Path decomposition of ruinous behavior for a general lÉvy insurance risk process

Philip S. Griffin, Ross A. Maller

Research output: Contribution to journalArticle

12 Scopus citations

Abstract

We analyze the general Levy insurance risk process for Levy measures in the convolution equivalence class S(α), α < 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u→∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u→∞, conditional on ruin occurring under our assumptions. Existing asymptotic results under the S(α) assumption are synthesized and extended, and proofs are much simplified, by comparison with previous methods specific to the convolution equivalence analyses. Additionally, limiting expressions for penalty functions of the type introduced into actuarial mathematics by Gerber and Shiu are derived as straightforward applications of our main results.

Original languageEnglish (US)
Pages (from-to)1411-1449
Number of pages39
JournalAnnals of Applied Probability
Volume22
Issue number4
DOIs
StatePublished - Aug 1 2012

Keywords

  • Convolution equivalence
  • Expected discounted penalty function
  • Lévy insurance risk process
  • Overshoot
  • Time to ruin

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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