Convolution equivalent lévy processes and first passage times

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Abstract

We investigate the behavior of Lévy processes with convolution equivalent Lévy measures, up to the time of first passage over a high level u. Such problems arise naturally in the context of insurance risk where u is the initial reserve. We obtain a precise asymptotic estimate on the probability of first passage occurring by time T . This result is then used to study the process conditioned on first passage by time T . The existence of a limiting process as uis demonstrated, which leads to precise estimates for the probability of other events relating to first passage, such as the overshoot. A discussion of these results, as they relate to insurance risk, is also given.

Original languageEnglish (US)
Pages (from-to)1506-1543
Number of pages38
JournalAnnals of Applied Probability
Volume23
Issue number4
DOIs
StatePublished - Aug 1 2013

Keywords

  • Convolution equivalence
  • First passage time
  • Insurance risk
  • Lévy process
  • Probability of ruin in finite time

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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