Fractional Generalizations of the Compound Poisson Process

TL;DR

This paper introduces the Generalized Fractional Compound Poisson Process (GFCPP), a unified framework that generalizes existing fractional processes, derives its properties, and explores special cases and simulations.

Contribution

It presents the GFCPP as a comprehensive fractional model encompassing various known processes and derives their distributional and differential properties.

Findings

GFCPP unifies multiple fractional processes.

Derived distributional properties and differential equations.

Simulated sample paths of various processes.

Abstract

This paper introduces the Generalized Fractional Compound Poisson Process (GFCPP), which claims to be a unified fractional version of the compound Poisson process (CPP) that encompasses existing variations as special cases. We derive its distributional properties, generalized fractional differential equations, and martingale properties. Some results related to the governing differential equation about the special cases of jump distributions, including exponential, Mittag-Leffler, Bernst\'ein, discrete uniform, truncated geometric, and discrete logarithm. Some of processes in the literature such as the fractional Poisson process of order $k$, P\'olya-Aeppli process of order $k$, and fractional negative binomial process becomes the special case of the GFCPP. Classification based on arrivals by time-changing the compound Poisson process by the inverse tempered and the inverse of inverse Gaussian subordinators are studied. Finally, we present the simulation of the sample paths of the above-mentioned processes.