Forward-Backward Stochastic Differential Equations and their Applications

This book is intended to give an introduction to the theory of forwa- backward stochastic di erential equations (FBSDEs, for short) which has received strong attention in recent years because of its interesting structure and its usefulness in various applied elds. 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 di erential equation and a (linear) backwardstochastic dieren tial equation (BSDE, for short). In 1983, Bensoussan [1] proved the well-posedness of general linear BSDEs by using martingale representation theorem. The r st 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 pr- abilistic interpretation of the solutions to some semilinear or quasilinear parabolic partial dieren tial 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 t- oretical areas such as stochastic control, mathematical n ance, dieren tial geometry, to mention a few. The study of (strongly) coupled FBSDEs started in early 90s. In his Ph.