The Study Of Discrete Solutions And Stability For Backward Stochastic Differential Equations

The theory of Backward Stochastic Differential Equations (BSDE) is developed in recent 20 years. Although the history is not long, the development is greatly rapid. Besides interesting mathematical property, the theory still has application prospect. In the financial theory, BSDE is usually used in the study of economic theory such as recursion utility, differential utility and the pricing of contingent claim. With the past decades development, BSDE has been penetrated in PDE, Financial Mathematics, Stochastic Control and Differential Geometry.The study above is only qualitative of the theory of BSDE. Because it is difficult for general nonlinear BSDE to obtain the explicit solutions, to discrete the solution of BSDE is becoming emphasis recently. In this thesis, we discrete the solutions of some BSDE, give rise to the discrete error and research the stability.There are five parts in this paper. The first chapter introduces the development of BSDE and the recent study of discrete solutions. In the second section, we discuss the discrete solution and its stability of a class of special BSDE, meanwhile we show its convergence speed. In the third section, we consider BSDE as the investment decision-making process, discuss general BSDE can be approached by discrete investment decision-making process, and estimate the approaching error. The next part is concerned with a class of investment decision-making process and a numerical method to solve this class of BSDE. In the last part, we discuss the stability of BSDE driven by continuous semi-martingale using the weak convergence of filtration.