Five-stage Milstein methods for SDEs

Abstract

In this paper, we consider the problem of computing numerical solutions for Itô stochastic differential equations (SDEs). The five-stage Milstein (FSM) methods are constructed for solving SDEs driven by an m-dimensional Wiener process. The FSM methods are fully explicit methods. It is proved that the FSM methods are convergent with strong order 1 for SDEs driven by an m-dimensional Wiener process. The analysis of stability (with multidimensional Wiener process) shows that the mean-square stable regions of the FSM methods are unbounded. The analysis of stability shows that the mean-square stable regions of the methods proposed in this paper are larger than the Milstein method and three-stage Milstein methods.

Keywords:

• stochastic differential equation
• explicit method
• mean convergence
• mean-square convergence
• stability

2000 AMS Subject Classifications :

• 60H10
• 65C20
• 65C30

Acknowledgements

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