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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, $f:\Bbb R^n\times S \to \Bbb R^n$, $g:\Bbb R^n\times S \to \Bbb R^{n\times m}$, $S=\{1,2,\dots ,N\}$ 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.

60H10Stochastic ordinary differential equations
93E15Stochastic stability
Full Text: DOI
[1] Anderson, W. J.: Continuous-time Markov chains. (1991) · Zbl 0731.60067
[2] Basak, G. K.; Bisi, A.; Ghosh, M. K.: Stability of a random diffusion with linear drift. J. math. Anal. appl. 202, 604-622 (1996) · Zbl 0856.93102
[3] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. (1994) · Zbl 0815.15016
[4] Ikeda, N.; Watanabe, S.: Stochastic differential equations and diffusion processes. (1981) · Zbl 0495.60005
[5] Ji, Y.; Chizeck, H. J.: Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE trans. Automat. control. 35, 777-788 (1990) · Zbl 0714.93060
[6] Mao, X.: Stability of stochastic differential equations with Markovian switching. Stochastic process. Appl. 79, 45-67 (1999) · Zbl 0962.60043
[7] Mao, X.; Matasov, A.; Piunovskiy, A. B.: Stochastic differential delay equations with Markovian switching. Bernoulli 6, 73-90 (2000) · Zbl 0956.60060
[8] Mariton, M.: Jump linear systems in automatic control. (1990)
[9] Shaikhet, L.: Stability of stochastic hereditary systems with Markov switching. Theory stochastic process. 2, 180-184 (1996) · Zbl 0939.60049
[10] Skorohod, A. V.: Asymptotic methods in the theory of stochastic differential equations. (1989)