Generalized Counting Process with Random Drift and Different Brownian Clocks
Mostafizar Khandakar, Manisha Dhillon, Kuldeep Kumar Kataria
TL;DR
This work extends the generalized counting process (GCP) via drifted and time-changed constructions, incorporating deterministic drift $b t$, random clocks built from stable subordinators and their inverses, and Brownian clocks such as first-passage times, Bessel and sojourn times. The authors derive exact distributional laws, pgfs, and moments for a variety of drifted and clocked GCPs, including explicit Laplace transforms and governing differential equations, and establish tail behavior and long-range dependence for certain tempered and incomplete-gamma subordinators. They also introduce a fractional integral of the GFCP, providing mean and variance formulas and linking special cases to known fractional Poisson processes. The results broaden the toolkit for modeling counting phenomena with heavy tails, anomalous diffusion, and long-range dependence in applications spanning finance, hydrology, and queueing theory.
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.