Exact and asymptotic \(n\)-tuple laws at first and last passage
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Publication:968775
DOI10.1214/09-AAP626zbMATH Open1200.60038arXiv0811.3075OpenAlexW1835046092MaRDI QIDQ968775
Author name not available (Why is that?)
Publication date: 6 May 2010
Published in: (Search for Journal in Brave)
Abstract: Understanding the space-time features of how a L'evy 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 cite{KD} a new quintuple law was established for a general L'evy 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 -tuple laws, for L'evy processes, L'evy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer 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 -tuple laws for L'evy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by inter-playing the role of a (conditioned) stable processes as both a (conditioned) L'evy 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 L'evy processes. This leads further to the introduction of a more general family of L'evy processes which we call hypergeometric L'evy processes, for which similar explicit identities may be considered.
Full work available at URL: https://arxiv.org/abs/0811.3075
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