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

Part of:Special processesMathematical 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 m δ(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 m δ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for m δ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m δ(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

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007, pp. 385 - 406
Copyright
Copyright © Applied Probability Trust 2007 

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.Google Scholar
Cartan, H. (1971). Differential Calculus. Hermann, Paris.Google Scholar
Cheng, Y. and Tang, Q. (2003). Moments of the surplus before ruin and the deficit at ruin in the Erlang(2) risk process. N. Amer. Actuarial J.. 7, 1–12.Google Scholar
Dickson, D. C. M and Hipp, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance Math. Econom. 29, 333–344.Google Scholar
Drekic, S., Dickson, D. C. M., Stanford, D. A. and Willmot, G. E. (2004). On the distribution of the deficit at ruin when claims are phase-type. Scand. Actuarial J.. 2004, 105–120.CrossRefGoogle Scholar
Dufresne, D. (2006). Fitting combinations of exponentials to probability distributions. Appl. Stoch. Models Business Industry 23, 23–48.CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T. (1958). Linear Operators. Part I. General Theory. John Wiley, New York.Google Scholar
Gerber, H. U. and Shiu, E. S. W. (1998). On the time value of ruin. N. Amer. Actuarial J.. 2, 48–78.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (2005). The time value of ruin in a Sparre Andersen model. N. Amer. Actuarial J.. 2, 49–84.CrossRefGoogle Scholar
Gerber, H. U., Goovaerts, M. J. and Kaas, R. (1987). On the probability and severity of ruin. ASTIN Bull. 17, 151–163.Google Scholar
Grübel, R. (1989). Stochastic models as functionals: some remarks on the renewal case. J. Appl. Prob. 26, 296–303.Google Scholar
Grübel, R. and Pitts, S. M. (1992). A functional approach to the stationary waiting time and idle period distributions of the GI/G/1 queue. Ann. Prob. 20, 1754–1778.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2004). On ruin for the Erlang(n) risk process. Insurance Math. Econom. 34, 391–408.Google Scholar
Lin, X. S. and Willmot, G. E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insurance Math. Econom. 25, 63–84.CrossRefGoogle Scholar
Lin, X. S. and Willmot, G. E. (2000). The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance Math. Econom. 27, 19–44.CrossRefGoogle Scholar
Pitts, S. M. (1998). A functional approach to approximations for the observed open times in single ion channels. J. Math. Anal. Appl. 219, 260–278.Google Scholar
Pitts, S. M. (1999). Approximations for semi-Markov single ion channel models. In Semi-Markov Models and Applications, eds Janssen, J. and Limnios, N., Kluwer, Dordrecht, pp. 103–115.CrossRefGoogle Scholar
Pitts, S. M. (2004). A functional approach to approximations for the individual risk model. ASTIN Bull. 34, 379–397.CrossRefGoogle Scholar
Politis, K. and Pitts, S. M. (1998). Approximations for solutions of renewal-type equations. Stoch. Process. Appl. 78, 195–216.Google Scholar
Rudin, W. (1986). Real and Complex Analysis. McGraw-Hill, New York.Google Scholar
Willmot, G. E. and Lin, X. S. (2001). Lundberg Approximations for Compound Distributions with Insurance Applications. Springer, New York.Google Scholar