Random evolutions in Poisson approximation scheme

Самойленко, І. В., Самойленко, Т. А., Довгай, Б. В. (2021). Random evolutions in Poisson approximation scheme. Bulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics(2), 69–77. https://doi.org/10.17721/1812-5409.2021/2.10

The operator approach in the study of random evolutions allows us to obtain the following results in the Poisson approximation scheme: functional limit theorems at increasing time intervals and the solution of the large deviations problem. We will focus on the last task.
To solve the problem, asymptotic analysis of nonlinear generators of random evolutions with Markov switching should be conducted in the series scheme. The specifics of asymptotic analysis is conditioned by the fact that the jump values of the stochastic system are split into two parts: a small jump taking values with probabilities close to one and a big jump taken values with probabilities tending to zero together with the series parameter $\varepsilon\to 0$. So, in the Poisson approximation principle the probabilities (or intensities) of jumps are normalized by the series parameter $\varepsilon >0$.
Having the limit nonlinear generator, we are able to construct the rate functional to solve the large deviations problem.
Pages of the article in the issue: 69 - 77
Language of the article: Ukrainian