Electronic Journal of Probability
- Electron. J. Probab.
- Volume 19 (2014), paper no. 30, 26 pp.
The hitting time of zero for a stable process
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.
Electron. J. Probab., Volume 19 (2014), paper no. 30, 26 pp.
Accepted: 9 March 2014
First available in Project Euclid: 4 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60G52: Stable processes
Secondary: 60G18: Self-similar processes 60G51: Processes with independent increments; Lévy processes
This work is licensed under a Creative Commons Attribution 3.0 License.
Kuznetsov, Alexey; Kyprianou, Andreas; Pardo, Juan Carlos; Watson, Alexander. The hitting time of zero for a stable process. Electron. J. Probab. 19 (2014), paper no. 30, 26 pp. doi:10.1214/EJP.v19-2647. https://projecteuclid.org/euclid.ejp/1465065672