Properties of Poisson processes directed by compound Poisson-Gamma subordinators

TL;DR

This paper investigates properties of Poisson processes modified by compound Poisson-Gamma subordinators, deriving hitting time expressions and analyzing processes with time changes involving such subordinators and their iterations.

Contribution

It introduces new analytical results for time-changed Poisson processes using compound Poisson-Gamma subordinators and explores their iterative behaviors.

Findings

Derived explicit formulas for hitting times.

Analyzed processes with linear and iterative time changes.

Provided insights into the structure of compound Poisson-Gamma subordinated processes.

Abstract

In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.