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Abstract
In this paper we return to the problem of Blumenthal-Getoor-Ray, published in 1961, which gave the law of the position of first entry of a symmetric alpha-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for a one dimensional alpha-stable process which enjoys two-sided jumps, and which is not necessarily symmetric. Our method is modern in the sense that we appeal to the relationship between alpha-stable processes and certain positive self-similar Markov processes. However there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Levy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for alpha-stable processes.
Original language | English |
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Pages (from-to) | 398-430 |
Number of pages | 33 |
Journal | Annals of Probability |
Volume | 42 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2014 |
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Dive into the research topics of 'Hitting distributions of α-stable processes via path censoring and self-similarity'. Together they form a unique fingerprint.Projects
- 1 Finished
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LEVY PROCESSES OPTIMAL STOPPING PROBLEMS AND STOCHASTIC GAME S
Engineering and Physical Sciences Research Council
1/01/07 → 31/12/09
Project: Research council
Profiles
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Andreas Kyprianou
- Department of Mathematical Sciences - Professor
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
- Probability Laboratory at Bath
- Institute for Mathematical Innovation (IMI) - Director of the Bath Institute for Mathematical Innovation
- Made Smarter Innovation: Centre for People-Led Digitalisation
Person: Research & Teaching, Teaching & Other