A Partially Truncated Euler-Maruyama Method For Solving Nonlinear Stochastic Differential Equations With Delays

Most of the stochastic delay differential equations are difficult to write analytical expressions,so the development of applicable numerical methods has both theoretical and practical value.Until now,the numerical solution of the nonlinear stochastic delay differential equation under global Lipschitz condition or local Lipschitz condition plus linear growth condition has been fully studied.However,for most stochastic delay differential equations,the linear growth conditions are still more stringent.Using Khasminskii condition to replace the linear growth condition can still guarantee the existence and uniqueness of the solution of a delay differential equation.It has been found that the numerical solutions of the Tamed method for a fixed-delay nonlinear stochastic differential equation with local Lipschitz condition and Khasminskii condition are mean-square convergence.The time-varying stochastic problems are more common in practical problems and have more research value.In this paper,we mainly use the partial truncated Euler-Maruyama method to study the mean-square convergence of nonlinear stochastic differential equations with timedelay under Khasminskii condition,and estimate the mean square convergence order.Partially truncated Euler-Maruyama method can reproduce the almost everywhere exponential stability of the original problem,and gives the equation that the speed of the numerical solution falls.Finally,numerical examples are used to verify the correctness of our theoretical results.