Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exact and asymptotic $n$-tuple laws at first and last passage   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 $n$-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.

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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: funct-an