Two Kinds Of Split-step Compoite θ Methods Of Solving -It(?) Stochastic Differential Equation

Stochastic differential equation (SDE) is a fundamental tool for the description of random phenomena treated in finance, physics, biology, and so on. And it has been widely used in many fields. On the other side, it is often difficult to obtain the solutions of stochastic differential equations (SDEs) explicitly, and hence there has been increasing interest in numerical analysis of SDEs. There are several features of simulation of SDEs, such as more computation and slow convergence. In recent decades, with the rapid development of computer technology and widely used in economics, much attention has been devoted to the development of numerical analysis of SDEs.Ito integral and stochastic Taylor expansion are fundamental tools of most of numerical methods for SDEs.In this paper, we give two types of split-step compositeθmethods for solving the Ito stochastic differential equations. We prove that these two schemes have the order of strong convergence 1/2 and 1 respectively under some assumption. Then we study the mean-square stability. By varying the values of compositeθmethod parameters-θandλ, we discuss the stability regions in detail and show that these methods generalize some existing methods effectively. Meanwhile, the second scheme can avoid solving a nonlinear algebraic equation so that it can be utilized easily. Further, we illustrate some stable examples with choosing the coefficients of Ito differential equations where the existing related schemes are unstable.