Electronic Journal of Probability

The hitting time of zero for a stable process

Alexey Kuznetsov, Andreas Kyprianou, Juan Carlos Pardo, and Alexander Watson

Full-text: Open access


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.

Article information

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

Levy processes stable processes positive self-similar Markov 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.

Export citation


  • Asmussen, Søren. Ruin probabilities. Advanced Series on Statistical Science & Applied Probability, 2. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. xii+385 pp. ISBN: 981-02-2293-9
  • Asmussen, Søren. Applied probability and queues. Second edition. Applications of Mathematics (New York), 51. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2003. xii+438 pp. ISBN: 0-387-00211-1
  • Bertoin, Jean; Yor, Marc. The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 (2002), no. 4, 389–400.
  • Blumenthal, R. M.; Getoor, R. K. Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29 Academic Press, New York-London 1968 x+313 pp.
  • Caballero, M. E.; Chaumont, L. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 (2006), no. 3, 1012–1034.
  • Caballero, M. E.; Chaumont, L. Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 (2006), no. 4, 967–983.
  • M. E. Caballero, J. C. Pardo, and J. L. Pérez. Explicit identities for Lévy processes associated to symmetric stable processes. phBernoulli, 17(1): 34–59, 2011. ISSN 1350-7265. DOI.
  • Carmona, Philippe; Petit, Frédérique; Yor, Marc. On the distribution and asymptotic results for exponential functionals of Lévy processes. Exponential functionals and principal values related to Brownian motion, 73–130, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1997.
  • Chaumont, L. Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 (1996), no. 1, 39–54.
  • Chaumont, Loic; Pardo, J. C. The lower envelope of positive self-similar Markov processes. Electron. J. Probab. 11 (2006), no. 49, 1321–1341.
  • X Symposium on Probability and Stochastic Processes and the First Joint Meeting France-Mexico of Probability. Proceedings of the symposium and the meeting held in Guanajuato, November 3–7, 2008. Edited by Ma. Emilia Caballero, Loïc Chaumont, Daniel Hernández-Hernández and Víctor Rivero. ESAIM Proceedings, 31. EDP Sciences, Les Ulis, 2011. front matter+115 pp. hal-00639336/fr/
  • Chaumont, Loic; Kyprianou, Andreas; Pardo, Juan Carlos; Rivero, Victor. Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40 (2012), no. 1, 245–279.
  • Chen, Zhen-Qing; Fukushima, Masatoshi; Ying, Jiangang. Extending Markov processes in weak duality by Poisson point processes of excursions. Stochastic analysis and applications, 153–196, Abel Symp., 2, Springer, Berlin, 2007.
  • Chybiryakov, Oleksandr. The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups. Stochastic Process. Appl. 116 (2006), no. 5, 857–872.
  • F. Cordero. On the excursion theory for the symmetric stable Lévy processes with index α in ]1,2] and some applications. PhD thesis, Université Pierre et Marie Curie – Paris VI, 2010.
  • Fitzsimmons, P. J.; Getoor, R. K. Occupation time distributions for Lévy bridges and excursions. Stochastic Process. Appl. 58 (1995), no. 1, 73–89.
  • Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. With one CD-ROM (Windows, Macintosh and UNIX). Seventh edition. Elsevier/Academic Press, Amsterdam, 2007. xlviii+1171 pp. ISBN: 978-0-12-373637-6; 0-12-373637-4
  • Hubalek, Friedrich; Kuznetsov, Alexey. A convergent series representation for the density of the supremum of a stable process. Electron. Commun. Probab. 16 (2011), 84–95.
  • J. Ivanovs. One-sided Markov additive processes and related exit problems. PhD thesis, Universiteit van Amsterdam, 2011.
  • Kuznetsov, Alexey. On extrema of stable processes. Ann. Probab. 39 (2011), no. 3, 1027–1060.
  • Kuznetsov, A.; Pardo, J. C. Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes. Acta Appl. Math. 123 (2013), no. 1, 113–139. arXiv:1012.0817v1
  • A. E. Kyprianou, J. C. Pardo, and A. R. Watson. Hitting distributions of α-stable processes via path censoring and self-similarity. arXiv:1112.3690v2, 2012. To appear in Ann. Probab.
  • Lamperti, John. Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225.
  • Lukacs, Eugene; Szász, Otto. On analytic characteristic functions. Pacific J. Math. 2, (1952). 615–625.
  • Maulik, Krishanu; Zwart, Bert. Tail asymptotics for exponential functionals of Lévy processes. Stochastic Process. Appl. 116 (2006), no. 2, 156–177.
  • H. Pantí. On Lévy processes conditioned to avoid zero. In preparation, 2012.
  • H. Pantí. A Lamperti type representation of real-valued self-similar Markov processes and Lévy processes conditioned to avoid zero. PhD thesis, CIMAT A.C., Guanajuato, Mexico, August 2012.
  • Peskir, Goran. The law of the hitting times to points by a stable Lévy process with no negative jumps. Electron. Commun. Probab. 13 (2008), 653–659.
  • Rivero, Véctor. Recurrent extensions of self-similar Markov processes and Cramér's condition. II. Bernoulli 13 (2007), no. 4, 1053–1070.
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
  • Vuolle-Apiala, J. Its excursion theory for self-similar Markov processes. Ann. Probab. 22 (1994), no. 2, 546–565.
  • Yano, Kouji; Yano, Yuko; Yor, Marc. On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes. Séminaire de Probabilités XLII, 187–227, Lecture Notes in Math., 1979, Springer, Berlin, 2009.
  • K. Yano, Y. Yano, and M. Yor. Penalising symmetric stable Lévy paths. phJ. Math. Soc. Japan, 61(3): 757–798, 2009natexlabb. ISSN 0025-5645. DOI.