Generalized Gamma Process: some results about composition and subordination

TL;DR

This paper explores the properties and compositions of generalized Gamma processes, providing explicit laws, representations via Fox's H-functions, and differential equations, with applications to processes like Cauchy and fractional Brownian motions.

Contribution

It offers new explicit laws and functional representations for compositions of generalized Gamma processes, and links these to differential equations and well-known stochastic processes.

Findings

Explicit laws for process compositions when possible

Representations in terms of Fox's H-functions

Differential equations governing the processes

Abstract

In this paper we deal with the generalized Gamma processes and their compositions. For the compositions of two or more than two generalized Gamma processes we give, when possible, the explicit law whereas, in the other cases the representations in terms of Fox's H-functions are given. We also study the connections between iteration and product of random processes by exploiting the properties of the generalized Gamma processes, such a study allows us to obtain some striking result about the compositions of the Cauchy processes or fractional Brownian motions. Furthermore, we find out the partial differential equations governing the generalized Gamma processes and their compositions