Fractional counting process at L\'evy times and its applications

TL;DR

This paper introduces a generalized fractional counting process (FCP) based on Laskin's recent work, explores its properties, variants, and connections to Bell polynomials, and demonstrates its application in a shock deterioration model.

Contribution

It extends the fractional counting process framework by introducing a time-changed version and explores its properties, variants, and applications, including connections to Bell polynomials.

Findings

Derived distributional properties of the TCFCP

Introduced multiplicative and additive variants of FCP and TCFCP

Applied FCP to a shock deterioration model

Abstract

Traditionally, fractional counting processes, such as the fractional Poisson process, etc. have been defined using fractional differential and integral operators. Recently, Laskin (2024) introduced a generalized fractional counting process (FCP) by changing the probability mass function (pmf) of the time fractional Poisson process using the generalized three-parameter Mittag-Leffler function. Here, we study some additional properties for the FCP and introduce a time-changed fractional counting process (TCFCP), defined by time-changing the FCP with an independent L\'evy subordinator. We derive distributional properties such as the Laplace transform, probability generating function, the moments generating function, mean, and variance for the TCFCP. Some results related to waiting time distribution and the first passage time distribution are also discussed. We define the multiplicative and additive compound variants for the FCP and the TCFCP and examine their distributional characteristics with some typical examples. We explore some interesting connections of the TCFCP with Bell polynomials by introducing subordinated generalized fractional Bell polynomials. It is shown that the moments of the TCFCP can be represented in terms of the subordinated generalized fractional Bell polynomials. Finally, we present the application of the FCP in a shock deterioration model.