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The density of the time of ruin in the classical risk model with a constant dividend barrier

Published online by Cambridge University Press:  04 November 2013

Shuanming Li  and
Yi Lu
Shuanming Li
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Australia
Yi Lu
Affiliation:
Department of Statistics and Actuarial Science, Simon Fraser University, Canada
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Abstract

In this paper, we investigate the density function of the time of ruin in the classical risk model with a constant dividend barrier. When claims are exponentially distributed, we derive explicit expressions for the density function of the time of ruin and its decompositions: the density of the time of ruin without dividend payments and the density of the time of ruin with dividend payments. These densities are obtained based on their Laplace transforms, and expressed in terms of some special functions which are computationally tractable. The Laplace transforms are being inverted using a magnificent tool, the Lagrange inverse formula, developed in Dickson and Willmot (2005). Several numerical examples are given to illustrate our results.

Keywords

Classical risk modelconstant dividend barrierfinite-time ruin probabilityLaplace transformstransform inversion relationship
Type
Papers
Information
Annals of Actuarial Science , Volume 8 , Issue 1 , March 2014 , pp. 63 - 78
Copyright
Copyright © Institute and Faculty of Actuaries 2013 

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References

Abramowitz, M., Stegun, I.A. (1972). Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Dover Publications, New York.Google Scholar
Asmussen, S., Albrecher, H. (2010). Ruin Probabilities, 2nd edition, World Scientific Publishing Co. Pte. Ltd., Singapore.CrossRefGoogle Scholar
De Vylder, F.E. (1978). A practical solution to the problem of ultimate ruin probability. Scandinavian Actuarial Journal, 114–119.CrossRefGoogle Scholar
Dickson, D.C.M. (2007). Some finite time ruin problems. Annals of Actuarial Science, 2, 217–232.Google Scholar
Dickson, D.C.M. (2008). Some explicit solutions for the joint density of the time of ruin and the deficit at ruin. ASTIN Bulletin, 38(1), 259–275.Google Scholar
Dickson, D.C.M. (2012). The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model. Insurance: Mathematics and Economics, 50, 334–337.Google Scholar
Dickson, D.C.M., Gray, J. (1984). Approximations to ruin probability in the presence of an upper absorbing barrier. Scandinavian Actuarial Journal, 105–115.Google Scholar
Dickson, D.C.M., Willmot, G.E. (2005). The density of the time to ruin in the classical Poisson risk model. ASTIN Bulletin, 35(1), 45–60.CrossRefGoogle Scholar
Drekic, S., Willmot, G.E. (2003). On the density and moments of the time of ruin with exponential claims. ASTIN Bulletin, 33(1), 11–21.Google Scholar
Erdélyi, A. (1937). Beitrag zur Theorie der konfluenten hypergeometrischen Funktionen von mehreren Veranderlichen. Wiener Sitzungsberichte, 146, 431–467.Google Scholar
Garcia, J.M.A. (2005). Explicit solutions for survival probabilities in the classical risk model. ASTIN Bulletin, 35(1), 113–130.Google Scholar
Gerber, H.U., Shiu, E.S.W. (1998). On the time value of ruin. North American Actuarial Journal, 2(1), 48–72.Google Scholar
Landriault, D., Willmot, G.E. (2009). On the joint distributions of the time to ruin, the surplus prior to ruin and the deficit at ruin in the classical risk model. North American Actuarial Journal, 13(2), 252–270.Google Scholar
Landriault, D., Shi, T., Willmot, G.E. (2011). Joint density involving the time to ruin in the Sparre Andersen model under exponential assumptions. Insurance: Mathematics and Economics, 49, 371–379.Google Scholar
Lin, X.S., Willmot, G.E., Drekic, S. (2003). The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function. Insurance: Mathematics and Economics, 33, 551–566.Google Scholar
Perry, D., Stadje, W., Zacks, S. (2005). A two-sided first-exit problem for a compound Poisson process with a random upper boundary. Methodology and Computing in Applied Probability, 7, 51–62.Google Scholar
Srivastava, H.M., Karlsson, A., Per, W. (1985). Multiple Gaussian Hypergeometric Series. Halsted Press, Ellis Horwood Ltd, Chichester, UK.Google Scholar