Mean Square Stability Analysis of Some Linear Stochastic Systems

Mean square stability analysis of some continuous and discrete time stochastic systems is carried out in this paper. We present a general approach to mean square stability investigation of systems with multiplicative noise and apply presented theory to discretized linear oscillators as often met in Mechanical Engineering. The analysis relies on the spectral theory of positive operators. As one of the results one obtains a simple and efficient criterion to decide the question of stability of equilibria of linear systems. Conclusions for practical usage and preference of numerical methods solving stochastic differential equations (SDEs) with white noise can be drawn too. For illustration and practical meaningfulness, we describe stability domains of stochastic θ-methods in terms of parametric restrictions