Asymptotic Properties of the Impulse Perturbation Process under the Poisson Approximation Conditions with the Equilibrium Point of Quality Criterion

Abstract

For a system of stochastic differential equations with Markov switchings and impulse perturbation under the Poisson approximation scheme and the conditions of the existence of a single equilibrium point of the quality criterion, the limit generators for the impulse process and the dynamic system are constructed. The complexity of the proposed evolutionary model lies in its three properties. First, the system is under conditions of an external random effect, which is modeled using the Markov switching process. Processes with independent increments, which also depend on the Markov switching process, have certain characteristics between the moments of its restoration, and at the moments of restoration, these characteristics change. Therefore, the so-called "gluing" of trajectories of processes with independent increments occurs. Second, the model contains a Poisson approximation scheme, which is a generalization of the classical averaging scheme and is determined by normalization depending on a small parameter. In the classical approximation scheme in the limit process, we do not see large jumps in the system. The maximum that we get is the shift of the deterministic trajectory. However in the Poisson approximation scheme, which was invented by Korolyuk and Limnios in the monograph of 2005, this problem is eliminated, therefore, both a deterministic shift and large jumps will take place in the limit. Third, the system has a control function, which is determined using the Robins-Monroe stochastic approximation procedure. This procedure solves the problem of determination of the equilibrium point of the regression function and consists in finding the unique solution to the equation relative to control. Assuming the existence of a unique control on every interval, we solve a two-level problem. The question, how the behavior of the limit process depends on the prelimit normalization of the stochastic system in an ergodic Markov environment is studied in the article.