Convergence And Stability Analysis Of Partially Truncated Euler-Maruyama Numerical Solutions For Neutral Stochastic Pantograph Differential Equations

Neutral pantograph stochastic differential equations are a class of stochastic differential equations with unbounded delay and stochastic systems associated with the derivatives of the delay term.In the research field,convergence and stability are two important characteristics of the solutions of stochastic differential equations.However,in practice,the exact solution of the equation is difficult to obtain,so numerical solutions are often constructed to analyze its statistical properties.When the coefficients satisfy the linear growth condition,the classical Euler-Maruyama(EM)type algorithm has received much attention from scholars.When the equation has nonlinear coefficients,the truncated EM method is commonly used to construct the numerical solution of the equation,which makes the numerical solution of the equation strongly convergent under the Khasminskii condition.However,when the coefficients of the equation have both linear and nonlinear growth parts,truncation of the linear growth part of the coefficients may destroy the stability of the numerical solution.The proposed partially truncated EM method solves this problem by truncating only the nonlinear growth part of the coefficients,so that the numerical solution maintains ^pL strong convergence and mean square exponential stability.For the neutral pantograph stochastic differential equations,there are fewer conclusions related to its explicit numerical method.In this paper,we will use the partially truncated EM numerical method to construct the numerical solution of the highly nonlinear neutral pantograph stochastic differential equations and make a comprehensive study of its boundedness,convergence,convergence rate,and stability.This paper will be divided into the following parts:firstly,the conditions satisfied by the coefficients of the equation are given,where the linear growth part satisfies the global Lipschitz condition and the nonlinear growth part satisfies the local Lipschitz condition and the Khasminskii condition,and the existence uniqueness and L^p-bounded of the exact solution are obtained;based on this,the partially truncated EM numerical solution of the equation is further constructed.The truncation idea is used to truncate only the nonlinear growth part of the equation coefficients,and the boundedness and strong convergence of the numerical solution are proved by using It(?) formula,several inequalities,and stopping time technique.Next,to prove the convergence rate of the numerical solution,a polynomial growth condition is added to the nonlinear growth part of the coefficients to obtain the convergence rate at a fixed point,and then the condition is enhanced for the diffusion term coefficients to prove that the convergence order of the numerical solution can reach p/2 in the L^p-sense on a finite interval.Subsequently,different assumptions on the coefficients of the original equation are re-given,and the iterative technique and the semi-martingale convergence theorem are used to ensure the mean-square and almost sure polynomial stability of the numerical solutions of this class of highly nonlinear neutral stochastic pantograph differential equations;on this basis,the mean-square and almost sure exponential stability of the numerical solutions are similarly proved by adding exponential terms to the coefficient conditions.Also several numerical simulations are given in this paper to verify the correctness of the conclusions.