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Ruin Probabilities and Deficit for the Renewal Risk Model with Phase-type Interarrival Times

Published online by Cambridge University Press:  17 April 2015

F. Avram  and
M. Usábel
F. Avram
Affiliation:
Dept. of Mathematics, Université de Pau, France. E-mail: Floris.Avram@univ-pau.fr
M. Usábel
Affiliation:
Dept. of Business Adm., Universidad Carlos III de Madrid, Spain. E-mail: usabel@emp.uc3m.es
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Abstract

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This paper shows how the multivariate finite time ruin probability function, in a phase-type environment, inherits the phase-type structure and can be efficiently approximated with only one Laplace transform inversion.

From a theoretical point of view, we also provide below a generalization of Thorin’s formula (1971) for the double Laplace transform of the finite time ruin probability, by considering also the deficit at ruin; the model is that of a Sparre Andersen (renewal) risk process with phase-type interarrival times.

In the case when the claims distribution is of phase-type as well, we obtain also an alternative formula for the single Laplace transform in time (or “exponentially killed probability’’), in terms of the roots with positive real part of the Lundberg’s equations, which complements Asmussen’s representation (1992) in terms of the roots with negative real part.

Keywords

Finite time ruin probabilityexponentially killed ruin probabilitydeficit at ruinSparre Andersen Modelphase-type distributions
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 34 , Issue 2 , November 2004 , pp. 315 - 332
Copyright
Copyright © ASTIN Bulletin 2004

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