On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators

TL;DR

This paper derives governing equations involving convolution-type derivatives for the marginal distributions of Poisson and Skellam processes that are modified by inverse subordinators, expanding understanding of their probabilistic structure.

Contribution

It introduces new governing equations in convolution-type derivatives for these time-changed processes, providing a novel analytical framework.

Findings

Derived convolution-type differential equations for Poisson processes

Extended equations to Skellam processes

Enhanced mathematical understanding of inverse subordinated processes

Abstract

In the paper we present the governing equations for marginal distributions of Poisson and Skellam processes time-changed by inverse subordinators. The equations are given in terms of convolution-type derivatives.