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Ruin Probabilities for Two Classes of Risk Processes

Published online by Cambridge University Press:  17 April 2015

Shuanming Li  and
José Garrido
Shuanming Li
Affiliation:
Centre for Actuarial Studies, Faculty of Economics and Commerce, University of Melbourne, Parkville 3052, Victoria, Australia, Email: shli@unimelb.edu.au
José Garrido
Affiliation:
Department of Mathematics and Statistics, Concordia University, 7141 Sherbrooke Street West, Montréal, Québec, H4B 1R7 Canada, Email: garrido@mathstat.concordia.ca
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Abstract

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We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

Keywords

Compound Poisson processgeneralized Erlang distributionSparre Andersen risk modelintegro-differential equationsmartingalesgeneralized Lundberg fundamental equationsupremum distributionWiener processdiffusion perturbed risk model
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 61 - 77
Copyright
Copyright © ASTIN Bulletin 2005

Footnotes

*

This research was funded by a 1SOA/CAS Ph.D. Grant and the 2Natural Sciences and Engineering Research Council of Canada (NSERC) operating grant OGP0036860.

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