Backward stochastic differential equations and quasilinear partial differential equations

Abstract

Abstract : In this paper we first review the classical Feynman-Kac formula and then introduce its generalization obtained by Pardoux-Peng via backward stochastic differential equations. It is because of the usefulness of the Feynman-Kac formula in the study of parabolic partial differential equations we see clearly how worthy to study the backward stochastic differential equations in more detail. We hence further review the work of Pardoux and Peng on backward stochastic differential equations and establish a new theorem on the existence and uniqueness of the adapted solution to a backward stochastic differential equation under a weaker condition than Lipschitz one. Key Words : Backward stochastic differential equation, parabolic partial differential equation, adapted solution, Bihari's inequality. Introduction In 1990, Pardoux & Peng initiated the study of backward stochastic differential equations motivated by optimal stochastic control (see Bensoussan, Bismut, Haussmann and Kushner). It is even more important that Pardoux & Peng, Peng recently gave the probabilistic representation for the given solution of a quasilinear parabolic partial differential equation in term of the solution of the corresponding backward stochastic differential equation. In other words, they obtained a generalization of the well-known Feynman-Kac formula (cf. Freidlin or Gikhman & Skorokhod). In view of the powerfulness of the Feynman-Kac formula in the study of partial differential equations e.g. K.P.P. equation (cf. Freidlin), one may expect that the Pardoux-Peng generalized formula will play an important role in the study of quasilinear parabolic partial differential equations. Hence from both viewpoints of the optimal stochastic control and partial differential equations, we see clearly how worthy to study the backward stochastic differential equations in more detail.