Abstract
Recently, several scholars discussed the question of under what conditions numerical solutions can reproduce exponential stability of exact solutions to stochastic delay differential equations, and some delay-independent stability criteria were obtained. This paper is concerned with delay-dependent stability of numerical solutions. Under a delay-dependent condition for the stability of the exact solution, it is proved that the backward Euler method is mean-square exponentially stable for all positive stepsizes. Numerical experiments are given to confirm the theoretical results.
Acknowledgements
The authors would like to thank two anonymous referees for their careful reading of the manuscript and valuable comments. This work was supported by the National Natural Science Foundation of China (Grant nos. 10971077 and 91130003).