Delay-dependent exponential stability of the backward Euler method for nonlinear stochastic delay differential equations

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.

Keywords:

• stochastic delay differential equation
• backward Euler method
• numerical solution
• mean square stability
• exponential stability
• delay-dependent stability

2010 AMS Subject Classification:

• 65L20
• 60H35

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).

Additional information

Notes on contributors

Xiaomei Qu

Present address: School of Mathematics, Statistics, Hubei Normal University, Huangshi 435002, China.