On some extensions of generalized counting processes

TL;DR

This paper explores fractional extensions of counting processes using inverse stable subordinators, deriving governing equations and closed-form solutions for their probability distributions and generating functions.

Contribution

It introduces new fractional models of counting processes with explicit governing equations and closed-form solutions, expanding the theoretical framework of stochastic processes.

Findings

Derived governing equations involving fractional derivatives.

Provided closed-form expressions for distributions and generating functions.

Extended the class of generalized counting processes with fractional dynamics.

Abstract

We study different fractional extensions of the Poisson process and generalized counting processes by introducing time-change represented by the inverse to the sums of stable and tempered stable subordinators. We state the governing equations for probability distributions and probability generating functions which involve fractional derivatives of different orders. Closed form expressions for probability distributions and probability generating functions are also provided for several considered models.