Variable Step Numerical Method For Backward Stochastic Differential Equations

Backward stochastic differential equations was first collaboratively proposed by Pardoux and Peng in 1990,from which they obtained the existence and uniqueness theorem under the condition that the generator satisfies the Lipschitz continuity condition.For the next 30 years,the research of backward stochastic differential equations obtained tremendous achievements,and has been applied to multiple fields including Chemistry,Biology,Economics and Mathematical Finance.Solving backward stochastic differential equations presents consequential significance to problems in various subject matters.In general,it is exceedingly difficult to obtain the analytical solutions of BSDEs.Under such circumstances,the numerical algorithms for backward stochastic differential equations hold great significance in both theoretical studies and practical applications.The major innovation of this paper lies in applying the variable step numerical method for backward stochastic differential equations.In the process of solving backward stochastic differential equations,reasonable selection of step size is of great significance to improving the accuracy and computational efficiency.Through the variable step method,the step size can be automatically adjusted according to the calculation in each step,thereby solving the problem of step size selection.In this paper,we provide a kind of type 2(1)variable step size numerical method,a kind of type 3(2)variable step size numerical method and two kinds of third-order numerical schemes that can be used to construct type 3(1)variable step size numerical method.The main framework and results of this paper are as follows:The first chapter introduces the research background and status of the numerical solution to backward stochastic differential equations and the variable step numerical method for ordinary differential equations.Furthermore,it discusses the feasibility of applying the variable step method to stochastic differential equations.The second chapter introduces the basic knowledge of stochastic analysis and results including the existence and uniqueness theorem for backward stochastic differential equations and so on.The third chapter introduces the type 2(1)variable step numerical method,the correlated proofs and the numerical experiments for backward stochastic differential equations.The fourth chapter introduces two kinds of third-order numerical schemes for backward stochastic differential equations under the condition that generator is independent of (5,and the numerical experiments.The fifth chapter introduces the type 3(2)variable step numerical method for backward stochastic differential equations under the condition that generator is independent of (5,and the numerical experiments.The sixth chapter introduces the analysis of the conclusion,the summary of the scheme and the prospect of the follow-up problems of the variable step numerical method for backward stochastic differential equations.The seventh chapter introduces the code of type 3(2)variable step size numerical method.There are totaly 7 figures,6 tables and 58 references in this paper.