Mean Square Stability Of The Balanced Implicit Method For Stochastic Differential Equations

Implicit methods are used to solve stiff stochastic differential equations. Balanced method and Balanced Milstein method are fully implicit methods whose strong convergence order are0.5and1respectively. In this paper we discuss the mean square stability of Balanced method and Balanced Milstein method, and give the conditions that they are mean stable, provide a way for the optimal parameters’s selection.This paper is composed of four chapters:The first chapter, we briefly Introduction the background of the stochastic differential equations, and make a brief introduction about this paper’s work.The second chapter introduce some basic knowledge that we will use, mainly involving probability theory and stochastic differential equations and so on.The third chapter, we discuss the mean stability of the Balanced method and weak Balanced method, give the sufficient condition for the mean square stability of Balanced method, in other words the relationship that parameters should satisfy. And we provide evidence for the optimal parameters selection.The fourth chapter, we discuss the mean stability of the Balanced Milstein method and weak Balanced Milstein method, give the sufficient condition for the mean square stability of Balanced Milstein method, in other words the relationship that parameters should satisfy. And provide evidence for the optimal parameters selection.