Densities of Inverse Tempered Stable Subordinators and Related Processes With Mellin Transforrm

TL;DR

This paper derives the probability densities of tempered stable and inverse tempered stable subordinators using Mellin transforms, providing new formulas and asymptotic behaviors, enriching the mathematical tools for these stochastic processes.

Contribution

It introduces an alternative Mellin transform-based method to derive densities of tempered stable and inverse tempered stable subordinators, including their products and quotients.

Findings

Derived infinite series form of densities using Mellin transform

Obtained asymptotic behaviors as x approaches 0+

Extended results to products and quotients of stable processes

Abstract

In this article, the infinite series form of the probability densities of tempered stable and inverse tempered stable subordinators are obtained using Mellin transform. Further, the densities of the products and quotients of stable and inverse stable subordinators are worked out. The asymptotic behaviours of these densities are obtained as $x \rightarrow 0^+$. Similar results for tempered and inverse tempered stable subordinators are discussed. Our results provide alternative methods to find the densities of these subordinators and complement the results available in literature.