Yarova O. Asymptotic analysis and transitional phenomena in Markov random evolutions

The dissertation is devoted to the study of the properties of homogeneous Markov processes and random evolutions in a time scale where when The first object of the study is the jump-like processes with independent increments in the schemes of two nonlinear approximations. In the dissertation work generators of Markov processes and Markov random evolution in schemes of Poisson and Levу approximation with nonlinear normalization are considered, solutions of the problem of large deviations in schemes of nonlinear approximations are studied and the connection between nonlinear normalization functions is determined. The purpose of the dissertation is to find nonlinear normalizing functions for generators of Markov processes and Markov random evolutions. The conditions of the nonlinear Poisson and Levу approximation are determined and the asymptotic image of generators of Markov processes is investigated. In the problem of large deviations, two nonlinear normalizing functions, which normalize the time and intensity of jumps, are investigated. In addition, impulse recurrence processes in the non-linear approximation scheme of Levy are considered. A theorem has been proved for such processes, which is based on the semimartingal representation of the process. In the dissertation work the following new scientific results were obtained: - nonlinear normalizing functions were found in the representation of generators of Markov processes in the scheme of Poisson and Levу approximation; - the existence of nonlinear normalizing functions is shown; - the solution of the problem of large deviations in the conditions of nonlinear approximations is found; and the connection between nonlinear normalizing functions is shown; - nonlinear normalizing functions for Markov random evolutions are found; - Investigated pulsed recurrent processes with nonlinear normalization in the Levy approximation scheme.