Forward-backward stochastic differential equations and their applications.

*(English)*Zbl 0927.60004
Lecture Notes in Mathematics. 1702. Berlin: Springer. xiii, 270 p. (1999).

The authors being the intellectual fathers of the forward-backward stochastic differential equations (FBSDE) in their general form, give with the present book a nice introduction to this theory. The eaerliest version of an FBSDE was introduced by J.-M. Bismut [J. Math. Anal. Appl. 44, 383-404 (1973; Zbl 0276.93060)], in a decoupled, linear form of a usual (forward) stochastic differential equation (SDE) and a linear backward stochastic differential equation (BSDE) in order to describe the adjoint equation in the Pontryagin-type maximum principle in the stochastic control theory. The real breakthrough in the theory of BSDEs was reached only more than fifteen years later, by E. Pardoux and S. G. Peng with the proof of existence and uniqueness of a solution of the general, nonlinear BSDE [Syst. Control Lett. 14, No. 1, 55-61 (1990; Zbl 0692.93064)] and by Peng (1992) who showed that the BSDE generalizes the Feynman-Kac formula and describes the solution of semilinear parabolic partial differential equations (PDE). Their discovery initiated an extensive study of BSDEs, motivated by a manifold of applications in stochastic control, theory of PDEs and in finance. These applications have led finally to FBSDEs, a coupled system of SDEs and BSDEs. After a first work by F. Antonelli [Ann. Appl. Probab. 3, No. 3, 777-793 (1993; Zbl 0780.60058)] who studied FBSDEs over a sufficiently small time interval which allows the application of the the point argument, the authors of the present book have started a systematic study which has led to two methods, the four-step-scheme [J. Ma, Ph. Protter and J. Yong, Probab. Theory Relat. Fields 98, No. 3, 339-359 (1994; Zbl 0794.60056)] and a method using a monotonicity condition [Y. Hu and S. Peng, ibid. 103, No. 2, 273-283 (1995; Zbl 0831.60065)], and it has also given rise to other subjects that are interesting in their own rights.

The book is organised as follows. In the introductory Chapter 1, the authors start with examples of solvable and nonsolvable FBSDEs in order to precise then the notion of a FBSDE. A comparison results for BSDEs is recalled and a comparison result for FBSDEs is established. Chapter 2 is entirely devoted to the study of the solvability of linear FBSDEs; the solvability of nonlinear FBSDEs is studied in Chapter 3 (by a method of optimal control), Chapter 4 (by the four-step-scheme) and Chapter 6 (by the continuation method). While Chapter 7 deals with FBSDEs with reflections, Chapter 8 collects some applications of FBSDEs as a nonlinear Feynman-Kac formula and, in finance, a stochastic Black-Scholes formula and a model of American game options. Finally, Chapter 9 presents a numerical method for FBSDEs.

The book addresses to specialists who are familiar with the stochastic ItĂ´ calculus and want to get a rather complete introduction into the theory of FBSDEs; in the first chapter they will find also a short introduction to BSDEs (with proofs). The book is very well written and completed by an exhaustive list of references.

The book is organised as follows. In the introductory Chapter 1, the authors start with examples of solvable and nonsolvable FBSDEs in order to precise then the notion of a FBSDE. A comparison results for BSDEs is recalled and a comparison result for FBSDEs is established. Chapter 2 is entirely devoted to the study of the solvability of linear FBSDEs; the solvability of nonlinear FBSDEs is studied in Chapter 3 (by a method of optimal control), Chapter 4 (by the four-step-scheme) and Chapter 6 (by the continuation method). While Chapter 7 deals with FBSDEs with reflections, Chapter 8 collects some applications of FBSDEs as a nonlinear Feynman-Kac formula and, in finance, a stochastic Black-Scholes formula and a model of American game options. Finally, Chapter 9 presents a numerical method for FBSDEs.

The book addresses to specialists who are familiar with the stochastic ItĂ´ calculus and want to get a rather complete introduction into the theory of FBSDEs; in the first chapter they will find also a short introduction to BSDEs (with proofs). The book is very well written and completed by an exhaustive list of references.

Reviewer: R.Buckdahn (Brest)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |