The hitting time of zero for a stable process

TL;DR

This paper derives an explicit law for the first hitting time of zero for two-sided stable processes with 1<α<2, expanding understanding beyond symmetric and one-sided cases using advanced Markov process representations.

Contribution

It provides a new explicit identity for the law of the hitting time of zero for two-sided stable processes, utilizing the Lamperti-Kiu representation and Mellin transform techniques.

Findings

Explicit law for hitting time of zero for two-sided stable processes.

Extension of previous results to non-symmetric cases.

Applications demonstrated for the derived identity.

Abstract

For any two-sided jumping $\alpha$-stable process, where $1 < \alpha < 2$, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Pant\'i-Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications.