Inverse Tempered Stable Subordinators

TL;DR

This paper studies the inverse tempered stable subordinator, deriving its density, series representation, moments, asymptotic behaviors, and governing PDE, extending known results for inverse stable subordinators.

Contribution

It provides the first explicit density, series representation, and PDE for the inverse tempered stable subordinator, broadening understanding of this stochastic process.

Findings

Explicit density function derived for ITS subordinator.

Series representation facilitates computational applications.

Asymptotic behaviors and PDE governing the density are established.

Abstract

We consider the first-hitting time of a tempered $\beta$-stable subordinator, also called inverse tempered stable (ITS) subordinator. The density function of the ITS subordinator is obtained, for the index of stability $\beta \in (0,1)$. The series representation of the ITS density is also obtained, which could be helpful for computational purposes. The asymptotic behaviors of the $q$-th order moments of the ITS subordinator are investigated. In particular, the limiting behaviors of the mean of the ITS subordinator is given. The limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained. The governing PDE for the ITS density is also obtained. The corresponding known results for inverse stable subordinator follow as special cases.