Asymptotic Stability Of Stochastic Differential Equations With General Decay Rate

This paper deals with the stability of the exact and the numerical solutions of the stochastic differential equation.The study of exact solutions includes the asymptotic stability with general decay rate of a nonlinear time-delay system,the stability with the general decay rate of stochastic differential equations,and the stability of neutral stochastic differential delay equations with Markovian switching.And the research of numerical solutions includes the formulation of the modified truncated EM method and its convergence theorem,its exponential stability and polynomial stability.These are the important contents of the qualitative research of stochastic differential equations,and they have extremely important significance in theoretical research and technical application.This paper first shows the significance,history,and status of the study of the stability of stochastic differential equations,especially focus on the current status of stability research.Then the article clarifies the research content.And then we briefly introduce the Innovation point.Finally,the basic knowledge of stochastic differential equations and stability,and some commonly theorems and inequalities are introduced.For the exact solution of stochastic differential equations,the stochastic stabilization and the stability of stochastic differential equations are studied.Stochastic stabilization,that is,through the construction of a suitable stochastic time-delay system,the existence and uniqueness of the solution and the almost surely stability of the general decay rate are investigated.For the stability of stochastic differential equations,the existence and uniqueness of the solutions for the general stochastic differential equations and the neutral stochastic delay differential equations with Markovian switching are given respectively,and the theorem of the moment and the almost surely stable is given.For the numerical solution of stochastic differential equations,we focus on the construction of a new numerical method of the modified truncated EM method and study the properties of the solutions.The convergence of the numerical method at fixed time and interval intervals are studied.The sufficient conditions for the numerical solution are given to maintain the exponential stability and polynomial stability.