The Numerical Algorithms For Bdsdes

In this paper we present two numerical schemes of approximating solutions for backward doubly stochastic differential equations(BDSDEs for short).We give a method to discretize a BDSDE.And we also give the proof of the convergence of these two kinds of solutions for BDSDEs,respectively.Since Pardoux and Peng introduced backward stochastic differential equation (BSDE),the theory of which has been widely used and developed,mainly because of a large part of problems in mathematical finance can be treated as a BSDE.However it is known that only a limited number of BSDE can be solved explicitly.To develop numerical method and numerical algorithm is very helpful,theoretically and practically.Recently many different types of discretization of BSDE and the related numerical analysis were introduced.On the other hand,Paroux and Peng(8)introduced a new class of backward stochastic differertial equations-backward "doubly" stochastic differential equations and also showed the existence and uniqueness of the solution of BDSDE.But until now little work is devoted to the numerical method and the related numerical analysis. Here following the approach of Memin,Peng and Xu(5),we present two numerical schemes of approximating solutions of BDSDE,and proved the convergence of these two kinds of solutions for BDSDEs,respectively.First of the proofs makes use of and extends Donsker-Type theorem.This paper is organized as follows.In chapter 2,we introduce some fundamental knowledge and assumptions of BDSDEs.In chapter 3,the discrete BDSDE and solutions are presented.In chapter 4,we will give our main results:the proof of convergence of numerical solutions for BDSDEs in two different schemes.