Convergence and stability of the balanced methods for stochastic differential equations with jumps

Abstract

This paper deals with the balanced methods which are implicit methods for stochastic differential equations with Poisson-driven jumps. It is shown that the balanced methods give a strong convergence rate of at least 1/2 and can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability analysed. Some numerical experiments are given to demonstrate the conclusions.

Keywords:

• stochastic differential equation
• jump
• balanced method
• convergence
• mean-square stability

2000 AMS Subject Classifications :

• 60H10
• 65C20
• 65L20

Acknowledgements

This research was supported with funds provided by the National Natural Science Foundation of China (No.10871207) and the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. We thank two anonymous reviewers for their very valuable comments and helpful suggestions which improved this paper significantly.