The hitting time of zero for a stable process

Alexey Kuznetsov (York University)
Andreas E Kyprianou (Bath University)
Juan Carlos Pardo (CIMAT)
Alexander R. Watson (University of Bath)


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-Panti-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.

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Pages: 1-26

Publication Date: March 9, 2014

DOI: 10.1214/EJP.v19-2647


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