The Gerber-Shiu function and the generalized Cramér-Lundberg model |
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Authors: | Chantal Labbé |
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Institution: | a Service de l’enseignement des méthodes quantitatives de gestion, HEC Montréal, 3000, chemin de la Côte-Sainte-Catherine, Montréal, QC, Canada H3T 2A7 b Department of Statistical and Actuarial Sciences, University of Western Ontario, 1151 Richmond St., London, ON, Canada N6A 5B7 |
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Abstract: | In this paper we extend some results in Cramér 7] by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber-Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang(n, β) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation. |
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Keywords: | Defective renewal equation Erlang(n β) annuity-related income distribution Expected discounted penalty function Probability of ruin Laplace transform of the time to ruin Two-sided jumps |
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