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

Анотація

This paper discusses the asymptotic behavior of the stochastic evolutionary system underthe Markov-modulated Poisson perturbations in an averaging schema. Such a perturbationprocess combines the Poisson process with the Markov process that modulates the intensityof jumps. This allows us to model systems with transitions between different modes orrare but significant jumps. Initially, the asymptotic properties of the Markov-modulated Poissonperturbation are investigated. For this purpose, we build the generator for the limitprocess solving the singular perturbation problem for the original process. Then we introducea 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 anadditional jump size function is described. We build the generator for an evolution process andinvestigate its asymptotic properties. Solving the singular perturbation problem we obtain theform of the limit process and its generator. This allows us to formulate and prove the theoremabout weak convergence of the evolution process to the averaged one. The limit process forthe stochastic evolutionary system at increasing time intervals is determined by the solutionof a deterministic differential equation. The obtained result makes it possible to study therate of convergence of the perturbed process to the limit one, as well as to consider stochasticapproximation and optimization procedures for problems in which the system is described byan evolutionary equation with the Markov-modulated Poisson perturbation.

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