The Stability And Convergence Of Numerical Method For Solving Stochastic Differential Equations

As an important mathematical model,the stochastic differential equations has been used in uncertain differential systems widely,such as fiance,cybernetics,ecology and neural network.However,the analytical solution of equations is not as easy as that of the ordinary differential equation.Generally,the solutions of equations are approximated by numerical solutions.So the validity of numerical method is the key to solve the problem,and the validity is measured by the stability and convergence of numerical method generally.This paper studies the stability and convergence of numerical methods for solving two types of stochastic differential equations mainly.1、For It(?) stochastic differential equations,The numerical method and its stability was studied mainly.First,θ-Heun method was obtained by improving the Heun method.Then,according to the definition of the mean square stability and exponential stability of numerical method,the mean square stability condition and the exponential stability condition of the θ-Heun method and its stability regions were gain.What’s more,the range of the θ that makes the stability of θ-Heun method was given,and the numerical validation was performed.Finally,the mean square stability and the asymptotic stability of these two methods were compared by numerical examples.2、The convergence of the θ-Heun method for solving one dimension scalar autonomousIt(?) stochastic differential equations was studied.According to the definition of the convergence of numerical method,several order of convergence of the θ-Heun method were proved,and the numerical validation was performed.3、Several numerical methods for solving Stratonovich stochastic differential equations was derived.By using the transformation rules of It(?) stochastic differential equations and Stratonovich stochastic differential equations,Stratonovich stochastic differential equations were converted into the corresponding It(?) stochastic differential equations,so that we obtained the Heun method and θ-Heun method for solving Stratonovich stochastic differential equations.Then,the drift step-by-step method and the diffusion step-by-step method of these two methods were obtained by using the step-by-step technique.So we have obtained six methods for solving Stratonovich stochastic differential equations.4、For Stratonovich stochastic differential equations,the stability of the above six methods was discussed mainly.First,according to the definition of the mean square stability and exponential stability of numerical method,stability conditions of these six methods for solving Stratonovich stochastic differential equations were verified.Then,the range of the θ that makes the stability of θ-Heun methods for solving Stratonovich stochastic differential equations was given,and the numerical validation was performed.Finally,the mean square stability and the asymptotic stability of these six methods were compared by numerical examples separately.