The Equivalence Between The Stability Of The Numerical Solution And The Stability Of The Analytical Solution Of The Stochastic Delay Differential Equation

In this paper,a general theorem on the equivalence of pth moment stability between stochastic differential delay equations(SDDEs)and their numerical methods is proved.We use the numerical methods to illustrate that the theorem indeed covers a large ranges of SDDEs.When the drift coefficients and diffusion coefficients of stochastic differential delay equa-tions satisfy the global Lipschitz condition,the equivalence of pth moment stability between stochastic differential delay equations(SDDEs)and Theta numerical method is proved.When the drift coefficients and diffusion coefficients of stochastic differential delay equa-tions satisfy the local Lipschitz condition and the Khasminskii-type condition,the equivalence of pth moment stability between stochastic differential delay equations(SDDEs)and the truncated Euler-Maruyama(EM)numerical method is proved.In the process of proving strong conver-gence,we relax the requirement of step size and prove that its convergence order is close to 1/2.Several numerical experiments are presented to demonstrate the theoretical results.