The improved split-step backward Euler method for stochastic differential delay equations

Abstract

A new, improved split-step backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.

Keywords:

• split-step backward Euler method
• strong convergence
• one-sided Lipschitz condition
• exponential mean-square stability
• mean-square linear stability

2000 AMS Subject Classifications :

• 60H35
• 65C20
• 65L20

Acknowledgements

This work was supported by NSF of China (No. 10871207) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and Hunan Provincial Innovation Foundation for Postgraduates (No. CX2010B118). The first author would like to express his gratitude to Prof. P.E. Kloeden for his kind help during the author's stay in University of Frankfurt am Main. The authors are grateful to the anonymous referees for their helpful comments and useful suggestions for improvement of this paper.