The improved stability analysis of the backward Euler method for neutral stochastic delay differential equations

Abstract

The aim of this paper is to improve some results obtained in our earlier paper [Z. Yu and M. Liu, Almost surely asymptotic stability of numerical solutions for neutral stochastic delay differential equations, Discrete Dyn. Nat. Soc. 2011 (2011), article id 217672, 11 p., doi:10.1155/2011/217672]. In this paper, we establish an improved theorem and show that the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to neutral stochastic delay differential equations. To obtain the desired result, some new proof techniques are adopted.

Keywords:

• neutral stochastic delay differential equations
• backward Euler method
• almost sure exponential stability
• mean square exponential stability
• numerical solution

2000 AMS Subject Classifications :

• 60H35
• 65C30
• 65L20

Acknowledgements

The authors thank the referees for their helpful comments and suggestions.