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 and Kees van Schaik

Insurance: Mathematics and Economics, 2012, vol. 51, issue 2, 382-392

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.

Keywords: Insurance risk process; Lévy process; Cramér condition; Convolution equivalent distributions; Ruin time; Overshoot; Undershoot; IM11; IM13 (search for similar items in EconPapers)
JEL-codes: C51 C52 G22 (search for similar items in EconPapers)
Date: 2012
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:insuma:v:51:y:2012:i:2:p:382-392

DOI: 10.1016/j.insmatheco.2012.06.005

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