Almost sure and mean square exponential stability of numerical solutions for neutral stochastic functional differential equations

Abstract

In this paper, we investigate the almost sure and mean square exponential stability of the Euler method and the backward Euler method for neutral stochastic functional differential equations (NSFDEs). Moreover, the almost sure and pth moment exponential stability of exact solutions for NSFDEs are considered. It is shown that the Euler method and the backward Euler method can reproduce the property of almost sure and mean square exponential stability of exact solutions to NSFDEs under suitable conditions. Numerical examples are demonstrated to illustrate the effectiveness of our theoretical results.

Keywords:

• neutral stochastic functional differential equations
• Euler method
• backward Euler method
• almost sure exponential stability
• pth moment exponential stability
• mean square exponential stability

2000 AMS Subject Classifications::

• 60H35
• 65C30
• 65L20

Acknowledgements

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