Asymptotic stability condition for stochastic Markovian systems of differential equations

summary:Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by ${\rm d} X(t) = A(\xi (t))X(t) {\rm d} t + H(\xi (t))X(t) {\rm d} w(t)$, where $\xi (t)$ is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition