Generalized Counting Process with Random Drift and Different Brownian Clocks

TL;DR

This paper introduces drifted and time-changed versions of the generalized counting process (GCP), deriving their probability laws, differential equations, and exploring various Brownian clocks and subordinators to extend GCP modeling.

Contribution

It presents novel drifted GCP models with stable subordinators and explores their dynamics under different Brownian clocks, including fractional and gamma-based time changes.

Findings

Derived probability laws and fractional differential equations for drifted GCPs.

Established governing equations for GCP time-changed with various Brownian clocks.

Analyzed fractional integrals and gamma-based time-changed GCPs.

Abstract

In this paper, we introduce drifted versions of the generalized counting process (GCP) with a deterministic drift and a random drift. The composition of stable subordinator with an independent inverse stable subordinator is taken as the random drift. We derive the probability law and its governing fractional differential equations for these drifted versions. Also, we study the GCP time-changed with different Brownian clocks, for example, the Brownian first passage-time with or without drift, elastic Brownian motion, Brownian sojourn time on positive half-line and the Bessel times. For these time-changed processes, we obtain the governing system of differential equation of their state probabilities, probability generating function, etc. Further, we consider a time-changed GCP where the time-change is done by subordinators linked to incomplete gamma function. Later, we study the fractional integral of GCP and its time-changed variant.