On the multivariate generalized counting process and its time-changed variants

In this paper, we study a multivariate version of the generalized counting process (GCP) and discuss its various time-changed variants. The time is changed using random processes such as the stable subordinator, inverse stable subordinator, and their composition, tempered stable subordinator, gamma subordinator etc. Several distributional properties that include the probability generating function, probability mass function and their governing differential equations are obtained for these variants. It is shown that some of these time-changed processes are Lévy and for such processes we derived the associated Lévy measure. The explicit expressions for the covariance and codifference of the component processes for some of these time-changed variants are obtained. An application of the multivariate generalized space fractional counting process to a shock model is discussed.