Stability And Convergence Of Two Numerical Methods For Solving Stochastic Differential Equations

As a discipline combining deterministic phenomena and uncertain phenomena,stochastic differential equations(SDE)can more accurately describe the motion laws of many objects in nature,and are now widely used in financial economy,physics,systems biology,engineering technology and other fields.However,compared with the ordinary differential equation,the analytical solution of the stochastic differential equation is more difficult to obtain,so in most cases,numerical methods are needed to approximate the solution.Therefore,finding effective numerical methods is especially important.In this paper,two numerical methods for solving stochastic differential equations are proposed,and the stability and convergence of the two numerical methods are discussed.1?Some concepts of the convergence and stability of numerical solutions of stochastic differential equations are summarized.Several common classical numerical methods are introduced.Then the numerical examples are used to verify the strong convergence order of Heun method.Finally,the mean square stability function of the Heun method and the ?-Heun method are given,and their mean square stability domains are drawn.2?Two new numerical methods are proposed,one is the hybrid Euler method which is improved by the trapezoid Euler method,and the other is the composite Heun method which is improved by the Heun method.3?The stability and convergence of the improved hybrid Euler method are discussed.Firstly,using the definition of mean square stability,the mean square stability of the hybrid Euler method is given,and the corresponding mean square stability domain is obtained.Secondly,using the definition of numerical solution convergence,several different convergence orders of the hybrid Euler method are given respectively.Finally,the mean square stability of the hybrid Euler method is verified by numerical examples.The numerical solutions obtained by the hybrid Euler method and the trapezoid Euler method are compared with the exact solutions.4?The stability and convergence of the improved composite Heun method are discussed.Firstly,the mean square stability and exponential stability of the composite Heun method are given.Secondly,the mutual equivalence between them is proved.Several different convergence orders of the composite Heun method are studied again.Finally,the mean square stability of the composite Heun method is verified by numerical examples.The numerical solutions obtained by the composite Heun method and the Heun method are compared with the exact solutions.