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

TL;DR

This paper develops new joint laws called n-tuple laws for Lévy processes and related processes at first and last passage times, providing explicit identities and asymptotic distributions relevant to various applied probability models.

Contribution

It introduces a family of n-tuple laws for Lévy and positive self-similar Markov processes at passage times, extending previous results and deriving explicit identities for Lamperti-stable and hypergeometric Lévy processes.

Findings

Established n-tuple laws for Lévy processes and conditioned processes.

Derived explicit first and last passage identities for Lamperti-stable Lévy processes.

Connected asymptotic overshoot and undershoot distributions to self-similar Markov processes.

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 $n$-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.