Multivariate Generalized Counting Process via Gamma Subordination

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.