The hitting time of zero for a stable process

Alexey Kuznetsov, Andreas E. Kyprianou, Juan Carlos Pardo, Alexander R. Watson

Published 2012-12-20, updated 2014-03-10Version 3

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.