Mean square stability for discrete linear stochastic systems.

Several results concerning asymptotical mean square stability of the null solution of specific linear stochastic systems are presented and proven. It is shown that the mean square stability of the implicit Euler method, taken from the monography of Kloeden and Platen (1992) and applied to linear stochastic differential equations, is necessary for the mean square stability of the corresponding implicit Milstein method (using the same implicitness parameter). Furthermore, a sufficient condition for the mean square stability of the implicit Euler method can be varified for autonomous systems. Additionally, the principle of 'monotonous inclusion' of the sequel of mean square stability domains holds for linear systems. The paper generalizes the results due to Schurz (1993) where one-dimensional linear complex systems with respect to asymptotical p-th mean stability have been investigated. Finally, a simple example confirms these assertions. The results can also be used to deduce recommendations for the practical implementation of numerical methods solving nonlinear systems by orienting on their linearization