Passage time and fluctuation calculations for subexponential Lévy processes

Ron Doney, Claudia Klüppelberg, Ross Maller

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

We consider the passage time problem for Lévy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to -∞a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.

Original languageEnglish
Pages (from-to)1491-1519
Number of pages29
JournalBernoulli
Volume22
Issue number3
DOIs
StatePublished - Aug 2016

Keywords

  • Fluctuation theory
  • Lévy process
  • Maximum domain of attraction
  • Overshoot
  • Passage time
  • Regular variation
  • Subexponential growth
  • Undershoot

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