## Hitting distributions of $$\alpha$$-stable processes via path censoring and self-similarity.(English)Zbl 1306.60051

Using potential theoretic methods, R. M. Blumenthal et al. [Trans. Am. Math. Soc. 99, 540–554 (1961; Zbl 0118.13005)] determined the distribution of the position of first entry into the unit interval of a symmetric $$\alpha$$-stable Lévy process. S. C. Port [J. Anal. Math. 20, 371–395 (1967; Zbl 0157.24702)] extended the result to stable Lévy processes with one-sided jumps. The paper under consideration is related to similar questions for the remaining class of strictly $$\alpha$$-stable Lévy processes $$(X_t)_{t\geq0}$$. More precisely, introducing the positivity parameter $$\rho=P\{X_t>0\}$$, the authors consider parameters in the range $(\alpha,\rho)\in(0,1)\times(0,1)\cup (1,2)\times(1-\tfrac1{\alpha},\tfrac1{\alpha})\cup \{(1,\tfrac12)\}.$ By a new method of proof, making use of the relation between the stable Lévy processes and certain positive selfsimilar Markov processes, the authors obtain explicit formulas for the distribution of the position of first entry into the unit interval, its corresponding hitting probability and the density of the killed potential measure. In a second first passage problem, they further determine the distribution of the position of first entry into $$(1,\infty)$$ before hitting zero. The main idea of proof is to remove negative components by a novel method of path censoring which leads to certain positive selfsimilar Markov processes. The Lamperti transform of such a Markov process can be represented as the exponential of a time-changed Lévy process. This Lévy process in turn is decomposed into the sum of two independent Lévy processes, a compound Poisson process and a Lamperti-stable process. A density formula for the jump distribution of the compound Poisson process enables the authors to derive an explicit Wiener-Hopf factorization of the Lévy process which is the main tool to prove their results on the original first passage problems.

### MSC:

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

### Citations:

Zbl 0118.13005; Zbl 0157.24702
Full Text:

### 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. · Zbl 0786.60101 [2] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003 [3] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Pure and Applied Mathematics 29 . Academic Press, New York. · Zbl 0169.49204 [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. · Zbl 0118.13005 [5] Bogdan, K., Burdzy, K. and Chen, Z.-Q. (2003). Censored stable processes. Probab. Theory Related Fields 127 89-152. · Zbl 1032.60047 [6] Caballero, M. E. and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 967-983. · Zbl 1133.60316 [7] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On Lamperti stable processes. Probab. Math. Statist. 30 1-28. · Zbl 1198.60022 [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. · Zbl 1284.60092 [9] Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948-961. · Zbl 1109.60039 [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. · Zbl 1170.60017 [11] Chaumont, L., Panti, H. and Rivero, V. (2011). The Lamperti representation of real-valued self-similar Markov processes. Preprint. Available at . [12] Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91-106. · Zbl 1101.60029 [13] Fitzsimmons, P. J. (2006). On the existence of recurrent extensions of self-similar Markov processes. Electron. Commun. Probab. 11 230-241. · Zbl 1110.60036 [14] Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75-90. · Zbl 0104.11203 [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. · Zbl 0138.40901 [16] Gnedin, A. V. (2010). Regeneration in random combinatorial structures. Probab. Surv. 7 105-156. · Zbl 1204.60028 [17] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals , Series , and Products , 7th ed. Elsevier, Amsterdam. · Zbl 1208.65001 [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. · Zbl 1138.60038 [19] Kuznetsov, A., Kyprianou, A. E. and Pardo, J. C. (2012). Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22 1101-1135. · Zbl 1252.60044 [20] Kuznetsov, A. and Pardo, J. C. (2010). Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Available at [math.PR]. · Zbl 1268.60060 [21] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001 [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. · Zbl 1200.60038 [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. · Zbl 1231.60031 [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. · Zbl 1193.60064 [25] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205-225. · Zbl 0274.60052 [26] Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. Anal. Math. 20 371-395. · Zbl 0157.24702 [27] Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 471-509. · Zbl 1077.60055 [28] Rogers, L. C. G. and Williams, D. (2000). Diffusions , Markov Processes , and Martingales. Vol. 1. Cambridge Univ. Press, Cambridge. · Zbl 0977.60005 [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. · Zbl 0269.60053 [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. · Zbl 0272.60050 [31] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68 . Cambridge Univ. Press, Cambridge. · Zbl 0973.60001 [32] Silverstein, M. L. (1980). Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 539-575. · Zbl 0459.60063 [33] Song, R. and Vondraček, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19 817-847. · Zbl 1119.60063 [34] Vuolle-Apiala, J. (1994). Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 546-565. · Zbl 0810.60067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.