Asymptotic properties of the impulse perturbation
process under Levy approximation conditions with the point of equilibrium of the quality
criterion

Y. M. Chabanyuk; A. V. Nikitin; U. T. Khimka

doi:10.30970/ms.52.1.96-104

For the system of stochastic differential equations with Markov switchings and impulse disturbance in the conditions of Levy approximation in the conditions of a single point of equilibrium of the quality criterion, limiting generators of the impulse process and dynamical system are constructed. In particular, we discuss how the behavior of the boundary process depends on the pre-limiting normalization of the stochastic evolutionary system in the ergodic Markov environment.

1. Jacod J., Shiryaev A.N., Limit theorems for stochastic processes, Springer-Verlag, Berlin, 2003, 601 p.

2. Korolyuk V.S., Korolyuk V.V., Stochastic models of systems, Kluwer, Dordrecht, 1999, 185 p.

3. Korolyuk V.S., Limnios N., Stochastic systems in merging phase space, World Scientific, 2005, 330 p.

4. Korolyuk V.S., Limnios N., Samoilenko I.V., Levy and Poisson approximations of switched stochastic systems by a semimartingale approach, Comptes Rendus Mathematique, 354 (2016), 723–728.

5. Papanicolaou G., Stroock D., Varadhan S.R.S., Martingale approach to some limit theorems, Duke turbulence conference (Durham, NC, April 23-25, 1976), Duke University Mathematics Series III, New York: Duke University, 1977, 120 p.

6. Samoilenko A.M., Stanzhytskyi O.M., Qualitative and asymptotic analysis of differential equations with random perturbations, World Scientific, Singapore, 2011, 323 p.

7. Samoilenko I.V., Chabanyuk Y.M., Nikitin A.V., Khimka U.T., Samoilenko A.M., Differential equations with small stochastic additions under Poisson approximation conditions, Cybernetics and System analysis, 53 (2017), №3, 410–416.

8. Samoilenko I.V., Nikitin A.V., Differential equations with small stochastic terms under the levy approximating conditions, Ukrainian Mathematical Journal, 69 (2018), №9, 1445–1454.

9. Nikitin A.V., Khimka U.T., Asymptotics of normalized control with Markov switchings, Ukrainian Mathematical Journal, 68 (2017), №8, 1252–1262.

10. Nevelson M.B., Khasminskii R.Z., Stochastic approximation and recurrent estimation, M.: Nauka, 1972.

	Asymptotic properties of the impulse perturbation process under Levy approximation conditions with the point of equilibrium of the quality criterion
Author	Y. M. Chabanyuk¹, A. V. Nikitin², U. T. Khimka³ yaroslav.chab@gmail.com¹, nikitin2505@gmail.com², ulyana.himka@gmail.com³ 1) Politechnika Lubelska, Lublin, Poland; 2) Taras Shevchenko National University of Kyiv, Kyiv, Ukraine; 3) Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	For the system of stochastic differential equations with Markov switchings and impulse disturbance in the conditions of Levy approximation in the conditions of a single point of equilibrium of the quality criterion, limiting generators of the impulse process and dynamical system are constructed. In particular, we discuss how the behavior of the boundary process depends on the pre-limiting normalization of the stochastic evolutionary system in the ergodic Markov environment.
Keywords	random evolution; Levy’s approximation conditions; point of equilibrium of the quality criterion
DOI	doi:10.30970/ms.52.1.96-104
Reference	1. Jacod J., Shiryaev A.N., Limit theorems for stochastic processes, Springer-Verlag, Berlin, 2003, 601 p. 2. Korolyuk V.S., Korolyuk V.V., Stochastic models of systems, Kluwer, Dordrecht, 1999, 185 p. 3. Korolyuk V.S., Limnios N., Stochastic systems in merging phase space, World Scientific, 2005, 330 p. 4. Korolyuk V.S., Limnios N., Samoilenko I.V., Levy and Poisson approximations of switched stochastic systems by a semimartingale approach, Comptes Rendus Mathematique, 354 (2016), 723–728. 5. Papanicolaou G., Stroock D., Varadhan S.R.S., Martingale approach to some limit theorems, Duke turbulence conference (Durham, NC, April 23-25, 1976), Duke University Mathematics Series III, New York: Duke University, 1977, 120 p. 6. Samoilenko A.M., Stanzhytskyi O.M., Qualitative and asymptotic analysis of differential equations with random perturbations, World Scientific, Singapore, 2011, 323 p. 7. Samoilenko I.V., Chabanyuk Y.M., Nikitin A.V., Khimka U.T., Samoilenko A.M., Differential equations with small stochastic additions under Poisson approximation conditions, Cybernetics and System analysis, 53 (2017), №3, 410–416. 8. Samoilenko I.V., Nikitin A.V., Differential equations with small stochastic terms under the levy approximating conditions, Ukrainian Mathematical Journal, 69 (2018), №9, 1445–1454. 9. Nikitin A.V., Khimka U.T., Asymptotics of normalized control with Markov switchings, Ukrainian Mathematical Journal, 68 (2017), №8, 1252–1262. 10. Nevelson M.B., Khasminskii R.Z., Stochastic approximation and recurrent estimation, M.: Nauka, 1972.
Pages	96-104
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
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Asymptotic properties of the impulse perturbation process under Levy approximation conditions with the point of equilibrium of the quality criterion