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.