A Structural Analysis of Asymptotic Mean-square Stability for Multi-dimensional Linear Stochastic Differential Systems

We are concerned with a linear mean-square stability analysis of numerical methods applied to systems of stochastic differential equations (SDEs) and, in particular, consider the Theta-Maruyama and the Theta-Milstein method in this context. We propose an approach, based on the vectorisation of matrices and the Kronecker product, that allows us to deal efficiently with the matrix expressions arising in this analysis and that provides the explicit structure of the stability matrices in the general case of linear systems of SDEs. For a set of simple test SDE systems, incorporating different noise structures but only a few parameters, we apply the general results and provide visual and numerical comparisons of the stability properties of the two methods.