Geodesic
Asymptotic stability condition for stochastic Markovian systems of differential equations
Mathematica Bohemica, Tome 135 (2010) no. 4, pp. 443-448

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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.
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.
DOI : 10.21136/MB.2010.140834
Classification : 34F05, 47B80, 60H25, 93E03
Keywords: jump parameter system; Markov process; asymptotic stability 
Shmerling, Efraim. Asymptotic stability condition for stochastic Markovian systems of differential equations. Mathematica Bohemica, Tome 135 (2010) no. 4, pp. 443-448. doi: 10.21136/MB.2010.140834
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