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Asymptotic stability condition for stochastic Markovian systems of differential equations

Efraim Shmerling — 2010

Mathematica Bohemica

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by $d X (t) = A (ξ (t)) X (t) d t + H (ξ (t)) X (t) d w (t)$ , where $ξ (t)$ 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.

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