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April 2010 Exact and asymptotic n-tuple laws at first and last passage
A. E. Kyprianou, J. C. Pardo, V. Rivero
Ann. Appl. Probab. 20(2): 522-564 (April 2010). DOI: 10.1214/09-AAP626


Understanding the space–time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) 91–106] a new quintuple law was established for a general Lévy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call n-tuple laws, for Lévy processes, Lévy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the n-tuple laws for Lévy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Lévy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Lévy processes. This leads further to the introduction of a more general family of Lévy processes which we call hypergeometric Lévy processes, for which similar explicit identities may be considered.


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A. E. Kyprianou. J. C. Pardo. V. Rivero. "Exact and asymptotic n-tuple laws at first and last passage." Ann. Appl. Probab. 20 (2) 522 - 564, April 2010.


Published: April 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1200.60038
MathSciNet: MR2650041
Digital Object Identifier: 10.1214/09-AAP626

Primary: 60G50 , 60G51

Keywords: conditioned Lévy process , First passage time , fluctuation theory , last passage time , Lévy process , n-tuple laws , overshoot , undershoot

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.20 • No. 2 • April 2010
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