The Annals of Applied Probability

Exact and asymptotic n-tuple laws at first and last passage

A. E. Kyprianou, J. C. Pardo, and V. Rivero

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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.

Article information

Ann. Appl. Probab., Volume 20, Number 2 (2010), 522-564.

First available in Project Euclid: 9 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G51: Processes with independent increments; Lévy processes 60G50: Sums of independent random variables; random walks

Fluctuation theory n-tuple laws Lévy process conditioned Lévy process last passage time first passage time overshoot undershoot


Kyprianou, A. E.; Pardo, J. C.; Rivero, V. Exact and asymptotic n -tuple laws at first and last passage. Ann. Appl. Probab. 20 (2010), no. 2, 522--564. doi:10.1214/09-AAP626.

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