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Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  21 June 2016

Erik J. Baurdoux ,
Juan Carlos Pardo ,
José Luis Pérez  and
Jean-François Renaud
Erik J. Baurdoux*
Affiliation:
London School of Economics
Juan Carlos Pardo*
Affiliation:
Centro de Investigación en Matemáticas
José Luis Pérez*
Affiliation:
Universidad Nacional Autónoma de Mèxico
Jean-François Renaud*
Affiliation:
Université du Québec à Montréal (UQAM)
*
* Postal address: Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, UK. Email address: e.j.baurdoux@lse.ac.uk
** Postal address: Centro de Investigación en Matemáticas, A.C. Calle Jalisco s/n, C.P. 36240, Guanajuato, Mexico. Email address: jcpardo@cimat.mx
*** Postal address: Department of Probability and Statistics, IIMAS, UNAM, C.P. 04510, Mexico, D.F., Mexico. Email address: garmendia@sigma.iimas.unam.mx
**** Postal address: Département de Mathématiques, Université du Québec à Montréal, 201 av. Président-Kennedy, Montréal, Québec, H2X 3Y7, Canada. Email address: renaud.jf@uqam.ca
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Abstract

Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).

Keywords

Scale functionParisian ruinLévy processexcursion theoryfluctuation theoryGerber–Shiu functionLaplace transform

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60J99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 2 , June 2016 , pp. 572 - 584
Copyright
Copyright © Applied Probability Trust 2016 

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