Split-step double balanced approximation methods for stiff stochastic differential equations

ABSTRACT

In the modelling of many important problems in science and engineering we face stiff stochastic differential equations (SDEs). In this paper, a new class of split-step double balanced (SSDB) approximation methods is constructed for numerically solving systems of stiff Itô SDEs with multi-dimensional noise. In these methods, an appropriate control function has been used twice to improve the stability properties. Under global Lipschitz conditions, convergence with order one in the mean-square sense is established. Also, the mean-square stability (MS-stability) properties of the SSDB methods have been analysed for a one-dimensional linear SDE with multiplicative noise. Therefore, the MS-stability functions of SSDB methods are determined and in some special cases, their regions of MS-stability have been compared to the stability region of the original equation. Finally, simulation results confirm that the proposed methods are efficient with respect to accuracy and computational cost.

Keywords:

• Stochastic differential equation
• mean-square convergence
• split-step balanced method
• numerical stability
• stiff equations

2010 AMS Subject Classifications:

• 65C30
• 65L07
• 65L04

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Amir Haghighi  http://orcid.org/0000-0001-8928-9557

Andreas Rößler  http://orcid.org/0000-0002-2740-2697