Almost Sure Stability of Linear Stochastic Systems with Poisson Process Coefficients.

Abstract

We consider the problem of determining the sample path stability of a linear stochastic differential equation with point process coefficients. Necessary and sufficient conditions are obtained which are similar in spirit to those derived by Khas'minskii and Pinsky for diffusion processes. The conditions are based on the deep theorems of Furstenburg on the asymptotic behavior of products of random matrices. Estimates on the probabilities of large deviations for stable processes are also given; together with a result on the stabilization of unstable systems by feedback controls.