Strong Convergence and Asymptotic Exponential Stability of Modified Truncated EM Method for Neutral Stochastic Differential Equations with Time-dependent Delay

In this paper, we consider asymptotic exponential stability of the exact solution and the corresponding modified truncated EM (MTEM) method for neutral stochastic differential equations (NSDEs) with time-dependent delay. We obtain sufficient conditions under which the MTEM method replicates the exponential stability of the exact solution no matter time-dependent delay δ(t) is bounded or not. To make sure that the stability conclusions are meaningful, we will obtain the strong convergence of the MTEM method at first. Two examples are presented to illustrate our conclusions.