### Heriot-Watt Mathematics Report Series

HWM09-13, 10 Dec 2009

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

### E Buckwar and T Sickenberger

#### 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. As a simple extension of the standard geometric Brownian motion as a test equation for the linear stability analysis, we 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.
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**Full text:** http://arxiv.org/abs/0912.1968

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