Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-19T02:00:52.846Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Explicit Solutions for the Joint Density of the Time of Ruin and the Deficit at Ruin

Published online by Cambridge University Press:  17 April 2015

David C.M. Dickson
David C.M. Dickson*
Affiliation:
Centre for Actuarial Studies, Department of Economics, University of Melbourne, Victoria 3010, Australia, E-Mail: dcmd@unimelb.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using probabilistic arguments we obtain an integral expression for the joint density of the time of ruin and the deficit at ruin. For the classical risk model, we obtain the bivariate Laplace transform of this joint density and invert it in the cases of individual claims distributed as Erlang(2) and as a mixture of two exponential distributions. As a consequence, we obtain explicit solutions for the density of the time of ruin.

Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 38 , Issue 1 , May 2008 , pp. 259 - 276
Copyright
Copyright © ASTIN Bulletin 2008

References

Abramowitz, M. and Stegun, I.A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
Cheung, E.C.K., Dickson, D.C.M. and Drekic, S. (2006) Optimal dividend threshold levels for two classes of claim size distributions. Unpublished manuscript.Google Scholar
Dickson, D.C.M. and Drekic, S. (2006) Optimal dividends under a ruin probability constraint. Annals of Actuarial Science, 1, 291–306.CrossRefGoogle Scholar
Dickson, D.C.M. and Waters, H.R. (1992) The probability and severity of ruin in finite and in infinite time. ASTIN Bulletin, 22, 177–190.CrossRefGoogle Scholar
Dickson, D.C.M. and Willmot, G.E. (2005) The density of the time to ruin in the classical Poisson risk model. ASTIN Bulletin, 35, 45–60.CrossRefGoogle Scholar
Dickson, D.C.M., Hughes, B.D. and Lianzeng, Z. (2005) The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scandinavian Actuarial Journal, 2005(5), 358–376 CrossRefGoogle 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.A. (2005) Explicit solutions for survival probabilities in the classical risk model. ASTIN Bulletin, 35, 113–130.CrossRefGoogle Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation, Philadelphia, PA.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
Willmot, G.E. and Woo, J.K. (2007) On the class of Erlang mixtures with risk theoretic applications. North American Actuarial Journal, 11(2), 99–115.CrossRefGoogle Scholar