Backward Doubly Stochastic Differential Equations With Stopping Time

In this paper, we give a sufficient condition on the coefficients of a class of backward doubly stochastic differential equations with stopping time (BDSDEs) under which the BDSDEs with stopping time have a unique solution for any given square integrable terminal values. We also discuss the continuous dependence theorem and convergence theorem for this class of equations.Since the backward stochastic differential equations (BSDEs in short) was introduced by Pardoux and Peng in 1990, the theory of BSDEs has been developed and widely used by many researchers. The main result is, for any givenξ∈L~2(Ω, F_T, P; R~k) , the following BSDEhas a unique solution pair (y_t,z_t) in the interval [0, T] under some condition. This class of equations provide a useful framework for formulating many problems in mathematical finance, and they are also useful for problems in stochastic control, stochastic differential game, and probabilistic formula for the solutions of quasi-linear partial differential equations.After Pardoux and Peng (1990) introduced the theory of BSDEs, Pardoux and Peng introduced a new class of backward stochastic differential equations-backward "doubly" stochastic differential equations and also showed the existence and uniqueness theorem of the solution of BDSDEs in a finite time interval. These research allow us to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDEs. In [3], Chen and Wang extended BSDEs from finite time interval [0,T] to infinite time interval [0,∞]. Chen also proved the existence and uniqueness theorem for BSDEs with stopping time in [4]. BDSDEs with stopping time are also very interesting to produce a probabilistic representation of certain quasi-linear stochastic partial differential equations. Here following the approach of Chen and Wang, we present the existence and uniqueness theorem for BDSDEs with stopping time and proved the continuous dependence theorem and convergence theorem.This paper is organized as follows. In chapter 2, we show some important theorems we used in our paper. In chapter 3, we present the setting of the problem and the main assumptions. In chapter 4, first we present and prove the main result-the existence and uniqueness theorem. Then we discuss the continuous dependence theorem and convergence theorem.