The hitting time of zero for a stable process. (English) Zbl 1293.60055

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

Electron. J. Probab. 19, Paper No. 30, 26 p. (2014).

Summary: 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. [K. Yano et al., Lect. Notes Math. 1979, 187–227 (2009; Zbl 1201.60043)] and [F. Cordero, “On the excursion theory for the symmetric stable Lévy processes with index \(\alpha\) in \(]1,2]\) and some applications”, Paris: Université Pierre et Marie Curie (PhD Thesis) (2010)], and [G. Peskir, Electron. Commun. Probab. 13, 653–659 (2008; Zbl 1193.60066)], 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.

MSC:

60G52	Stable stochastic processes
60G18	Self-similar stochastic processes
60G51	Processes with independent increments; Lévy processes

Keywords:

Lévy processes; stable processes; hitting times; positive self-similar Markov processes; Lamperti representation; real self-similar Markov processes; Lamperti-Kiu representation; generalised Lamperti representation; exponential functional; conditioning to avoid zero

Citations:

Zbl 1201.60043; Zbl 1193.60066