Abstract
In the paper we study the models of time-changed Poisson and Skellam-type processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. We obtain explicitly the probability distributions of considered time-changed processes and discuss their properties.
Highlights
Stochastic processes with random time and more general compositions of processes are quite popular topics of recent studies both in the theory of stochastic processes and in various applied areas
Models with random time appear in reliability and queuing theory, biological, ecological and medical research, note that for solving some problems of statistical estimation sampling of a stochastic process at random times or on the trajectory of another process can be used
In the present paper we study time-changed Poisson and Skellam processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse processes
Summary
Stochastic processes with random time and more general compositions of processes are quite popular topics of recent studies both in the theory of stochastic processes and in various applied areas. In the present paper we study various compositions of Poisson and Gamma processes. Interesting models of processes are based on the use of the difference of two Poisson processes, so-called Skellam processes, and their generalizations via time change Investigation of these models can be found, for example, in [2], and we refer to the paper [10], where fractional Skellam processes were introduced and studied. In the present paper we study time-changed Poisson and Skellam processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. 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 Poisson and Skellam processes are presented in terms of generalized Mittag-Leffler functions.
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