Stability Of Nonlinear Stochastic Delay Differential Equations

In nature,a lot of dynamic systems depend not only on the current state,but also on the past history.In order to better describe the objective world,much attention has been paid to the study of the stochastic delay differential equations.This paper mainly discusses a class of highly nonlinear stochastic delay differential equationsdx(t)=ax(t)+bx^?1(t)x^?1(t-?)+cx^?(t)dt+mx^?2(t)x^?2(t-?)dw(t).On the basis of the existence of globally unique solution to the equation,sufficient conditions are given to guarantee the bounded moments and stability of the solution.Due to the analytical solution of general nonlinear stochastic differential equations is difficult to obtain,this paper further studies the relevant properties of the backward Euler-Maruyama numerical solution of the equation,and presents a numerical simulation verify its rationality.This paper consists of the following parts:In the first chapter,we introduce the research background and significance of this article,as well as the basic concepts and theoretical knowledge involved,and summarizes the main research contents of the article.The second chapter,we discuss the conditions for the existence and uniqueness of global solution for highly nonlinear stochastic delay differential equations.Then we take Young's inequality,BDG inequality and non-negative semimartingale convergence theorem as primary tools,giving the sufficient conditions of the solution's p-moment boundedness,uniformly boundedness,asymptotic stability,and exponential stability.In the third chapter,under the condition that the drift term of the equation satisfies the One-sided Lip-schitz condition,imposing differernt additional conditions,we show that the backward Euler-Maruyama approximate holds the moment boundedness,uniformly boundedness,the almost surely exponential sta-bility and mean square exponential stability.In Chapter Four,taking three examples of highly nonlinear equations and numerical simulations,we verify the rationality of the conclusions in this paper.Furthermore,reasonably analyze the possibility of generalizing the conclusions to a more general form.