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Zbl 0962.60043
Mao, Xuerong
Stability of stochastic differential equations with Markovian switching.
(English)
[J] Stochastic Processes Appl. 79, No.1, 45-67 (1999). ISSN 0304-4149

Stochastic differential equations of the form $$dx(t)=f(x(t),t,r(t)) dt + g(x(t),t,r(t)) dw(t)$$ are considered where $w(t)$ is an $m$-dimensional Brownian motion, $r(t)$ is a right-continuous Markov chain with values in $S:=\{1,2,\dots{},N\}$, and $f:\Bbb R^n\times \Bbb R_+\times S\to \Bbb R^n$, $g:\Bbb R^n\times \Bbb R_+\times S\to \Bbb R^{n\times m}$ satisfy suitable Itô-type conditions for the existence and uniqueness of the solution. Note that this equation can be regarded as a result of $N$ equations $$dx(t)=f(x(t),t,i) dt + g(x(t),t,i) dw(t),\quad 1\le i\le N,$$ switching from one to other according to the movement of the Markov chain. Criteria for exponential stability of the moments and for a.s. exponential stability are given, special attention being devoted to the linear equations and to nonlinear deterministic jump equations.
[B.Maslowski (Praha)]
MSC 2000:
*60H10 Stochastic ordinary differential equations

Keywords: Lyapunov exponent; Markovian switching; Brownian motion

Cited in: Zbl 1077.60512

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