Numerical Analysis Of Two Kinds Of Stochastic Delay Differential Equations In Statistical Dynamics

Stochastic delayed differential equations have been a hot topic in recent years and can be widely applied to many fields such as natural sciences,finance,and engineering technology.The study of numerical solutions has important scientific significance and practical engineering value.This paper focuses on the numerical study of two types of stochastic delay differential equations.First,the semi-implicit Milstein method is applied to a class of stochastic integral delay differential equations with a Markov chain.Theoretical analysis proves the stability of the semi-implicit Milstein method for solving the equations.The sufficient conditions for the mean-square stability and generally mean-square stability of the equation are given for different ranges of parameters ?.Numerical experiments verify the feasibility of the semi-implicit Milstein method,and the stability of the semi-implicit Milstein method is compared with the existing Milstein method,showing the superiority of the semi-implicit Milstein method.Secondly,the Heun scheme is constructed for a class of nonlinear neutral stochastic delay differential equations,and theoretical analysis proves the stability of the stochastic Heun method.Numerical experiments confirm that the numerical solution of the stochastic Heun method of the equation is mean-square stable when the step size satisfies the step size limit condition.Numerical experiments validate the theoretical analysis.Finally,a composite Milstein scheme is constructed for nonlinear neutral stochastic delay differential equations.The composite Milstein method is constructed by combining explicit Milstein method and implicit Milstein method.Theoretical analysis proves the mean-square stability of the composite Milstein method of the equation,and gives the stability conditions of the composite Milstein method of the equation with step limit.Numerical experiments confirm the effectiveness of the composite Milstein method,and the numerical stability of the composite Milstein method is better than that of the Heun method.