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Asymptotic stability in distribution of stochastic differential equations with Markovian switching. (English) Zbl 1075.60541
Asymptotic stability in distribution for a stochastic differential equation of the form \[ dX(t)= f(X(t), r(t))\,dt + g(X(t), r(t))\,dB(t) \] is studied where \(B(t)\) is an \(m\)-dimensional Brownian motion, \(f:\mathbb R^n\times S \to \mathbb R^n\), \(g:\mathbb R^n\times S \to \mathbb R^{n\times m}\), \(S=\{1,2,\dots ,N\}\) and \(r(t)\) is a right-continuous, \(S\)-valued Markov chain. Sufficient criteria for the asymptotic stability are given in terms of Lyapunov functions and \(M\)-matrices.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
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