On stochastic stability of Markov evolution associated with impulse Markov dynamical systems

Korolyuk, V.; Carkovs, Je.

doi:10.1090/S0094-9000-08-00714-X

On stochastic stability of Markov evolution associated with impulse Markov dynamical systems

Authors: V. Korolyuk and Je. Carkovs
Journal: Theor. Probability and Math. Statist. 75 (2007), 65-69
MSC (2000): Primary 37H10, 34D20
DOI: https://doi.org/10.1090/S0094-9000-08-00714-X
Published electronically: January 24, 2008
MathSciNet review: 2321181
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Abstract: This paper deals with the family of Cauchy matrices of a linear differential equation dependent on a step Markov process and an impulse type dynamical system rapidly switched by the above process. Applying the stochastic and deterministic averaging procedures according to the invariant measures of the Markov process one achieves a simpler linear differential equation dependent on simpler dynamical systems such as an ordinary differential equation, a differential equation with the right hand side switched by a merger Markov process or a stochastic Itô differential equation. It is proved that under some hypotheses one may successfully apply these resulting evolution families not only to analyzing the initial family on an arbitrary finite time interval but also to describing a time asymptotic of this family.

References

B. V. Anisimov, Random Processes with Discrete Component. Limit Theorems, Kiev Univ., Kiev, 1988. (Russian)

E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der Mathematischen Wissenschaften, Band 121, vol. 122, Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR 0193671

Lambros Katafygiotis and Yevgeny Tsarkov, Mean square stability of linear dynamical systems with small Markov perturbations. I. Bounded coefficients, Random Oper. Stochastic Equations 4 (1996), no. 2, 133–154. MR 1399076, DOI https://doi.org/10.1515/rose.1996.4.2.133

Lambros Katafygiotis and Yevgeny Tsarkov, Mean square stability of linear dynamical systems with small Markov perturbations. II. Diffusion coefficients, Random Oper. Stochastic Equations 4 (1996), no. 3, 251–272. MR 1414878, DOI https://doi.org/10.1515/rose.1996.4.3.251

V. S. Koroljuk and A. F. Turbin, Limit theorems for Markov random evolutions in the scheme of asymptotic state lumping, Probability theory and mathematical statistics (Tbilisi, 1982) Lecture Notes in Math., vol. 1021, Springer, Berlin, 1983, pp. 327–332. MR 735999, DOI https://doi.org/10.1007/BFb0072929

V. Korolyuk and A. Swishchuk, Semi-Markov random evolutions, Mathematics and its Applications, vol. 308, Kluwer Academic Publishers, Dordrecht, 1995. Translated from the 1992 Russian original by V. Zayats and revised by the authors. MR 1472977

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280

A. V. Skorokhod, Asymptotic methods in the theory of stochastic differential equations, Translations of Mathematical Monographs, vol. 78, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 1020057

Ye. Tsarkov, Asymptotic methods for stability analysis of Markov impulse dynamical systems, Advances of Stability Theory of the End of XXth Century. Stability and Control: Theory, Methods and Applications, Gordon and Breach Science Publishers, London, 2000, pp. 251–264.

V. S. Korolyuk, Stability of an autonomous dynamical system with fast Markov switchings, Ukrain. Mat. Zh. 43 (1991), no. 9, 1176–1181 (Russian, with Ukrainian summary); English transl., Ukrainian Math. J. 43 (1991), no. 9, 1101–1105 (1992). MR 1149579, DOI https://doi.org/10.1007/BF01089209