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Exact and asymptotic \(n\)-tuple laws at first and last passage. (English) Zbl 1200.60038
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 [R. A. Doney and A. E. Kyprianou, Ann. Appl. Probab. 16, No. 1, 91–106 (2006; Zbl 1101.60029)], a new quintuple law was established for a general Lévy process at first passage below a fixed level.
In this article, the quintuple law is used to establish a family of related joint laws, which are called \(n\)-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, asymptotic overshoot and undershoot distributions are considered and related 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, a suite of completely explicit first and last passage identities are obtained for so-called Lamperti-stable Lévy processes. This leads further to the introduction of a more general family of Lévy processes which are called hypergeometric Lévy processes, for which similar explicit identities may be considered.

MSC:
60G51 Processes with independent increments; Lévy processes
60G50 Sums of independent random variables; random walks
60G40 Stopping times; optimal stopping problems; gambling theory
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