A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

TL;DR

This paper compares the mean-square stability of Theta-Maruyama and Theta-Milstein methods for stochastic differential equations, revealing that Milstein's stability is more restrictive and depends on noise terms, with implications for method parameter tuning.

Contribution

It introduces an extended test equation for stability analysis and explores how implicitness and noise influence the stability regions of these numerical methods.

Findings

Milstein method has more restrictive stability conditions than Maruyama.

Stability regions of Milstein depend explicitly on noise terms.

Partially implicit diffusion terms can control stability properties.

Abstract

In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.