×

zbMATH — the first resource for mathematics

Stability of stochastic differential equations with Markovian switching. (English) Zbl 0962.60043
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 \(f:\mathbb R^n\times \mathbb R_+\times S\to \mathbb R^n\), \(g:\mathbb R^n\times \mathbb R_+\times S\to \mathbb R^{n\times m}\) 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 \[ dx(t)=f(x(t),t,i) dt + g(x(t),t,i) dw(t),\quad 1\leq i\leq N, \] 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.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnold, L., 1972. Stochastic Differential Equations: Theory and Applications. Wiley, New York.
[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., 1994. Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, PA. · Zbl 0815.15016
[4] Friedman, A., 1976. Stochastic Differential Equations and Their Applications. vol. 2. Academic Press, New York. · Zbl 0323.60057
[5] Has’minskii, R.Z., 1981. Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen.
[6] 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
[7] Kolmanovskii, V.B., Myshkis, A., 1992. Applied Theory of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht. · Zbl 0917.34001
[8] Ladde, G.S., Lakshmikantham, V., 1980. Random Differential Inequalities, Academic Press, New York. · Zbl 0466.60002
[9] Mao, X., 1991. Stability of Stochastic Differential Equations with Respect to Semimartingales. Longman Scientific and Technical, New York.
[10] Mao, X., 1994. Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York. · Zbl 0806.60044
[11] Mao, X., 1997. Stochastic Differential Equations and Applications. Horwood, New York. · Zbl 0892.60057
[12] Mariton, M., 1990. Jump Linear Systems in Automatic Control. Marcel Dekker, New York.
[13] Minkowski, H., 1907. Diophantische Approximationen. Teubner, Leipzig. · JFM 38.0220.15
[14] Mohammed, S.-E.A., 1986. Stochastic Functional Differential Equations. Longman Scientific and Technical, New York.
[15] Skorohod, A.V., 1989. Asymptotic Methods in the Theory of Stochastic Differential Equations, American Mathematical Society, Providence, RI.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.