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Abstract
Understanding the space time features of how a Levy 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) 91106] a new quintuple law was established for a general Levy 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 ntuple laws, for Levy processes, Levy processes conditioned to stay positive and positive selfsimilar 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 selfsimilar Markov processes issued from the origin. Although the relation between the ntuple laws for Levy processes and positive selfsimilar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Levy processes and a positive selfsimilar Markov processes, we obtain a suite of completely explicit first and last passage identities for socalled Lampertistable Levy processes. This leads further to the introduction of a more general family of Levy processes which we call hypergeometric Levy processes, for which similar explicit identities may be considered.
Original language  English 

Pages (fromto)  522564 
Number of pages  43 
Journal  Annals of Applied Probability 
Volume  20 
Issue number  2 
DOIs  
Publication status  Published  Apr 2010 
Keywords
 last passage time
 undershoot
 conditioned Levy process
 ntuple laws
 Levy process
 fluctuation theory
 overshoot
 first passage time
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Projects
 2 Finished

Fluctuaction Theory of Positive SelfSimilar Markov and Levy
1/04/08 → 30/06/08
Project: Research council

LEVY PROCESSES OPTIMAL STOPPING PROBLEMS AND STOCHASTIC GAME S
Engineering and Physical Sciences Research Council
1/01/07 → 31/12/09
Project: Research council