## The Annals of Probability

### Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity

#### Abstract

We consider two first passage problems for stable processes, not necessarily symmetric, in one dimension. We make use of a novel method of path censoring in order to deduce explicit formulas for hitting probabilities, hitting distributions and a killed potential measure. To do this, we describe in full detail the Wiener–Hopf factorization of a new Lamperti-stable-type Lévy process obtained via the Lamperti transform, in the style of recent work in this area.

#### Article information

Source
Ann. Probab. Volume 42, Number 1 (2014), 398-430.

Dates
First available in Project Euclid: 9 January 2014

https://projecteuclid.org/euclid.aop/1389278528

Digital Object Identifier
doi:10.1214/12-AOP790

Mathematical Reviews number (MathSciNet)
MR3161489

Zentralblatt MATH identifier
1306.60051

#### Citation

Kyprianou, Andreas E.; Pardo, Juan Carlos; Watson, Alexander R. Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity. Ann. Probab. 42 (2014), no. 1, 398--430. doi:10.1214/12-AOP790. https://projecteuclid.org/euclid.aop/1389278528

#### References

• [1] Bertoin, J. (1993). Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 17–35.
• [2] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
• [3] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Pure and Applied Mathematics 29. Academic Press, New York.
• [4] Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 540–554.
• [5] Bogdan, K., Burdzy, K. and Chen, Z.-Q. (2003). Censored stable processes. Probab. Theory Related Fields 127 89–152.
• [6] Caballero, M. E. and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 967–983.
• [7] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On Lamperti stable processes. Probab. Math. Statist. 30 1–28.
• [8] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2011). Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17 34–59.
• [9] Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948–961.
• [10] Chaumont, L., Kyprianou, A. E. and Pardo, J. C. (2009). Some explicit identities associated with positive self-similar Markov processes. Stochastic Process. Appl. 119 980–1000.
• [11] Chaumont, L., Panti, H. and Rivero, V. (2011). The Lamperti representation of real-valued self-similar Markov processes. Preprint. Available at http://hal.archives-ouvertes.fr/hal-00639336.
• [12] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91–106.
• [13] Fitzsimmons, P. J. (2006). On the existence of recurrent extensions of self-similar Markov processes. Electron. Commun. Probab. 11 230–241.
• [14] Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75–90.
• [15] Getoor, R. K. (1966). Continuous additive functionals of a Markov process with applications to processes with independent increments. J. Math. Anal. Appl. 13 132–153.
• [16] Gnedin, A. V. (2010). Regeneration in random combinatorial structures. Probab. Surv. 7 105–156.
• [17] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th ed. Elsevier, Amsterdam.
• [18] Kadankova, T. and Veraverbeke, N. (2007). On several two-boundary problems for a particular class of Lévy processes. J. Theoret. Probab. 20 1073–1085.
• [19] Kuznetsov, A., Kyprianou, A. E. and Pardo, J. C. (2012). Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22 1101–1135.
• [20] Kuznetsov, A. and Pardo, J. C. (2010). Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Available at arXiv:1012.0817v1 [math.PR].
• [21] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
• [22] Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic $n$-tuple laws at first and last passage. Ann. Appl. Probab. 20 522–564.
• [23] Kyprianou, A. E. and Patie, P. (2011). A Ciesielski–Taylor type identity for positive self-similar Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 47 917–928.
• [24] Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13 1672–1701.
• [25] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205–225.
• [26] Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. Anal. Math. 20 371–395.
• [27] Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 471–509.
• [28] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 1. Cambridge Univ. Press, Cambridge.
• [29] Rogozin, B. A. (1971). The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Probab. Appl. 16 575–595.
• [30] Rogozin, B. A. (1972). The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Probab. Appl. 17 332–338.
• [31] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
• [32] Silverstein, M. L. (1980). Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 539–575.
• [33] Song, R. and Vondraček, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19 817–847.
• [34] Vuolle-Apiala, J. (1994). Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 546–565.