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Phase-type Approximations to Finite-time Ruin Probabilities in the Sparre-Andersen and Stationary Renewal Risk Models

Published online by Cambridge University Press:  17 April 2015

D.A. Stanford ,
F. Avram ,
A.L. Badescu ,
L. Breuer ,
A. Da Silva Soares  and
G. Latouche
D.A. Stanford
Affiliation:
Dept. of Statistical & Actuarial Sciences, WSC 262, University of Western Ontario, London, Ontario N6A 5B7, Canada, E-mail: stanford@stats.uwo.ca, abadescu@math.uwaterloo.ca
F. Avram
Affiliation:
Département de Mathématiques, Université de Pau, Bat. IPRA, Avenue de l’université, 64000 Pau, France, E-mail: Florin.Avram@univ-pau.fr
A.L. Badescu
Affiliation:
Dept. of Statistical & Actuarial Sciences, WSC 262, University of Western Ontario, London, Ontario N6A 5B7, Canada, E-mail: stanford@stats.uwo.ca, abadescu@math.uwaterloo.ca
L. Breuer
Affiliation:
Department IV – Computer Science, University of Trier, D-54286 Trier, Germany, E-mail: breuer@info04.uni-trier.de
A. Da Silva Soares
Affiliation:
Département d’Informatique, Université Libre de Bruxelles, Campus Plaine, CP 212, B-1050 Bruxelles, Belgium, E-mail: andasilv@ulb.ac.be, latouche@ulb.ac.be
G. Latouche
Affiliation:
Département d’Informatique, Université Libre de Bruxelles, Campus Plaine, CP 212, B-1050 Bruxelles, Belgium, E-mail: andasilv@ulb.ac.be, latouche@ulb.ac.be
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Abstract

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The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

Keywords

Sparre-Andersen modelladder heightmaximal aggregate lossdeficit at ruinphase-type distribution
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 131 - 144
Copyright
Copyright © ASTIN Bulletin 2005

Footnotes

†

Presently in Dept. of Statistics & Actuarial Science, U. Waterloo

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