Time-Changed Poisson Processes

TL;DR

This paper derives difference-differential equations for various time-changed Poisson processes, including those with inverse Gaussian, stable, and tempered stable subordinators, extending existing results in the field.

Contribution

It introduces new governing equations for Poisson processes time-changed by inverse Gaussian, stable, and tempered stable subordinators, including a novel PDE for the tempered stable case.

Findings

Derived DDEs for inverse Gaussian time-changed Poisson processes

Established DDEs for stable and iterated stable subordinators

Presented a new PDE for tempered stable subordinators with rational index

Abstract

We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDE's. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDE's corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index $0<\beta<1,$ when $\beta $ is a rational number. We then use this result to obtain the governing DDE for the mass function of Poisson process time-changed by tempered stable subordinator. Our results extend and complement the results in Baeumer et al. \cite{B-M-N} and Beghin et al. \cite{BO-1} in several directions.