Research On The Convergence Of ? Methods Of Stochastic Differential Equations

Stochastic differential equations(SDEs) have been proven to be a powerful way to model and solve the problems in real applications, which can take the influence of random factors on the motion of the system into consideration. Thus, SDEs have been widely used in various fields, such as electronic engineering, physics, finance, control theory and so on.However, most of the stochastic differential equations are nonlinear, which make it hard to obtain the analytical solution. Therefore, the numerical method has long been an powerful tool for us to study the solution of SDEs. Constructing appropriate numerical methods and studying the convergence of numerical solutions of SDEs have become two important problems in the numerical analysis of SDE.In recent years, many researchers study the strong or weak convergence of numerical solution of SDEs when the drift coefficient and diffusion coefficient of SDE meet the global lipschitz conditions. However, a lot of important SDEs can not meet the global lipschitz conditions. In this paper, we focus on studying the convergence and convergence order of numerical solution of SDEs with ? method when the drift coefficient of SDEs do not satisfy the global lipschitz condition.This paper is divided into two parts to study the convergence and convergence order of numerical solutions of SDEs. In the first part, we focus on studying the convergence and the optimal convergence order of the numerical solution of the SDEs for different avalues of ? under non-global condition. In the second part, we focus on studying the convergence and the optimal convergence order of the numerical solution of the stochastic delay differential e?uation under non-global condition, which can be obtained by introducing time-continuous interpolation process and taking advantage of the good properties of ? method. The proofs of this paper are mainly dependent on the two good properties of the ? method:(a) The numerical schemes of method can be converted into the equation without ? through the deformation and proper substitution.(b) Themoments of numerical solutions obtained directly by the ? method or its variant are bounded.