Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris 1702 Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Jin Ma Jiongmin Yong Forward-Backward Stochastic Differential Equations and Their Applications Springer Authors Jin Ma Department of Mathematics Purdue University West Lafayette, IN 47906-1395 USA e-mail: majin @ math.purdue.edu Jiongmin Yong Department of Mathematics Fudan University Shanghai, 200433, China e-mail: jyong@fudan.edu.cn Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ma, Jin: Foreward backward stochastic differential equations and their applications / Jin Ma ; Jiongmin Yong. - Berlin, Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan, Paris ; Singapore, Tokyo : Springer, 1999 (Lecture notes in mathematics ; 1702) ISBN 3-540-65960-9 Mathematics Subject Classification (1991): Primary: 60H10, 15, 20, 30; 93E03; Secondary: 35K15, 20, 45, 65; 65M06, 12, 15, 25; 65U05; 90A09, 10, 12, 16 ISSN 0075-8434 ISBN 3-540-65960-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. 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Typesetting: Camera-ready TEX output by the authors SPIN: 10650174 41/3143-543210 - Printed on acid-free paper 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 prob- abilistic 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 the- oretical 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 counterexam- ple showing that for coupled FBSDEs, large time duration might lead to non-solvability. In 1993, the present authors started a systematic investiga- tion 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 con- sidered 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 essen- tially 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 Preface 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 solvabil- ity, 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 solvability 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 de- veloped. 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, ex- ample, 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, (5.4), where 5 is the sec- tion 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. Duffle, P. Protter, with whom we Preface ix enjoyed wonderful collaboration on this subject; to N. E1 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 #N00014-96-1-0262; and JY would like 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 w Some Examples 1 w A first glance 1 w A stochastic optimal control problem 3 w Stochastic differential utility 4 w Option pricing and contingent claim valuation 7 w Definitions and Notations 8 w Some Nonsolvable FBSDEs 10 w Well-posedness of BSDEs 14 w Solvability of FBSDEs in Small Time Durations 19 w Comparison Theorems for BSDEs and FBSDEs 22 Chapter 2. Linear Equations 25 w Compatible Conditions for Solvability 25 w Some Reductions 30 w Solvability of Linear FBSDEs 33 w Necessary conditions 34 w Criteria for solvability 39 w A Riccati Type Equation 45 w Some Extensions 49 Chapter 3. Method of Optimal Control 51 w Solvability and the Associated Optimal Control Problem 51 w An optimal control problem 51 w Approximate Solvability 54 w Dynamic Programming Method and the HJB Equation 57 w The Value Function 60 w Continuity and semi-concavity 60 w Approximation of the value function 64 w A Class of Approximately Solvable FBSDEs 69 w Construction of Approximate Adapted Solutions 75 Chapter 4. Four Step Scheme 80 w A Heuristic Derivation of Four Step Scheme 80 w Non-Degenerate Case Several Solvable Classes 84 w A general case 84 w The case when h has linear growth in z 86 w The case when m : 1 88 w Infinite Horizon Case 89 xii Contents w The nodal solution 89 w Uniqueness of nodal solutions 92 w The limit of finite duration problems 98 Chapter 5. Linear, Degenerate Backward Stochastic Partial Differential Equations 103 w Formulation of the Problem 103 w Well-posedness of Linear BSPDEs 106 w Uniqueness of Adapted Solutions 111 w Uniqueness of adapted weak solutions 111 w An It5 formula 113 w Existence of Adapted Solutions 118 w A Proof of the Fundamental Lemma 126 w Comparison Theorems 130 Chapter 6. The Method of Continuation 137 w The Bridge 137 w Method of Continuation 140 w The solvability of FBSDEs linked by bridges 140 w A priori estimate 143 w Some Solvable FBSDEs 148 w A trivial FBSDE 148 w Decoupled FBSDEs 149 w FBSDEs with monotonicity conditions 151 w Properties of Bridges 154 w Construction of Bridges 158 w A general consideration 158 w A one dimensional case 161 Chapter 7. FBSDEs with Reflections 169 w Forward SDEs with Reflections 169 w Backward SDEs with Reflections 171 w Reflected Forward-Backward SDEs 181 w A priori estimates 182 w Existence and uniqueness of the adapted solutions 186 w A continuous dependence result 190 Chapter 8. Applications of FBSDEs 193 w An Integral Representation Formula 193 w A Nonlinear Feynman-Kac Formula 197 w Black's Consol Rate Conjecture 201 w Hedging Options for a Large Investor 207 w Hedging without constraint 210 w Hedging with constraint 219 w A Stochastic Black-Scholes Formula 226 [...]... t}), and the recursive relation (1. 18) is replaced by a differential equation: (1. 19) dY(t) dt - - f ( c ( t ) , Y ( t ) ) , where the function f is the aggregator We note that the negative sign in front of f reflects the time-reverse feature seen in (1. 18) Again, once a solution of (1. 19) can be determined, then U(c) = Y(0) defines a unitiliy function An interesting variation of (1. 18) and (1. 19) is their. .. ~rT-measurable in general Now let us assume that (1. 13) admits an adapted solution (Y(.), Z(.)) Then, applying ItS's formula to Y(t)~(t), one has E IX (T)~(T)] = E [Y(T)~(T)] (1. 14) =E =E /o { [ - aY(t) - X(t)] ~(t) + Y(t)[a~(t) + bv(t)] }dt [ - X(t)~(t) + bY(t)v(t)]dt Hence, (1. 11) becomes (1. 15) 0 o-adapted It is clear that (1. 13) is a BSDE with a more general form than the one we saw in w since Y(.) is specified at t = T, and X ( T ) is ~rT-measurable... (1. 13)!) Substituting (1. 16) into the state equation (1. 9), we finally obtain the following optimality system: dX(t) = laX(t) - b2Y(t)]dt + dW(t), (1. 17) dY(t) = - [ a Y ( t ) + X(t)]dt + Z(t)dW(t), X(O) = x, Y(T) = X(T) t e [0, 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, (1. 17)... following representation: (1. 4) f0t Z ( s ) d W ( s ) , Y ( t ) = Y(O) + V t e [0, T], a.s., where Z(-) E L~(0, T; ~), the set of all {SL-t}t_>0-adapted square integrable processes Writing (1. 4) in a differential form and combining it with (1. 3) (note that ~ is UT-measurable), we have dY(t)=/(t)dW(t), (1. 5) t e [0, T], Y(T) = In other words, if we reformulate (1. 2) as (1. 5); and more importantly, instead... Euclidean norm, see w then (1. 8) EISI 2 - E I Y ( t ) I 2 + EIZ(s )12 ds, V t e [0,T] Thus ~ = 0 implies that Y _ 0 and Z 0 Note that equation (1. 7) is linear, relation (1. 8) leads to the uniqueness of the {~t}t>0-adapted solution (Y(-), Z(.)) to (1. 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 (1. 7), as we expect In the... stochastic differential equation (FBSDE, for short) It is clear that if we can prove that (1. 17) admits an adapted solution (X(-), Y(.), Z(.)), then (1. 16) gives an optimal control, solving the original stochastic optimal control problem Further, if the adapted solution (X(.), Y(.), Z(.)) of (1. 17) is unique, so is the optimal control u(-) w Stochastic differential utility Two of the most remarkable applications. .. ( T ) = ~ so that (1. 1) has a unique solution Y ( t ) - ~ However, if we consider (1. 1) as a stochastic differential equation (with null drift and diffusion coefficients) in ItS's sense, things will become a little more complicated First note that a solution of an It5 SDE has to be {gct}t_>0-adapted Thus specifying Y(0) and Y ( T ) will have essential difference Consider again (1. 1), but as a terminal... (backward) difference equation (1. 18) with terminal condition VT = u(c(T)) can be solved uniquely Likewise, we may pose (1. 19), the continuous counterpart of (1. 18), as a terminal value problem with given Y ( T ) = u(c(T)), or equivalently, /, T (1. 20) Z(t) = u(c(T)) + / t f(c(s), Y(s))ds, t C [0,T] In a stochastic model (model with uncertainty) one assumes that both consumption c and utility Y are stochastic... ff's and the terminal utility functions ui's, the process ca takes the form: ca(t) = K(A(t),e(t),Y(t)), for some ~m-valued function K , and A = (A1, , Am), derived from a first-order necessary condition of the optimization problem (1. 24), satisfies the differential equation: (1. 25) d)~i(t) = Ai(t)bi(t, A(t),Y(t))dt; with bi(t,A,y,w) = ~ an FBSDE w y" t e [0, T], c=(K~(X,e(t,w),y))" Thus (1. 23) and (1. 25)form . Partial Differential Equations 10 3 w Formulation of the Problem 10 3 w Well-posedness of Linear BSPDEs 10 6 w Uniqueness of Adapted Solutions 11 1 w Uniqueness of adapted weak solutions 11 1. mathematics ; 17 02) ISBN 3-540-65960-9 Mathematics Subject Classification (19 91) : Primary: 60H10, 15 , 20, 30; 93E03; Secondary: 35K15, 20, 45, 65; 65M06, 12 , 15 , 25; 65U05; 90A09, 10 , 12 , 16 ISSN. feature seen in (1. 18). Again, once a solution of (1. 19) can be determined, then U(c) = Y(0) defines a unitiliy function. An interesting variation of (1. 18) and (1. 19) is their finite horizon