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Fourier/Laplace Transforms and Ruin Probabilities

Published online by Cambridge University Press:  29 August 2014

Fátima D.P. Lima ,
Jorge M.A. Garcia  and
Alfredo D. Egídio Dos Reis
Fátima D.P. Lima
Affiliation:
SGF - Sociedade Gestora de Fundos de Pensões, Lisbon, E-mail:flima@sgf.pt.
Jorge M.A. Garcia
Affiliation:
SGF - Sociedade Gestora de Fundos de Pensõe, Lisbon, E-mail:jorgegarcia@sgf.pt.
Alfredo D. Egídio Dos Reis
Affiliation:
Department of Mathematics, Technical University of Lisbon, Rua do Quelhas 6, 1200-781 Lisboa, Portugal, Tel: +351-213925867, Fax: +351-213922781, E-mail:alfredo@iseg.utl.pt.
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Abstract

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In this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.

By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.

To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

Keywords

Risk theoryRuin theoryFourier transformLaplace transformProbability of ultimate ruinSeverity of ruinErlang(2) risk process
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 32 , Issue 1 , May 2002 , pp. 91 - 105
Copyright
Copyright © International Actuarial Association 2002

References

Dickson, D.C.M. (1998) On a class of renewal risk processes, North American Actuarial Journal, 2(3), 60–73.CrossRefGoogle Scholar
Dickson, D.C.M. and Hipp, C. (1998) Ruin probabilities for Erlang(2) risk processes, Insurance: Mathematics and Economics, 22, 251–262.Google Scholar
Dickson, D.C.M., Egídio dos Reis, A.D. and Waters, H.R. (1995) Some stable algorithms in ruin theory and their applications, Astin Bulletin, 25(2), 153–175.CrossRefGoogle Scholar
Dufresne, F. and Gerber, H.U. (1989) Three methods to calculate the probability of ruin, Astin Bulletin, 19, 71–90.CrossRefGoogle Scholar
Egídio dos Reis, A.D. (1993) How long is the surplus below zero?, Insurance: Mathematics and Economics, 12, 23–38.Google Scholar
Garcia, J.M.A. (1999) Characteristic functions and their application to risk processes, Cemapre Working Paper no. 6/99, ISEG.Google Scholar
Garcia, J.M.A. (2000) Other properties of the cosine transform, Cemapre Working Paper no. 6/99, ISEG.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory, S.S. Huebner Foundation for Insurance Education, University of Pennsylvania, Philadelphia.Google Scholar
Gerber, H.U., Goovaerts, M.J. and Kaas, R. (1979) On the probability and severity of ruin, Astin Bulletin, 17, 151–163.CrossRefGoogle Scholar
Poularikas, A. (1996) The Transforms and Applications Handbook, CRC Press.Google Scholar
Ramsay, C.M. and Usábel, M.A. (1997) Calculating ruin probabilities via product integration, Astin Bulletin, 27(2), 263–271.CrossRefGoogle Scholar
Seal, H. (1977) Numerical inversion of characteristic functions, Scandinavian Actuarial Journal, 48–53.CrossRefGoogle Scholar
Sparre Andersen, E. (1957) On the collective theory of risk in the case of contagion between the claims, Transactions of the XV International Congress of Actuaries, 2, 219–229.Google Scholar
Usábel, M.R. (2001) Ultimate ruin probabilities for generalized Gamma-convolutions claim sizes, Astin Bulletin, 31(1), 59–79.CrossRefGoogle Scholar