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Explicit Solutions for Survival Probabilities in the Classical Risk Model

Published online by Cambridge University Press:  17 April 2015

Jorge M.A. Garcia
Jorge M.A. Garcia*
Affiliation:
CEMAPRE, ISEG, Technical University of Lisbon, Rua do Quelhas, 2 1200-781 Lisboa, Portugal, E-mail: [email protected]
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Abstract

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The purpose of this paper is to show that, for the classical risk model, explicit expressions for survival probabilities in a finite time horizon can be obtained through the inversion of the double Laplace transform of the distribution of time to ruin. To do this, we consider Gerber and Shiu (1998) and a particular value for their penalty function. Although other methods to address the problem exist, we find this approach, perhaps, more direct and simple. For the analytic inversion, we have applied twice, after some algebra, the Laplace complex inversion formula.

Keywords

Classical risk modeldouble Laplace transformTime to ruinRuin probabilities
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 113 - 130
Copyright
Copyright © ASTIN Bulletin 2005

References

Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk processes: Insurance: Mathematics and Economics, 29, 333–344.Google Scholar
Dickson, D.C.M., Hughes, B.D. and Lianzeng, Z. (2003) The Density of the Time to Ruin for a Sparre Andersen Process with Erlang Arrivals and Exponential Claims. University of Melbourne, Research Papers Series No 111.Google Scholar
Dickson, D.C.M. and Willmot, G.E. (2004) The density of the time to ruin in the classical Poisson risk model. University of Melbourne, Research Papers Series No 115.Google Scholar
Drekic, S. and Willmot, G.E. (2003) On the Density and Moments of the Time of Ruin with Exponential Claims. Astin Bulletin, 33.CrossRefGoogle Scholar
Garcia, J.M.A. (1999) Other properties of the cosine transform. Cemapre working paper, ISEG, Technical University of Lisbon.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation Monograph series No.8. Irwin, Homewood, IL.Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. North American Actuarial Journal, 2, 48–78.CrossRefGoogle Scholar
Marsden, J.E. and Hoffman, M.J. (1999) Basic complex analysis. W.H. Freeman, New York.Google Scholar
Lima, F.D.P., Garcia, J.M.A. and Egidio Reis, A.D. (2002) Fourier/Laplace transforms and ruin probabilities. Astin Bulletin, 32(1), 91–105.CrossRefGoogle Scholar
Poularikas, A.D. (1996) The Transforms and Applications Handbook, CRC Press.Google Scholar