Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Several results concerning asymptotical mean square stability of an equilibrium point (here the null solution) of specific linear stochastic systems given at discrete time-points are presented and proven. It is shown that the mean square stability of the implicit Euler method, taken from the monograph of Kloeden and Platen (1992) and applied to linear stochastic differential equations, is necessary for the mean square stability of the corresponding implicit Mil 'shtein method (using the same implicitness parameter). Furthermore, a sufficient condition for the mean square stability of the implicit Euler method can be verified for autonomous systems, while the principle of 'monotonic nesting' of the mean square stability domains holds for linear systems. The Euler method taking any integration step size with drift-implicitness 0.5 is able to indicate mean square stability of any equilibrium point of the continuous time system. As a practicable alternative for controlling the temporal mean square evolution, the class of Balanced methods with deterministic, positive scalar correction provides the most mean square stable numerical solution known under 'low smoothness conditions' so far. The paper summarizes and continues the stability examinations of Schurz (1993). The results can also be used to deduce recommendations for the practical implementation of numerical methods solving nonlinear systems in terms of their linearization. Finally, effects of the presented mean square calculus are shown by the Kubo oscillator perturbed by white noise and a simplified system of noisy Brusselator equations