Strong Convergence And Stability Of Stochastic Differential Delay Equations

This paper mainly studies the strong convergence and stability of the numerical solution of neutral stochastic differential delay equations.The convergence study includes the strong convergence of the modified truncated EM method at fixed time T and interval[O,T]and the convergence of the numerical solution of the continuous stochastic 0 method.The stability study includes the almost sure asymptoic exponential stability of the backward Euler method and the forward-backward Euler method.Stability and convergence are important evaluation topics for the numerical solution of stochastic differential equations.They also have important significance in the practical application of stochastic differential equations.This paper will first introduce the research background of the convergence and stability of stochastic differential equations and the existing research results,and then briefly explain the basic definitions,theorems and formulas used in this paper,and then the convergence and stability respectively.This paper introduces the properties of numerical solutions of stochastic differential equations with different numerical methods under different numerical methods,and combines the existing research contents to illustrate the innovation of this paper.In the part of convergence,this paper first considers the strong convergence of the modified truncated Euler-Maruyama method for stochastic differential equations with neutral delays.The strong convergence rate of the numerical solution of a given numerical solution at a fixed time T is obtained.In addition,under the condition of polynomial growth of diffusion terms,there is no weak monotonic condition(usually the standard assumption for obtaining convergence rate),and the convergence rate on time interval[0,T]is also obtained,and two examples are proposed to explain our conclusion.Next,the convergence of the numerical solution of the continuous stochastic O method is considered.Compared with the existing research results of the discrete stochastic method,the global lipschitz condition is changed to the local lipschitz condition,and the continuous stochastic method can be obtained according to the probability convergence.In the part of stability study,this paper considers the almost everywhere stability of the two numerical solutions of the backward Euler method and the Euler method before and after.In the constraint condition,the drift term and the diffusion term coefficient are put into the same inequality.Reducing the limit on the growth of each coefficient,and considering the case that the lag term is time-dependent,yields stronger results under weaker conditions.Finally,the effectiveness of the result is further proved by a concrete example analysis.