Numerical Analysis Of General Predictor-Corrector Methods For Stochastic Differential Equations

In recent years, the theory of stochastic differential equations and stochastic analysis has been developed rapidly, and is widely used in many areas such as physics, engineering, biology and economics, but Solutions of most stochastic differential equations are difficult to be obtained, so exploring efficient and stable numerical algorithm has important theoretical and practical significance. Firstly, the general predictor-corrector methods for stochastic differential equations are constructed, and the convergence and stability of the numerical methods are explored. Then the specific methods such as Euler-θ predictor-corrector meth-ods and Milstein-Euler predictor-corrector methods are constructed. Then we discuss their convergence and stability. Finally numerical experiments are taken for verifying the theo-rems. This paper is divided into six parts.The first chapter is an introduction. We mainly introduce the background status of stochastic differential equations, the innovation of this paper and the main content.In the second chapter, we propose test class of problems, and introduce the general predictor-corrector methods for stochastic differential equations.The third chapter, convergence of general predictor-corrector methods for stochastic differential equations is studied, and convergence theorem is given there. Then we discuss the stability f general predictor-corrector methods for stochastic differential equations, and prove that general predictor-corrector methods for stochastic differential equations are zero stable under certain conditions.The fourth chapter, we construct Euler-θ predictor-corrector methods. we get predic-tive value by Euler method, then get correction value by θ method. we also discusses the convergence order and stability of Euler-θ predictor-corrector methods, and do the numerical experiments by nonlinear equations.The fifth chapter, we construct Milstein-Euler predictor-corrector methods. We get predictive value by Milstein methods, then get correction value by Euler methods. We also discuss the convergence order and stability of Milstein-Euler predictor-corrector methods, and do the numerical experiments by nonlinear equations.Finally, we make a summary and outlook.