Asymptotic distributions of the overshoot and undershoots for the Lévy insurance risk process in the Cramér and convolution equivalent cases

Philip S. Griffin, Ross A. Maller, Kees van Schaik

Research output: Contribution to journalArticle

7 Scopus citations

Abstract

Recent models of the insurance risk process use a Lévy process to generalise the traditional Cramér-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Lévy process which drifts to -∞ and satisfies a Cramér or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramér case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding these quantities, the "medium-heavy" tailed convolution equivalent model segues into the "light-tailed" Cramér model in a natural way. This suggests a usefully expanded flexibility for modelling the insurance risk process. We illustrate this relationship by comparing the asymptotic distributions obtained for the overshoot and undershoots, assuming the Lévy process belongs to the "GTSC" class.

Original languageEnglish (US)
Pages (from-to)382-392
Number of pages11
JournalInsurance: Mathematics and Economics
Volume51
Issue number2
DOIs
StatePublished - Sep 2012

Keywords

  • Convolution equivalent distributions
  • Cramér condition
  • Insurance risk process
  • Lévy process
  • Overshoot
  • Ruin time
  • Undershoot

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty

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