Stochastic evolutionary system with Markov-modulated Poisson perturbations in the averaging schema

  • S. A. Semenyuk Lviv Polytechnic National University Lviv, Ukraine
  • Ya. M. Chabanyuk Ivan Franko National University of Lviv Lviv, Ukraine Lublin University of Technology Lublin, Poland

Анотація

This paper discusses the asymptotic behavior of the stochastic evolutionary system under
the Markov-modulated Poisson perturbations in an averaging schema. Such a perturbation
process combines the Poisson process with the Markov process that modulates the intensity
of jumps. This allows us to model systems with transitions between different modes or
rare but significant jumps. Initially, the asymptotic properties of the Markov-modulated Poisson
perturbation are investigated. For this purpose, we build the generator for the limit
process solving the singular perturbation problem for the original process. Then we introduce
a compensated Poisson process with a zero mean value, and it is used to center the jumps.
The stochastic evolutionary system perturbed by the compensated Poisson process with an
additional jump size function is described. We build the generator for an evolution process and
investigate its asymptotic properties. Solving the singular perturbation problem we obtain the
form of the limit process and its generator. This allows us to formulate and prove the theorem
about weak convergence of the evolution process to the averaged one. The limit process for
the stochastic evolutionary system at increasing time intervals is determined by the solution
of a deterministic differential equation. The obtained result makes it possible to study the
rate of convergence of the perturbed process to the limit one, as well as to consider stochastic
approximation and optimization procedures for problems in which the system is described by
an evolutionary equation with the Markov-modulated Poisson perturbation.

Біографії авторів

S. A. Semenyuk, Lviv Polytechnic National University Lviv, Ukraine

Lviv Polytechnic National University
Lviv, Ukraine

Ya. M. Chabanyuk, Ivan Franko National University of Lviv Lviv, Ukraine Lublin University of Technology Lublin, Poland

Ivan Franko National University of Lviv
Lviv, Ukraine,
Lublin University of Technology
Lublin, Poland

Посилання

J. Landon, S. Ozekici, R. Soyer, A markov modulated poisson model for software reliability, Eur. J. Oper. Res., 229 (2013), №2, 404–410. https://doi.org/10.1016/j.ejor.2013.03.014

S. Mews, B. Surmann, L. Hasemann, S. Elkenkamp, Markov-modulated marked Poisson processes for modeling disease dynamics based on medical claims data, Stat. Med., 42 (2023), 3804–3815. https://doi.org/10.1002/sim.9832

W. Fischer, K. Meier-Hellstern, The Markov-modulated Poisson process (MMPP) cookbook, Perf. Evaluation, 18 (1993), №2, 149–171. https://doi.org/10.1016/0166-5316(93)90035-S

V.S. Korolyuk, N. Limnius, Stochastic systems in merging phase space, World Scientific, 2005. https://doi.org/10.1142/5979

S. Ozekici, R. Soyer, Semi-Markov modulated Poisson process: probabilistic and statistical analysis, Math. Meth. Oper. Res., 64 (2006), 125–144. https://doi.org/10.1007/s00186-006-0067-3

S.A. Semenyuk, Y.M. Chabanyuk, Fluctuations of a stochastic system under an asymptotic diffusive perturbation, Cybern. Syst. Anal., 44 (2008), №5, 716–721. https://doi.org/10.1007/s10559-008-9042-8

Y.M. Chabanyuk, A.V. Nikitin, U.T. Khimka, Asymptotic properties of the impulse perturbation process under Levy approximation conditions with the point of equilibrium of the quality criterion, Mat. Stud., 52 (2019), №1, 96–104. https://doi.org/10.1007/s10559-008-9042-8

Y. Chabanyuk, A. Nikitin, U. Khimka, Asymptotic analyses for complex evolutionary systems with Markov and semi-Markov switching using approximation schemes, Mathematics and Statistics. John Wiley & Sons, 2020.

Опубліковано
2024-09-15
Як цитувати
Semenyuk, S. A., & Chabanyuk, Y. M. (2024). Stochastic evolutionary system with Markov-modulated Poisson perturbations in the averaging schema. Математичні студії, 62(1), 102-108. https://doi.org/10.30970/ms.62.1.102-108
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