Numerical methods for backward stochastic differential equations with applications to stochastic optimal control

Abstract

The concept of backward stochastic differential equation (BSDE) was initially brought up by Bismut when studying the stochastic optimal control problem. And it has been applied to describe various problems particularly to those in finance. After the fundamental work by Pardoux and Peng who proved the well-posedness of the nonlinear BSDE, the BSDE has been investigated intensively for both theoretical and practical purposes. In this thesis, we are concerned with a class of numerical methods for solving BSDEs, especially the one proposed by Zhao et al.. For this method, the convergence theory of the semi-discrete scheme (the scheme that discretizes the equation only in time) was already established, we shall further provide the analysis for the fully discrete scheme (the scheme that discretizes in both time and space). Moreover, using the BSDE as the adjoint equation, we shall construct the numerical method for solving the stochastic optimal control problem. We will discuss the situation when the control is deterministic as well as when the control is feedback.