Time-changed generalized fractional Skellam process

TL;DR

This paper introduces two novel time-changed variants of the generalized fractional Skellam process, analyzing their probabilistic properties, dependence structures, and differential equations, expanding the understanding of fractional stochastic processes.

Contribution

It presents the first study of time-changed generalized fractional Skellam processes, deriving their key properties and establishing their long-range dependence and differential equations.

Findings

Derived probability generating and moment generating functions.

Established long-range dependence property.

Analyzed special cases and governing differential equations.

Abstract

In this paper, we introduce and study two time-changed variants of the generalized fractional Skellam process. These are obtained by time-changing the generalized fractional Skellam process with an independent L\'evy subordinator with finite moments of any order and its inverse, respectively. We call the introduced processes the time-changed generalized fractional Skellam process-I (TCGFSP-I) and the time-changed generalized fractional Skellam process-II (TCGFSP-II), respectively. The probability generating function, moment generating function, moments, factorial moments, variance, covariance, {\it etc.}, are derived for the TCGFSP-I. We obtain a variant of the law of the iterated logarithm for it and establish its long-range dependence property. Several special cases of the TCGFSP-I are considered, and the associated system of governing differential equations is obtained. Later, some distributional properties and particular cases are discussed for the TCGFSP-II.