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February 2016 Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér–Lundberg and convolution equivalent conditions
Philip S. Griffin
Ann. Appl. Probab. 26(1): 360-401 (February 2016). DOI: 10.1214/14-AAP1094

Abstract

Recent studies have demonstrated an interesting connection between the asymptotic behavior at ruin of a Lévy insurance risk process under the Cramér–Lundberg and convolution equivalent conditions. For example, the limiting distributions of the overshoot and the undershoot are strikingly similar in these two settings. This is somewhat surprising since the global sample path behavior of the process under these two conditions is quite different. Using tools from excursion theory and fluctuation theory, we provide a means of transferring results from one setting to the other which, among other things, explains this connection and leads to new asymptotic results. This is done by describing the evolution of the sample paths from the time of the last maximum prior to ruin until ruin occurs.

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Philip S. Griffin. "Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér–Lundberg and convolution equivalent conditions." Ann. Appl. Probab. 26 (1) 360 - 401, February 2016. https://doi.org/10.1214/14-AAP1094

Information

Received: 1 September 2013; Revised: 1 October 2014; Published: February 2016
First available in Project Euclid: 5 January 2016

zbMATH: 1334.60076
MathSciNet: MR3449321
Digital Object Identifier: 10.1214/14-AAP1094

Subjects:
Primary: 60F17, 60G51
Secondary: 62P05, 91B30

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 1 • February 2016
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