Polynomial stability of exact solution and a numerical method for stochastic differential equations with time-dependent delay

We consider a stochastic differential equations with time-dependent delay in this paper. We first obtain the existence, uniqueness and polynomial stability of exact solution to this equation under suitable conditions. Then for the numerical method of the corresponding SDDE, we present a so called modified truncated Euler-Maruyama(MTEM) method and consider the almost sure and mean square polynomial stability of this numerical method. By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay δ to ensure the polynomial stability of the corresponding numerical approximation. Results suggest that the MTEM method replicates the polynomial stability of given SDDE under suitable conditions. Examples are presented to illustrate the conclusion.