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The Density of the Time to Ruin in the Classical Poisson Risk Model

Published online by Cambridge University Press:  17 April 2015

David C.M. Dickson  and
Gordon E. Willmot
David C.M. Dickson
Affiliation:
Centre for Actuarial Studies, Department of Economics, University of Melbourne, Victoria 3010, Australia
Gordon E. Willmot
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1
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Abstract

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We derive an expression for the density of the time to ruin in the classical risk model by inverting its Laplace transform. We then apply the result when the individual claim amount distribution is a mixed Erlang distribution, and show how finite time ruin probabilities can be calculated in this case.

Keywords

Time to ruinLaplace transformLagrange’s implicit function theoremCompound geometric distributionMixed Erlang distributionFinite time ruinTransient M/G/1 waiting time
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 1 , May 2005 , pp. 45 - 60
Copyright
Copyright © ASTIN Bulletin 2005

References

Asmussen, S. (2000) Ruin probabilities. World Scientific, Singapore.CrossRefGoogle Scholar
Asmussen, S., Avram, F. and Usabel, M. (2002) Erlangian approximations for finite-horizon ruin probabilities. ASTIN Bulletin 32, 267–281.CrossRefGoogle Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1997) Actuarial Mathematics, 2nd edition. Society of Actuaries, Itasca, IL.Google Scholar
Dickson, D.C.M. and dos Reis, A.D.E. (1996) On the distribution of the duration of negative surplus. Scandinavian Actuarial Journal, 148–164.CrossRefGoogle Scholar
Dickson, D.C.M. and Hipp, C. (2001) On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics & Economics 29, 333–344.Google Scholar
Dickson, D.C.M., Hughes, B. and Zhang, L. (2003) The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Centre for Actuarial Studies Research Paper Series No. 111, University of Melbourne.Google Scholar
Dickson, D.C.M. and Waters, H.R. (2002) The distribution of the time to ruin in the classical risk model. ASTIN Bulletin 32, 299–313.CrossRefGoogle Scholar
Drekic, S., Stafford, J.E. and Willmot, G.E. (2004) Symbolic calculation of the moments of the time to ruin. Insurance: Mathematics & Economics 34, 109–120.Google Scholar
Drekic, S. and Willmot, G.E. (2003) On the density and moments of the time to ruin with exponential claims. ASTIN Bulletin 33, 11–21.CrossRefGoogle Scholar
Garcia, J.M. (2002) Explicit solutions for survival probabilities in a finite time horizon. Presented to the 6th International IME Conference, Lisbon. http://pascal.iseg.utl.pt/~cemapre/ime2002/index.html (No. 49).Google Scholar
Gerber, H.U. and Shiu, E.S.W. (1998) On the time value of ruin. North American Actuarial Journal 2(1), 48–78.CrossRefGoogle Scholar
Goulden, I.P and Jackson, D.M. (1983) Combinatorial Enumeration. Wiley, New York.Google Scholar
Lin, X.S. and Willmot, G.E. (1999) Analysis of a defective renewal equation arising in ruin theory. Insurance: Mathematics & Economics 25, 63–84.Google Scholar
Lin, X.S. and Willmot, G.E. (2000) The moments of the time to ruin, the surplus before ruin and the deficit at ruin. Insurance: Mathematics & Economics 27, 19–44.Google Scholar
Seal, H.L. (1978) Survival Probabilities. John Wiley and Sons, New York.Google Scholar