Stability Of Numerical Methods For Stochastic Differential Equations With Piecewise Continuous Arguments

Stochastic differential equations with piecewise continuous arguments is an important class of stochastic delay differential equations.It has been widely used in economics,control science,physics and many other disciplines.The stability of exact and numerical solutions plays an important role in the study of differential equations.However,since only a few stochastic differential equations can be solved accurately,the study on the stability of numerical methods and the stability relationship between numerical solutions and exact solutions becomes particularly important.In this paper,the exact and numerical solutions of stochastic differential equations with piecewise continuous arguments are unified into a stochastic impulsive differential equation with piecewise continuous arguments.Accordingly,the problem of the numerical solutions reproduce the stability of the exact solutions is transformed into the stability of the corresponding stochastic impulsive equations with piecewise continuous arguments.This is totally different from the method used in the past.In this paper,chapter one introduces the development and research status of the stability of solutions for stochastic differential equations,stochastic delay differential equations and stochastic differential equations with piecewise continuous arguments.The second chapter introduces some basic concepts.The third chapter introduces the existence and uniqueness of the solutions and different numerical methods when the equations' coefficients satisfy different conditions,and then constructs the corresponding stochastic impulsive differential equations with piecewise continuous arguments.Chapter 4 focuses on the p-th moment exponential stability of solution for stochastic impulsive differential equations with piecewise continuous arguments first.Then we obtained the conditions that numerical solutions reproduce the stability of exact solutions for stochastic differential equations with piecewise continuous arguments by using the conclusion above.