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December 2009 The first passage event for sums of dependent Lévy processes with applications to insurance risk
Irmingard Eder, Claudia Klüppelberg
Ann. Appl. Probab. 19(6): 2047-2079 (December 2009). DOI: 10.1214/09-AAP601

Abstract

For the sum process X=X1+X2 of a bivariate Lévy process (X1, X2) with possibly dependent components, we derive a quintuple law describing the first upwards passage event of X over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undershoot of the previous maximum. The dependence between the jumps of X1 and X2 is modeled by a Lévy copula. We calculate these quantities for some examples, where we pay particular attention to the influence of the dependence structure. We apply our findings to the ruin event of an insurance risk process.

Citation

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Irmingard Eder. Claudia Klüppelberg. "The first passage event for sums of dependent Lévy processes with applications to insurance risk." Ann. Appl. Probab. 19 (6) 2047 - 2079, December 2009. https://doi.org/10.1214/09-AAP601

Information

Published: December 2009
First available in Project Euclid: 25 November 2009

zbMATH: 1209.60029
MathSciNet: MR2588239
Digital Object Identifier: 10.1214/09-AAP601

Subjects:
Primary: 60G51
Secondary: 60G50, 60J75, 91B30

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 6 • December 2009
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