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Journal of Applied Probability Journal of Applied Probability

Article contents

A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

Part of: Stochastic processes Mathematical economics

Published online by Cambridge University Press:  04 April 2017

Onno J. Boxma ,
Esther Frostig  and
David Perry
Onno J. Boxma
Affiliation:
Eindhoven University of Technology
Esther Frostig
Affiliation:
University of Haifa
David Perry
Affiliation:
University of Haifa Western Galilee College
Corresponding
E-mail address:
frostig@stat.haifa.ac.il
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Abstract

We consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

Keywords

Spectrally negative Lévy process scale function ruin probability claim refraction

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 267 - 285
Copyright
Copyright © Applied Probability Trust 2017 

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References

Asmussen, S. and Albrecher, H. (2010).Ruin Probabilities, 2nd edn. (Adv. Ser. Statist. Sci. Appl. Prob. 14).World Scientific,Hackensack, NJ.CrossRefGoogle Scholar
Boxma, O. J., Frostig, E., Perry, D. and Yosef, R. (2016).Partial coverage by a rich uncle until ruin: a reinsurance model. Res. Rep. Eurandom Vol. 2016-001.Google Scholar
Dickson, D. C. M. and Drekic, S. (2006).Optimal dividends under a ruin probability constraint.Ann. Actuar. Sci. 1,291–306.CrossRefGoogle Scholar
Gerber, H. U. and Shiu, E. S. W. (2006).On optimal dividends: from reflection to refraction.J. Comput. Appl. Math. 186,4–22.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012).The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II, (Lecture Notes Math. 2061).Springer,Heidelberg, pp.97–186.CrossRefGoogle Scholar
Kyprianou, A. E. (2006).Introductory Lectures on Fluctuations of Lévy Processes with Applications.Springer,Berlin.Google Scholar
Kyprianou, A. E. (2013).Gerber–Shiu Risk Theory.Springer,Cham.CrossRefGoogle Scholar
Kyprianou, A. E. and Loeffen, R. L. (2010).Refracted Lévy processes.Ann. Inst. H. Poincaré. Prob. Statist. 46,24–44.CrossRefGoogle Scholar
Lin, S. X. and Pavlova, K. P. (2006).The compound Poisson risk model with a threshold dividend strategy.Insurance Math. Econom. 38,57–80.CrossRefGoogle Scholar
Loeffen, R. L. (2014).On obtaining simple identities for overshoots of spectrally negative Lévy processes. Res. Rep. 12, Probability and Statistics Group, The University of Manchester. Available at https://arxiv.org/abs/1410.5341.Google Scholar
Wan, N. (2007).Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion.Insurance Math. Econom. 40,509–523.CrossRefGoogle Scholar
Zhang, H. Y., Zhou, M. and Gou, J. Y. (2006).The Gerber–Shiu discounted penalty function for classical risk model with two-step premium rate.Statist. Prob. Lett. 76,1211–1218.CrossRefGoogle Scholar

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