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Exact and asymptotic $n$-tuple laws at first and last passage | | A. Kyprianou
; J.C. Pardo
; V. Rivero
; | | Date: |
19 Nov 2008 | | 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 above 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 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 $n$ 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 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. | | Source: | arXiv, 0811.3075 | | Services: | Forum | Review | PDF | Favorites | |
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