The hitting time of zero for a stable process

Abstract

For any two-sided jumping α-stable process, where 1 < α < 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–Pantí–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.

Publication
Electronic Journal of Probability 19, no. 30
Date