Study Of Stochastic Differential Equation Of Two Types Of Equilibrium Method

Stochastic differential equations (SDEs) has come to play more and more important role in many fields, such as in financial systems, biology, control systems, statistical physics and so on. Because of the complexity of stochastic systems, except some special cases, it is hard to obtain the explicit solution of SDEs. So the construction of numerical methods is of great importance.The numerical methods for solving stochastic differential equations include Euler method, Milstein method and so on. But these methods are not fit to solve stiff system. Some effective implicit methods are employed to solve stiff stochastic differential equations.Balanced method and balanced Milstein method are very effective implicit method for solving stiff stochastic differential equations. In this paper, firstly, we investigate balanced method and give a criteria for the optimal selection of parameters; secondly, we take a splitting technique to balanced method and construct split-step backward balanced method. We also prove the strong order convergence and error of the method; lastly, we improve split-step backward balanced Milstein method (Wang [1]) and investigate the mean square stability of method.This paper is composed of four parts.In Chapter1, many applications of stochastic differential equations in different fields are presented. The development of stochastic differential equations and numerical analysis for stochastic differential equations in the past decades is introduced.In Chapter2, some elementary concepts for this paper are presented, mainly involving, probability theory, stochastic integral, Brownian motion, stochastic differential equations.Chapter three investigates balanced method and gives a criteria for the optimal selection of parameters. We also construct split-step balanced method and prove the convergence of the strong split-step balanced method.In Chapter4, for the split-step backward balanced Milstein methods [1] we give a modify version. We also study the stability of our method. The numerical experiments show that the mean square stability region of the modified version is larger than that of the original one.