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

In this article we compare the mean-square stability properties of the θ-Maruyama and θ-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 θ-Milstein method and thus, for some choices of θ, the conditions on the step-size, are much more restrictive than those for the θ-Maruyama method; (ii) the precise stability region of the θ-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partial 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 σ. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.