Compensated stochastic theta methods for stochastic differential delay equations with jumps

Abstract

In this article, a class of compensated stochastic theta methods (CSTM) are constructed for stochastic differential delay equations with Poisson jumps. Compared with the usual stochastic theta methods (STM), the compensated methods proposed here have the same strong convergence rate , but promise much better mean-square stability properties. More precisely, the CSTM with ≤θ≤1 is mean-square P-stable. This result gives a natural extension of deterministic P-stability. In contrast, the STM with 0≤θ≤1 all fail to possess this excellent mean-square stability. Finally, some numerical experiments are reported to illustrate the theoretical results.

Keywords:

• stochastic differential delay equations
• Poisson jumps
• stochastic theta methods
• compensated stochastic theta methods
• mean-square stability
• P-stability

2000 AMS Subject Classifications:

• 65C20
• 65L20
• 60H35

Acknowledgements

The authors thank the referees and the editors for their valuable detailed comments and helpful suggestions. This study was supported by NSF of China (No. 11171352).