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Asymptotic stability condition for stochastic Markovian systems of differential equations. (English) Zbl 1224.34182
Summary: Asymptotic stability of the zero solution of the system \[ {\text d} X(t) = A(\xi (t))X(t)\,{\text d} t + H(\xi (t))X(t)\,{\text d} w(t) \] is considered, where \(\xi (t)\) is a finite-valued Markov process and \(w(t)\) is a standard Wiener process. 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 condition for necessary asymptotic stability.
34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34D20 Stability of solutions to ordinary differential equations
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