Split-step Adams–Moulton Milstein methods for systems of stiff stochastic differential equations

Abstract

In this paper we discuss new split-step methods for solving systems of Itô stochastic differential equations (SDEs). The methods are based on a L-stable (strongly stable) second-order split Adams–Moulton Formula for stiff ordinary differential equations in collusion with Milstein methods for use on SDEs which are stiff in both the deterministic and stochastic components. The L-stability property is particularly useful when the drift components are stiff and contain widely varying decay constants. For SDEs wherein the diffusion is especially stiff, we consider balanced and modified balanced split-step methods which posses larger regions of mean-square stability. Strong order convergence one is established and stability regions are displayed. The methods are tested on problems with one and two noise channels. Numerical results show the effectiveness of the methods in the pathwise approximation of stiff SDEs when compared to some existing split-step methods.

Keywords:

• Stochastic differential equations
• split-step method
• predictor–corrector method
• mean-square stability
• stiff equations

2010 AMS Subject Classifications::

• 60H10
• 65L06
• 65L20

Acknowledgements

The authors thank the anonymous referees whose comments improved the quality of our paper.