Fractional Skellam Process of Order $k$

TL;DR

This paper introduces the fractional Skellam process of order k (FSPoK), a new stochastic process obtained by time-changing the classical process with an inverse stable subordinator, and explores its properties and variants.

Contribution

It presents the first fractional version of the Skellam process of order k, including its distributional equations, generating function, moments, and dependence structure, along with time-changed extensions.

Findings

Derived integral representation and fractional differential equations for FSPoK

Established long-range dependence property of FSPoK

Analyzed distributional properties of time-changed variants

Abstract

We introduce and study a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order $k$ (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of the FSPoK which are utilized to establish its long-range dependence property. Later, we considered two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent L\'evy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.