Buchak K. Analytical properties of the Poisson and Skellam processes with random time-change

Models of stochastic processes with random time-change are used in various applied areas: biological, ecological and medical research, reliability and queuing theory, analysis of financial data, when it appears that observed data and systems can be better described not with respect to the calendar time but rather with respect to some non-decreasing stochastic process taken as random time. In the dissertation time-changed Poisson processes are investigated, where the role of time is played by compound Poisson-Gamma subordinators and their inverse processes. For these processes the formulas for marginal distributions, moments and covariance functions are presented. During the research, new classes of distributions given by special functions such as the generalized Wright function and the three-parameter generalized Mittag-Leffler function are obtained. Hitting times and first passage times are investigated. Differential equations for marginal distributions are presented. Poisson processes with iterated time-change, namely with time given by the iterated Bessel transforms are studied. For the processes with such a time-change, the formulas for marginal distributions are derived and the system of difference-differential equations for marginal distributions are presented. The time-changed Skellam processes were studied, where the role of time is played by compound Poisson-Gamma subordinators and their inverse processes. We obtain explicitly the probability distributions of considered time-changed Skellam processes and their first and second order moments. In particular, for the case, where time-change is taken by means of compound Poisson-exponential subordinator and its inverse process, corresponding probability distributions of time-changed Skellam processes are presented in terms of generalized Mittag-Leffler functions. The governing equations are obtained for marginal distributions of Poisson and Skellam processes time-changed by inverse subordinators. The equations are given in terms of convolution-type derivatives which are generalizations of classical fractional derivatives in the sense of Caputo-Djrbashian.