Multivariate Generalized Counting Process via Gamma Subordination

TL;DR

This paper introduces a multivariate gamma subordinator driven by a negative binomial process, deriving its mathematical properties and applying it to a shock model, advancing stochastic process modeling techniques.

Contribution

It presents a novel multivariate gamma subordinator with explicit transforms, differential equations, and a time-changed process with practical applications.

Findings

Explicit joint Laplace-Stieltjes transform derived

Probability density function and differential equations obtained

Application to shock modeling demonstrated

Abstract

In this paper, we study a multivariate gamma subordinator whose components are independent gamma processes subject to a random time governed by an independent negative binomial process. We derive the explicit expressions for its joint Laplace-Stieltjes transform, its probability density function and the associated governing differential equations. Also, we study a time-changed variant of the multivariate generalized counting process where the time is changed by an independent multivariate gamma subordinator. For this time-changed process, we obtain the corresponding L\'evy measure and probability mass function. Later, we discuss an application of the time-changed multivariate generalized counting process to a shock model.