Multistep Numerical Methods Of Backward Stochastic Differential Equations

In 1973,Bismut proposed a class of backward stochastic differential equations with random terms in linear form.In 1990,Pardoux and Peng proposed the more general form of nonlinear backward stochastic differential equations,and on this basis obtained the existence and uniqueness theorem of solutions when the generator satisfies the integrability condition and Lipschitz condition.After more than 30 years of development,many scholars have made a lot of achievements in the field of backward stochastic differential equations,and applied it to many practical problems.As far as practical applications are concerned,it is difficult to obtain the analytical solutions of most nonlinear backward stochastic differential equations,so it is of great significance to construct numerical solution methods to solve related problems.In this paper,the explicit multistep schemes and generalized multistep schemes are proposed based on the numerical multistep methods for solving backward stochastic differential equations.By introducing the adjustment coefficients,the generalized multistep schemes solves the problem that it is impossible to construct multistep schemes of more than 3 order for in existing studies and enables the convergence order of to reach 4 or above.At the same time,the error estimates of two numerical schemes are given respectively in the case that the generator 1)of the backward stochastic differential equations does not depend on .Some numerical experiments are given respectively,and the results of numerical experiments are in agreement with the theoretical results.This paper is divided into six chapters.The first chapter is the introduction,mainly introduces the research status of numerical methods of backward stochastic differential equations.The second chapter is the preliminary knowledge,mainly introduces the basic knowledge and some stochastic analysis related knowledge.In chapter 3,a class of explicit multistep methods is discussed,error estimates and some numerical experiments are given.In chapter 4,a class of generalized multistep methods is discussed,and error estimates are completed.Numerical experiments show that the methods can indeed make the convergence order of reach more than 3.The fifth chapter is the summary of the whole paper and the prospect of the subsequent problems such as the methods of variable stepsize.In chapter 6,the numerical experiments code of the generalized multistep methods and some coefficients tables are appended.