'',Xi) = u\ — w'l. We have the following theorem.
Theorem 2.2. Assume (A1)—(A3). Then
sup|Cf|-0(/i + Ai).
k,i
Proof. First, by subtracting B.7) from B.12), we see that {Cf} satisfies
the difference equation
rcf-(«r-<-^) 1
B.18)
Since
At
+ m^-bn + e^;
U? = o.
w.
k-l
*:-! ;^k\
= [uit''-\5tf)-u{t''-\x'!)] + [uit''-\xf)
ik-l -k\
,fc-l/=fc
in)]
= cr' + Ht''-\^i)-u{t''-\xt)]

§2, Numerical approximation of the quasilinear PDE 243
where Cf~^ = u{t''~'^,xf) - w*^~^(ff^), and
we can rewrite B.18) as
B.19) ^^^F = ^K^mC)t+/f + ef,
C=0,
where
,^ u{t''-\5t1)-u{t''-\x'^)
' At
+ l[a'{t'',x,,ut') - a'{t\x,,wt'Mi^)' + [^(«)' "^'l-
It is clear that, by A.6) and B.11), for some constant C > 0 that is
independent of k and i, it holds that
^-[a'{t\x,,ut')-<r'{t'',xuwt')]Sl{u)'', + [6(«)^ - 6f]| < C\Ct'l
u{t''-\5t'!)-uit''-\x1) ^^^^
At
Consequently we have
B.20) |/f|<C(|Cf-'| + Ai).
Now by B.19) we have
Cf = Cf-' + {\{^frSl{Ol+l!+el}At.
Considering the "localized" solution of u (described in the previous
subsection) if necessary, we assume without loss generality that the
maximum absolute value of Cf occurs at an "interior" mesh point x^,|.^, where
-R < i{k)h < R for some large R > 0. Now, if we set HC^^H = maxj ICl'l,
then at i{k) we have E^(C)f(fc) < 0- Applying Lemma 2.1 and B.20) we
have
||C'=|| < max|Cf-^|+max{|/^| + leflJAi
B.21) ' * •■ ^
< max \Ct'I + C\\C''~^W^t + C{h + At)At,
i
where C is again a generic constant. Note that the constant C is
independent of the localization, therefore by taking the limit we see that B.21)
should hold for the "global solution" as well.

244 Chapter 9. Numerical Methods of FBSDEs
In order to estimate max.i\(f~^\, we let /i(u)(i*^,-) denote the linear
interpolate of the grid values {u^}'^-oo ^^^ w*^(-) the linear interpolate of
{^,^}fc-oo. then
B.22) max\C^-^ < ma.x\d-'\ + max\u{t''-\x'l) - Ii{u){t''-\x'^)\.
i it
Apply the Peano Kernel Theorem (cf. e.g., [ ]) to show that
ma.x\u{t''-\x'l) - Ii{u){t''-\xl)\ < Ch*h,
i
where h* = 0{At) and C > 0 is independent of k and i. This, together
with C.27), amounts to saying that B.21) can be rewritten as
IIC'II < IIC'-^II + C||C'=-^||Ai + C{h + At)At,
= \\C''-'\\{l + CAt) + Cih + At)At,
where C is independent of fc. It then follows from the Gronwall lemma and
the bound on ||C°|| that HC^'H < C{h + At), proving the theorem. D
§2.1.3. The approximating solutions {u'"'}^i
We now construct for each n an approximate solution u'"' as follows, for
each n e IN let Ai = T/n, and h = 2||6||ooAi. Since h > CAt implies that
l^i - Xi\ < ||6||ooAi < h, xf do not go beyond the interval (a;^_i,a;f+i) for
each i. Now define
B.24) w("'(i,a;) =w^"-^^'"(i,a;), {t,x) e [0,T]xWl,
where defined by B.9). Our main theorem of this section is the
following.
Theorem 2.3. Suppose tha.t (Al)—(A3) hold. Then, the sequence
{«'"'(-,■)} enjoys the following properties:
A) for fixed x G IR, u'"'(-,a;) is left continuous;
B) for fixed t G [0,T], u^"^{t, ■) is Lipschitz, uniformly in t and n (i.e.,
the Lipschitz constant is independent of t and n);
C)snp,^Jui-\t,x)-uit,x)\=0{i^).
Proof. The property A) is obvious by definition B.9). To see C), we
note that
N
u^"^{t,x)-u{t,x) = [w''{x)-u{0,x)]l[o}{t)+Y,[w''{x)-u{t,x)]l^t.-i^t.-i{t).
k=l
Since for each fixed t G {t''~^,t''], fc > 0 or i = 0, we have u'"'(i,a;) = w*^(a;)
for fc > 0 or fc = 0 if i = 0. Thus,
sup|w*^(a;) - u{t,x)\
X
< ||C*^||+sup|/l(u)(i^a;)-w(i^a;)|+sup|u(i^a;)-u(i,a;)|
X X
< IIC^II + o{h + At) + ||«t||ooAi = 0{h + At) = 0{-),
n

§2. Numerical approximation of the quasilinear PDE 245
by virtue of Theorem 2.2 and the definitions of h and At. This proves C).
To show B), let n and t be fixed, and assume that t G (i*^, i*^+^]. Then,
u'"'(i, a;) = w*^(a;) is obviously Lipschitz in x. So it remains to determine
the Lipschitz constant of every w''. Let a;^ and x'^ be given. We may assume
that a;^ £ [xi,Xi^i) and a;^ £ [a;j,a;j+i), with i < j. For i < £ < j — 1,
Theorem 2.2 implies that
|w*(a;^) - w''{xi+i)\ < \w''{xi) - u(i^a;^)|
B.25) + |^i^a;,) - «(i^a;,+l)| + \u{t'',xe+,)-w''{xe+,)\
< 2||C'II + Il«xlloo|a;^ - xe+,\ < Kh = K{xi+i - a;,),
where isT is a constant independent of fc, f and n. Further, for a;^ £
\Xi, a;j_|_i j.
w^ix^) = w''{xi+i) + —^ + ^_ ^^(a;^ - a;i+i).
Hence,
k^a;^) -^'(a^i+OI = ^'(^-+i)_^'(^-) la;! -a:,+i| < i^la;! -a:,+i|,
a^i+l Xi
where K is the same as that in B.25). Similarly,
|w'=(a;2)-w'=(a;j)| <K\x''-Xj\.
Combining the above gives
|w'=(a;i)-w'=(a;2)|
■ <\w\x') - w\xi+^)\ + Y, \y^\xi) - w\xt+^)\ + \w''{xj) - w\x^)\
i=\
j-1
<K^{xi+i -a;^) + ^(a;^+i - a;^) + (a;^ - Xj+i)] = K\x^ - x^\.
i=i
Since the constant K is independent of t and n, the theorem is proved.
D
§2.2. General case
In order to approximate the adapted solution 0 = (X, Y, Z) to the general
FBSDE A.1), we need to approximate also component Z. In fact this
component is particularly important in some application, for jnstance, it is the
hedging strategy in an option pricing problem (see Chapter 1). The main
difficulty is, in light of the Four Step Scheme, we need also to approximate
the derivative of the solution 6 of the PDE B.4), which in general is more
difficult. Our idea is to reduce the PDE B.4) to a system of PDEs so that
6^ becomes a part of the solution but not the derivative of the solutions.

246
Chapter 9. Numerical Methods of FBSDEs
To be more precise let us assume that b and b both depend on z, thus
PDE B.4) becomes
B.25)
0 = dt+ 2^' (*. a;, 6N^^ + 6(i, x, 6, -a{t, x, 6N^N^
+ b{t,x,e,-a{t,x,e)e^y,
[e{T,x)=g{x).
Define bo and 6o by
B.26)
bo{t,x,y,z) = b{t,x,y,-a{t,x,y)z)]
bo{t, X, y, z) = b{t, X, y, -a{t, x, y)z).
One can check that, if ct, b and b satisfy (A1)-(A3), then so do the functions
a, bo and 6o- Further, if we again set u{t,x) = 6{T — t,x), V(i,a;), then
B.25) becomes
. . I ut = ■-a'^{t,x,u)u^^ + bo{t,x,u,u^)u^ + bo{t,x,u,u^y,
[u{0,x) ^g{x).
We will again drop the sign """ in the sequel. Now define v{t, x) = Ux{t, x).
Using standard "difference quotient" argument (see, e.g., Gilbarg-Trudinger
[1]) one can show that under (A1)-(A3) ?; is a solution to the
"differentiated" equation of B.27). In other words, (u, v) satisfies a parabolic system:
B.28)
where
Ut = ■-a^{t,x,u)uxx + bo{t,x,u,v)ux + bo{t,x,u, v);
vt = ■^^'^{t,x,u)vxx + Boit,x,u,v)vx + Bo{t,x,u,v);
u{0,x) = g{x), v{0,x) = g'{x),
B.29) <
Bo{t,x,y,z) =a{t,x,y)[ax{t,x,y) +ay{t,x,y)z] + b{t,x,y,z)
+ bz{t, X, y, z) + bz{t, X, y, z);
Bo{t,x,y,z) = [bx{t,x,y,z) + by{t,x,y,z)z]z + b{t,x,y,z)z
+ bx{t,x,y,z).
We should point out that, unlike in the previous case, the functions Bo
and Bo in B.29) are neither uniformly bounded nor uniformly Lipschitz,
thus more careful consideration should be given before we make arguments
parallel to the previous special case. First let us modify B.29) as follows.
Let K be the constant in A.6) and let (fK £ C°°AR) be a "truncation
function" such that ^k{z) = z for \z\ < K, ^k{z) = 0 for \z\ > K +1, and
l¥'i<-(-2^)l < C for some (generic) constant C > 0. Define
Bq {t,X,y,z) = Bo{t,X,y,^k{z)); B^{t,x,y,z) = Bo{t,x,y,<pk{z)).

§2. Numerical approximation of the quasilinear PDE 247
Then B^ and B^ are uniformly bounded and uniform Lipschitz in all
variables. Now consider the "truncated" version of B.28), that is, we replace Bq
and Bq by Bq and Bq in B.28). Applying Lemma 4.2.1 we know that this
truncated version of B.28) has a unique classical solution, say, {u^,v^),
that is uniformly bounded. But since ||?^||oo < -^^^ by A.6), {u,v) is also a
(classical) solution to the truncated version of B.28), thus we must have
{u,v) = {u^,v^) by uniqueness. Consequently, we need only approximate
the solution to the truncated version of B.28), which reduces the technical
difficulty considerably. For notational simplicity, from now on we will not
distinguish B.28) and its truncated version unless specified. In fact, as we
will see later, such a truncation will be used only once in the error analysis.
§2.2.1. Numerical scheme
Following the idea presented in §2.1, we first determine the characteristics
of the first order system
j ut -bo{t,x,u,v)u^ = 0;
\^vt - Bo{t, X, u, v)vx = 0.
It is easy to check that the two characteristic curves d : {t, Xi{t)), i = 1,2,
are determined by the ODEs
J dxi{t) = bo{t,Xi{t),u{t,xi{t)),v{t,Xi{t)))dt;
\dX2{t) =Bo{t,X2{t),u{t,X2{t)),v{t,X2{t)))dt.
Let Ti and T2 be the arc-lengths along Ci and C2, respectively. Then,
dn = xpi {t, xi {t))dt; dT2 = V'2 (*, X2 {t))dt,
where
Ui{t,x) = [l + bl{t,x,u{t,x),v{t,x))Y^^;
\Mt,x) = [1 + B^it,x,u{t,x),v{t,x))f/\
Thus, along Ci and C2, respectively.
and B.28) can be simplified to
B.30)
du 1 n , T , s
ipi— = -a [t,X,u)u^^ + bo{t,X,u,V);
'4'2-K— = ■^(^'^{t,X,u)v^^ +Bo{t,X,U,v).
Numerical Scheme.
For any n £ IN, let At = T/n. Let h > 0 be given. Let t'' = kAt,
fc = 0,1,2, • • •, and Xi = ih, i = ■ ■ ■, —1,0,1, ■ • •, as before.

248
Chapter 9. Numerical Methods of FBSDEs
Step 0: Set Uf = g{xi), V^° = g'{xi), Vi, and extend C/° and V° to all a; £ K.
by linear interpolation.
Next, suppose that U''~^, V''~'^ are defined such that U''~^ (xi) — U^~^,
y*^-i (si) = y/-\ and let
B.31)
f Fo)f =6o(<^a:i,C/^^V,'=-^);
{Bo)f = Bo{t'',Xi,Ut\V^-'y,
a^ = a{t'',Xi,Ut');
;^k
•^ =Xi + bo{t'',Xi,Ur\V^-')At;
{ ^1=Xi + Bo{t\xuUt\V,^-')M,
and U^-^ = C/'=-i(ff), y^-i = V^-^{^'y).
Step k: Determine the fc-th step grid values {U'',V'') by the system of
difference equations
B.32)
We then extend the grid values {17^} and {V/} to the functions U''{x) and
y*^(a;), a; e IR, by linear interpolation.
§2.2.2. Error analysis
We follow the arguments in §2.1. First, we evaluate the first equation in
B.30) along Ci and the second one along C2 to get an analogue of B.12):
where uf = u{t'',Xi), vf = v{t'',Xi) (recall that {u,v) = (w,Wx) is the true
solution of B.29)), and uf^~^ = u{t''-^,Si), vf = v{t''-^,Sii), with
xf =Xi + bo{t'',Xi,u'^-\vt')^t; xf = Xi+Boit'',Xi,ut\vt')^t-
Also, <T(u)f, bo{u,v)i and Bo{u,v)i are analogous to af, Fo)f and (-Bo)f,
except that U^~^ and V^''~^ are replaced by u^~^ and !;f~^
Estimating the error {(eOf} and {F2)^^} in the same fashion as in
Lemma 2.1 we obtain that
B.33)
sup{\{e^)n + \{e2)l\}<0{h + At).
k.i

§2. Numerical approximation of the quasilinear PDE
249
We now define as we did in B.9) the approximate solutions C/'"' and
y(") by
C/("'(i,a;)
B.34)
y ("'(<, a;) = <
^C/'=(a;)l((^-iiT_^j(i), te{0,T];
k=l
[C/0(rr), i = 0;
n
^y'=(a;)l(i^-^_^j(i), te{0,T];
k=l
[vyx],
t = 0.
Let ^{t,x) = u{t,x) - U"{t,x) and C{t,x) = v{t,x) - V"{t,x). We can
derive the analogue of B.19):
^'' ^' ' - -Mrs'M + {hfi + {ei)\;
( (k _ ?k-l
where
{hf, =
u{t''-\xf)-u{t''-\x'f)
At
+ [bo{u,v)^-{bo)n
+ ^-[a'{t'',Xi,ur')-a'it\x,,UtWMi-,
ih)! =
T^{t'',Xi,u'^-^)-a^{t''
j.k—1 ^k\ _ ,,/ik—l zyk\
vit^-^yif) - vif'-^^tf)
1,
At
..k-l^
+ [Bo{u,v)'^-{Bo)'i]
+ ^[a'{t'',Xi,vt')-cT'{t'',xuVr')]Sl{v)l;
Using the uniform Lipschitz property of bo in y and z, one shows that
B.35) |(/^)^|<C2{|^^l| + |C^M}+C3(/l + Ai), Vfc,j.
To estimate {h)^, we will assume that {{U^},{Vj'}) is uniformly bounded,
otherwise we consider the truncated version version of B.28). Thus B^ is
uniform Lipschitz. Thus,
\Bo{u,v)^ - {Bo)'!\ < Ci{\^t'\ + \(i~'\), Vfc,j,
where d depends only on the bounds of u, v, {U^}, {V/}, and that of a,
b, b and their partial derivatives. Consequently,
B.36) \{l2)l\<C!,{\^t'\ + \(i~'\} + C's{h + At), yk,i.
Use of the maximum principle and the estimates B.33), B.35) and B.36)
leads to
ik'ii < ii^'-^i + c2{\\e-'\\ + wc'-'mt+c,{h+At)At;
IIC'II < IIC'-'ll + Cm'-'W + IIC'-'IDAi + C',{h + At)At;

250 Chapter 9. Numerical Methods of FBSDEs
Add the two inequalities above and apply Gronwall's lemma; we see that
sup{\\e\\ + \\C''\\)=0{h + At).
k
Applying the arguments similar to those in Theorem 2.3 we can derive the
following theorem.
Theorem 2.4. Suppose tha.t (A1)-~(A3) hold. Then,
sup{\U^"\t,x)-u{t,x)\ + \V^"Ht,x)-u,{t,x)\} = 0{-).
{t,x) "
Moreover, for each fixed a; £ IR, [/'"'(-, a;) and y'"' (•, a;) are ieft-continuous;
for hxed t e [0, T], [/<"' {t, ■) and y'"' {t, ■) are uniformly Lipschitz, with the
same Lipschitz constant that is independent of n.
§3. Numerical Approximation of the Forward SDE
Having derive the numerical solution of the PDE A.5), we are now ready
to complete the final step: approximating the Forward SDE A.4). Recall
that the FSDE to be approximated has the following form:
C.1) Xt = x+ f b{s,Xs)ds + [ a{s,Xs)dWs,
Jo Jo
where
b{t,x) = b{t,x,6{t,x), —a{t,x,6{t,xNx{t,x)) = bo{t,x,6{t,x),6x{t,x));
a{t,x) = a{t,x,6{t,x)).
for {t, x) e [0, T] X IR.
To define the approximate SDEs, we need some notations. For each
n e IN, set Ai„ = T/n, t"''' = kAt„, fc = 0,1,2, ■ ■ ■, n, and
n-l
C.2)
,,"(*) = ^^".^[.^....^..^.((i), te[0,T);
k=0
[77"(T)=T.
Next, for each n, let ([/*"',y'"') be the approximate solution to the PDE
A.5), defined by B.35) (in the special case we may consider only u'"'
defined by B.24)). Set
C.3) r(i,a;) = C/("'(T-i,a;), ^^(i,a;) = y("'(T - i,a;),
and
6"(i,a;) = boit,x,e''{t,x),e2it,x)); aF"(i,a;) = a{t,x,e''{t,x)).
By Theorem 2.4 we know that ^" is right continuous in t and uniformly
Lipschitz in x, with the Lipschitz constant being independent of t and n;

§3. Numerical approximation of the FSDE 251
thus, so also are the functions 6" and a". We henceforth assume that there
exists a constant K such that, for all t and n,
C.4) |6"(i,a;)-6"(i,a;')| + |5"(<,a;)-5"(i,a;')| <i<r|a;-a;'|, x,x'eU.
Also, from Theorem 3.4,
C.5) sup\b"{t,x)~b{t,x)\+sup\a"{t,x)-a{t,x)\ = 0{-).
t,X t,X "^
We now introduce two SDEs: the first one is a discretized SDE given
by
C.6) X^ = x+ f 6"(-,X."),„(,)d5+ / aF"(.,X."),„(,)dW^„
Jo Jo
where 77" is defined by C.2). The other is an intermediate approximate
SDE given by
C.7) X^ = x+ [ 6"(s,X^)ds + [ aF"(s,X;)dW,.
Jo Jo
It is clear from the properties of 6" and ?" mentioned above that both
SDEs C.6) and C.7) above possess unique strong solutions.
We shall estimate the differences X" — X" and X" — X, separately.
Lemma 3.1. Assume (Al)—(A3). Then,
e{ sup \X^-Xn'}=0{-).
Proof. To simplify notation, we shall suppress the sign "~" for the
coefficients in the sequel. We first rewrite C.6) as follows:
X^ = Xo+u^+ f b"{s,X^)ds+ [ c7"(s,X;)dW„
Jo Jo
where
< = [\b"{.,Xnv.is)-b"{s,X-)]ds+ [\a"{;Xnr,.^,)-a"{s,X^)]dW,.
Jo Jo
Applying Doob's inequality, Jensen's inequality, and using the Lipschitz
property of the coefficients C.4) we have
e{sup|X,"-X,"|2|
•■ s<t ->
C.8) <3E\snp\u^\''}+3KH [ E{\X^ - X^\''}ds
'^ s<t ' Jo
ft
+ 12K'^ / E{\X^ - X^\^}ds.
Jo

252 Chapter 9. Numerical Methods of FBSDEs
Now, set anit) = E{ sup^<4 \X^ ~ X^\'^}. Then, from C.8),
a„(i) <3E{supK|2}+3i<:2(T+4) / a„(s)ds,
s<t Jo
and Gronwall's inequality leads to
C.9) e|sup|X,"-X;P} <3e3-^'(^+^'E{supKp|.
We now estimate E{sup^<j Kl^}. Note that if s £ [i"'*^,i"'*^+i), for
some 1 < fc < n, then ??"(s) = fcAi„ (whence T — ??"(s) = (n — fc)Ai„, as
T = nAi„) and T - s £ ((n - fc - l)Ai„, (n - fc)Ai„]. Thus, by definitions
B.9) and C.2), for every a; £ IR
6l"G7"(s),a;) = w("'(T - rf{s),x) = «<"'((« - k)At„,x)
= w("'(T-s,a;)=r(s,a;).
More generally, for all {s,x) £ [0,T] x IR,
b"{s,x) = b{s,x,e"{s,x)) = b{s,x,e"{r]"{s),x)).
Using this fact, it is easily seen that
[ b^{;X")r,„(s)-b''{s,X^)ds
Jo
< f |6G7»E),X;„(,),^"G7"E),X;„(,))) -6E,X,",^"E,X,"))
Jo '
< r{|6G?"(s),X;„,,),r(s,A7„(,)))-6(s,X;,^"(s,X;„,,)))
JO '■'
+ \b{s,x^,e^{s,x^^^^^))-b{s,x^,e^{s,x^))\]ds
Using the boundedness of the functions bt, b^ and by, we see that
/i < ^ {\\bt\Uv"is) -s\ + ||6x||oo|x;^.(.) - X^\]ds,
Thus,
|'6"(-,X."),„(,) -6"(s,X,")ds| < ^1 {|»?"(s) -s| + |^,"„(,) -X^\}ds,
where K depends only on K, \\bt\
OO 5 11 ^X 11OO
and ||6y||oo- Since
/ \r{s)~s\ds = Y, (s-i'=)d5<-^(Ai„J^—,
■^° fc=o •'t'°^t k=o "

§3. Numerical approximation of the FSDE 253
Sfsup f 6"(-,X."),„(,)-6"(s,X,")dsf)
C.10) "^* ^° , ^\
Using the same reasoning for a with Doob's inequality, we can see that
Efsup / a"(-,X."),"(.)-c7"(s,X;)dW^/|
•■ u<t Jo ' ^
C.11) < 8^2{ |*E|X;„,,) - X^l'ds + j\s - n''{s)?ds}
Combining C.10) and C.11), we get
E{sup \u1\'} < k\AT + 16) £ E|X;„(,) - X^\^ds + k'T{T + y)^.
Thus, by C.9),
£;{sup|x,"-x;n
C.12) < 3e3^'(^+4) [k^{AT + 16) T i;!^;;^.,^) - X^fds
+ ^^T(T+f)^}.
Finally, noting that |X;„(^) - X,"| < |X;„(^) - X,"| + |X," - X^\ and that
X;;"^,,) - X," = 6"(.,X."),„(,)(s - 77"(s)) + a(-,X."),.(.)(W^. - W^,"(.)),
we see as before that
ft
^ E|x;„(,)-x,"|2d5<2^ {lHlL(«-^"(«))' + IHlLk-^"(«)l}rf«
Therefore, C.12) becomes
C.13) E|sup|X;-X,"n <Ci-+C2^+C3 / E|sup|X,"-X,"|')ds,
•- s<t n n^ Jo "■ r-<s ->
where Ci, C2 and C3 are constants depending only on the coefficients 6,
a and isT and can be calculated explicitly from C.12). Now, we conclude
from C.13) and Gronwall's inequality that

254 Chapter 9. Numerical Methods of FBSDEs
where Pn = C\n^^ + C2n~'^ and Ct = C3T. In particular, by slightly
changing the constants, we have
an{T)=E{ sup |Xf-X;|n<,^ + % = 0(-),
•■ o<t<T inn'' n
proving the lemma. □
The main result of this chapter is the following theorem.
Theorem 3.2. Suppose that the standing assumptions (Al)—(A3) hold.
Then, the adapted solution (X, Y, Z) to the FBSDE A.1) can be
approximated by a sequence of adapted processes (X",Y", Z"), where X" is the
solution to the discretized SDE C.6) and, for t £ [0,r],
y," := r(i,xf); z^ := -a{t,x^,e"{t,x,"))C(i,^D,
with 9" and 6^ being defined by C.3) and [/<"' and V'"' by B.34).
Furthermore,
C.14) e{ sup |Xf-Xt|+ sup \Yt"-Yt\-^ sup \Z^-zA = 0{^).
"■ o<t<T o<t<T o<t<T -' yn
Moreover, if f is C^ and uniformly Lipschitz, then for n large enough.
C.15) E{f{X^, Z?)} - E{f{XT, Zt)}
K
for a constant K.
Proof. Recall that at the beginning of the proof of Lemma 3.1, we have
suppressed the sign "~" for h and a to simplify notation. Set
£"(i) = f sup|6"(i,a;) -6(i,a;)|2 +sup|CT"(i,a;) -CT(i,a;)|2),
where 6, 6", a and ct" are defined by C.1) and C.3). Then, from C.5) we
know that supj |£"(i)| = 0{^). Now, applying Lemma 3.1, we have
e{ sup |X; - X,|21 < 2e{ sup \X^ - X;|21 + 2e{ sup \X^ - X,!^|
= 0(i) + 2E|sup|X,"-X,|2).
n y ^<-t ■ J
s<t

§3. Numerical approximation of the FSDE 255
Further, observe that
E{snp\X^~X,f]
^ s<t J
<2T E\b"{s,X^)-b{s,X,)\'^ds + 8 E\a"{s,X^)-a{s,Xs)\^ds
Jo Jo
j-t
<4T E\b"{s,X^)-b"{s,X,)\'ds
Jo
+ 16/ E|c7"(s,X,")-c7"(s,X,)|'ds+4(r + 4) / en{s)ds
Jo Jo
<A{T + A)K'^ [ e\ sup \X^ - Xr\^}ds + 4{T + 4) f en{s)ds.
Jo ^ r<s -' Jo
Applying Gronwall's inequality, we get
C.16) e|sup|X^"-X,|2| <4(r + 4) ( £„(s)ds ■ e^'^+^t^' < -^,
•■ s<t ^ Jo "
where C is a constant depending only on K and T. Now, note that the
functions 6 and ^" are both uniformly Lipschitz in x. So, if we denote their
Lipschitz constants by the same L, then
e{ sup \Yt-Yr\^]
<2e{ sup \e{t,Xt)-e^{t,xn^]
^ 0<t<T '
+ 2e{ sup \e^{t,xit)-e{t,Yn?]
"■ 0<t<T -'
<2L'e\ sup |Xt-Xf|2| +2sup |^(i,a;)-^"(i,a;)|2 =©(-),
•- 0<t<T -' (t,x) "
by Theorem 3.4 and C.16). The estimate C.14) then follows from an
easy application of Cauchy-Schwartz inequality. To prove C.15), note that
Theorem 2.3 implies that, for n large enough, sup^j ,j.) \6^{t,x) — 6{t,x)\ <
Cn~^, for some (generic) constant C > 0. We modify X^ as defined by
C.6) by fixing n and approximating the solution X" of C.7) by a standard
Euler scheme indexed by k:
Xt'' = ^+ I 6(-,X."''=),.(,)ds+ / a{-,X:^'%.(,)dWs.
Jo Jo
It is then standard (see, for example, Kloeden-Platen [1, p.460]) that
C.17) E{/(X?)}-E{/(XJ''=)}
<
- k

256 Chapter 9. Numerical Methods of FBSDEs
On the other hand, we have
|E{/(Xt)} - E{f{X^)}\ < KE{\Xt - X^\}
C.18)
<e\ sup |Xt-X,"|)<-^
for Lipschitzian /, by C.16). Therefore, noting that X" as defined by C.6)
is just X"'", the triangle inequality, C.17) and C.18) lead to C.15). Q

Comments and Remarks
The main body of this book is built on the works of the authors, with
various collaboration with other researchers, on this subject since 1993.
Some significant results of other researchers are also included to enhance
the book. However, due to the limitation of our information, we inevitably
might have overlooked some new development in this field while writing
this book, for which we deeply regret.
In Chapter 1, the results on the pure BSDEs, especially the
fundamental well-posedness result, are based on the method introduced in the seminal
paper of Pardoux-Peng [1]. The results on nonsolvability of FBSDEs are
inspired by the example of Antonelli [1]. The well-posedness results of
FBSDEs over small duration is also based in the spirit of the work of Antonelli
[1]. The whole Chapter 2 is based on the paper of Yong [4].
In Chapter 3 we begin to consider a general form of the FBSDE A)
with an arbitrarily given T > 0. The main references for this chapter
are based on the works of Ma-Yong [1], virtually the first result regarding
solvability of FBSDE in this generality; and Ma-Yong [4], in which the
notion of approximate solvability is introduced. A direct consequence of the
method of optimal control is the Four Step Scheme presented in Chapter 4.
The finite horizon case is initiated by Ma-Protter-Yong [1]; and the infinite
horizon case is the theoretical part of the work on "Black's Consol Rate
Conjecture" presented later in Chapter 8, by Duffie-Ma-Yong [1].
Chapter 5 can be viewed either as a tool needed to extend the Four Step
Scheme to the situation when the coefficients are allowed to be random, or
as an independent subject in stochastic partial differential equations. The
main results come from the papers of Ma-Yong [2] and [3]; and the
applications in finance (e.g, the stochastic Black-Scholes formula) are collected
in Chapter 8.
The method of continuation of Chapter 6 is based on the paper of Hu-
Peng [2], and its generalization by Yong [1]. The method adopted a widely
used idea in the theory of partial differential equations. Compared to the
Four Step Scheme, this method allows the randomness of the coefficients
and the degeneracy of the forward diffusion, but requires some analysis
which readers might find difficult in a different way.
Chapter 7 is based on the work of Cvitanic-Ma [2]. The idea for the
forward SDER using the solution mapping of Skorohod problem is due
to Anderson-Orey [1], while the Lipschitz property of such solution
mapping is adopted from Dupuis-Ishii [1]. The proof of the backward SDER
is a modification of the arguments of Pardoux-Rascanu [1], [2], as well as
some arguments from Buckdahn-Hu [1]. The proof of the existence and
uniqueness of FBSDER adopted the idea of Pardoux-Tang [1], a
generalized method of contraction mapping theorem, which can be viewed as an
independent method for solving FBSDE as well.

258 Comments and Remarks
Chapter 8 collects some successful applications of the FBSDEs
developed so far. The integral representation theorem is due to Ma-Protter-Yong
[1]; the Nonlinear Feynman-Kac formula is in the spirit of Peng [4], but the
argument of the proof follows more closely those of Cvitanic-Ma [2]. The
Black's consol rate conjecture is due to Duffie-Ma-Yong [1]; while hedging
contingent claims for large investors comes from Cvitanic-Ma [1] for uncon-
straint case, and from Buckdahn-Hu [1] for constraint case. The section on
stochastic Black-Scholes formula is based on the results of Ma-Yong [2] and
[3], and the American game option is from Cvitanic-Ma [2].
Finally, the numerical method presented in Chapter 9 is essentially
the paper of Douglas-Ma-Protter [1], with slight modifications. We should
point out that, to our best knowledge, the scheme presented here is the
only numerical method for (strongly coupled) FBSDEs discovered so far,
and even when reduced to the pure BSDE case, it is still one of the very
few existing numerical methods that can be found in the literature.
In summary, FBSDE is a new type of stochastic differential equations
that has its own mathematical flavor and many applications. Like a usual
two-point boundary value problem, there is no generic theory for its
solvability, and many interesting insights of the equations has yet to be
discovered. In the meantime, although the theory exists only for such a short
period of time (recall that the first paper on FBSDE was published in
1993!), many topics in theoretical and applied mathematics have already
been found closely related to it, and its applicability is quite impressive.
It is our hope that by presenting a lecture notes in the series of LNM,
more attention would be drawn from the mathematics community, and the
beauty of the problem would be further exposed.

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Index
adapted solution, 10,25,138
approximate, 54
classical, 105
strong, 105
weak, 105
adjoint equation, 4
aggregator, 5
allocation, 6
cc-efficient, 6
American game option, 232
appreciation rate, 7,208
approximate solvability, 54
Bellman's principle of optimality, 58
Black-Scholes formula
stochastic, 228
robustness of, 231
Black's conjecture, 202
bond, 7
bridge, 139
BSDE, 3
BSDER, 171
BSPDE, 103
Brownian motion, 1
E
equation
adjoint, 4
Hamilton-Jcicobi-Bellman, 60
Riccati, 46
state, 3
stochastic differential, 1
backward, 3
forward-backward, 1,4
European option, 7
exercise price, 7
expiration date, 7
F
FBSDE, 1,4
FBSDER, 181
formula
stochastic Black-Scholes, 228
Feynman-Kcic, 197
FSDER, 169
Four Step Scheme, 81
function
representing, 90
utility, 5
value, 58
charasteristics, 237,247
comparison theorem, 22,130,214,217
condition
monotone, 151
Novikov, 23
consumption, 7,208
contingent claim, 7
control, 3,33,52
controllability, 33,52
cost functional, 3,53
current utility function, 5
D
dynamic programming method, 57
differential utiUty, 5
Gelfand triple. 111
H
Hamiltonian, 59
hedging strategy, 8,209
indicator, 175
large investor, 207
M
maturity date, 7
method of
continuation, 137
optimal control, 51

270
Index
N
nodal set, 53
nodal solution, 90
nonsolvable, 10,139
O
option, 7
call, 7
optimal control, 53
optimal control problem, 3,52
optimality system, 4
ordinally equivalent, 6
P
parabolicity, 104
degenerate, 104
super, 104
portfolio, 7
price
exercise, 7
fair, 8
problem.
optimal control, 3,52
Skorohod, 169
R
rate
appreciation, 7,208
interest, 7
recursive utlity, 5
reflected
BSDE, 171
FBSDE, 181
FSDE, 169
risk
-neutral, 227
premium, 8,227
S
semi-concave, 61
solution
adapted, 10,25,138
approximate, 54
nodal, 90
viscosity, 66
sub-, 66
super-, 66
solvable, 10,139
approximately, 54
stock, 7
sub differential, 174
T
target, 52
Theorem
comparison, 22,214
martingale representation, 2
U
utility
current, 5
process, 5
recursive, 5
V
value
function, 58
lower, 233
upper, 233
viscosity
solution, 66
subsolution, 66
supersolution, 66
volatility, 7,208
W
weight vector, 6
Y
Yosida approximation, 174