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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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##### References:
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