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Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

Part of: Special processes Mathematical economics

Published online by Cambridge University Press:  01 July 2016

Susan M Pitts  and
Konstadinos Politis
Susan M Pitts*
Affiliation:
University of Cambridge
Konstadinos Politis*
Affiliation:
University of Piraeus
*
∗ Postal address: Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK. Email address: s.pitts@statslab.cam.ac.uk
∗∗ Postal address: Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Demetriou Street, Piraeus, 185 34, Greece.
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Abstract

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In the classical risk model with initial capital u, let τ(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function mδ(u) =E[e−δ τ(u)w(X+(u), Y+(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

Keywords

Time of ruindeficit at ruinsurplus prior to ruindefective renewal equationfunctional approach

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.) 91B30: Risk theory, insurance
Secondary: 60K05: Renewal theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 385 - 406
Copyright
Copyright © Applied Probability Trust 2007 

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