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

Andreas E Kyprianou, J C Pardo, V Rivero

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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) 91-106] 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 n-tuple laws, for Levy processes, Levy 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 Levy processes and positive self-similar 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 self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable 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 languageEnglish
Pages (from-to)522-564
Number of pages43
JournalAnnals of Applied Probability
Volume20
Issue number2
DOIs
Publication statusPublished - Apr 2010

Fingerprint

Lévy Process
Self-similar Processes
Markov Process
n-tuple
Stable Process
Overshoot
Applied Probability
Lévy process
Queueing Theory
Optimal Stopping Problem
Branching process
Integer
Markov process

Keywords

  • last passage time
  • undershoot
  • conditioned Levy process
  • n-tuple laws
  • Levy process
  • fluctuation theory
  • overshoot
  • first passage time

Cite this

Exact and asymptotic n-tuple laws at first and last passage. / Kyprianou, Andreas E; Pardo, J C; Rivero, V.

In: Annals of Applied Probability, Vol. 20, No. 2, 04.2010, p. 522-564.

Research output: Contribution to journalArticle

Kyprianou, Andreas E ; Pardo, J C ; Rivero, V. / Exact and asymptotic n-tuple laws at first and last passage. In: Annals of Applied Probability. 2010 ; Vol. 20, No. 2. pp. 522-564.
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