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

Abstract

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.