Fully Implicit Taylor And Runge-Kutta Numerical Methods For Solving Stochastic Differential Equation

Since the Japanese mathematician Kiyoshi Ito established the theory of stochastic differential equations in the middle of the twentieth century,stochastic differential equations have been widely used in modeling in various fields of life,especially in the field of financial theory,and they are widely known.This thesis will focus on the numerical methods of stochastic differential equations,including the fully implicit Taylor method and the Runge-Kutta method.The existing studies of these two types of methods are only low-order cases.This article will give the high-order methods of these two types of methods.The research content of the thesis is specifically arranged as follows:Chapters 2 and 3 propose a backward stochastic Taylor expansion,and based on this expansion,a fully implicit Taylor strong approximation method is given.Depending on the truncation term,the method converges strongly to the solution of stochastic differential equations with order of 1,2,3,….The second chapter considers the situation of Ito points,and the third chapter considers the situation of Stratonovich points.Chapters 4 and 5 propose a family of stochastic Taylor expansions,and based on this expansion,a family of fully implicit Taylor strong approximation methods are given.Depending on the truncation term,the method strongly converges to the solution of stochastic differential equations with order of 1,2,3,….The fourth chapter considers the situation of Ito points,and the fifth chapter considers the situation of Stratonovich points.Chapter 6 extends the methods from Chapters 2 to 5 to weak approximations.Chapter 7 proposes a family of stochastic Runge-Kutta method.Depending on the selection of stochastic multiple Ito integrals,the method strongly converges to the solution of stochastic differential equations with the order of 1/2,1,3/2,2,….Chapter 8 extends the method of Chapter 7 to the weak approximation case.Chapter 9 and Chapter 10 are the stability research and numerical experiment results of the method described in this paper,respectively.Stability research shows that the fully implicit Taylor method described in this paper has better mean square stability than existing methods.The numerical experiment results are consistent with the convergence theory and stability analysis.