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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
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