Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T03:10:18.255Z Has data issue: false hasContentIssue false
  • English
  • Français

An identity based on the generalised negative binomial distribution with applications in ruin theory

Published online by Cambridge University Press:  10 September 2018

David C. M. Dickson
David C. M. Dickson*
Affiliation:
Centre for Actuarial Studies, Department of Economics, University of Melbourne, Melbourne, Victoria 3010, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study, we show how expressions for the probability of ultimate ruin can be obtained from the probability function of the time of ruin in a particular compound binomial risk model, and from the density of the time of ruin in a particular Sparre Andersen risk model. In each case evaluation of generalised binomial series is required, and the argument of each series has a common form. We evaluate these series by creating an identity based on the generalised negative binomial distribution. We also show how the same ideas apply to the probability function of the number of claims in a particular Sparre Andersen model.

Keywords

Generalised negative binomial distributionGeneralised binomial seriesRuin probabilityTime of ruinErlang distribution
Type
Paper
Information
Annals of Actuarial Science , Volume 13 , Issue 2 , September 2019 , pp. 308 - 319
Copyright
© Institute and Faculty of Actuaries 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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 & Economics, 50, 334–337.Google Scholar
Dickson, D.C.M., Hughes, B.D. & Zhang, L. (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.Google Scholar
Dickson, D.C.M. & Li, S. (2010). Finite time ruin problems for the Erlang(2) risk model. Insurance: Mathematics & Economics, 46, 12–18.Google Scholar
Dickson, D.C.M. & Qazvini, M. (2016). Gerber-Shiu analysis of a risk model with capital injections. European Actuarial Journal, 6, 409–440.Google Scholar
Drekic, S. & Willmot, G.E. (2003). On the density and moments of the time to ruin with exponential claims. ASTIN Bulletin, 33, 11–21.Google Scholar
Frostig, E., Pitts, S.M. & Politis, K. (2012). The time to ruin and the number of claims until ruin for phase-type claims. Insurance: Mathematics & Economics, 51, 19–25.Google Scholar
Gerber, H.U. (1988). Mathematical fun with the compound binomial process. ASTIN Bulletin, 18, 161–168.Google Scholar
Graham, R.L., Knuth, D.E. & Patashnik, O. (1994). Concrete Mathematics, 2nd edition. Addison-Wesley, Upper Saddle River, NJ.Google Scholar
Grandell, J. (1991). Aspects of risk theory. Springer-Verlag, New York.Google Scholar
Jain, G.C. & Consul, P.C. (1971). A generalized negative binomial distribution. SIAM Journal of Applied Mathematics, 21, 501–513.Google Scholar
Landriault, D., Shi, T. & Willmot, G.E. (2011). Joint densities involving the time to ruin in the Sparre Andersen risk model under exponential assumptions. Insurance: Mathematics & Economics, 49, 371–379.Google Scholar
Li, S. & Sendova, K.P. (2013). The finite-time ruin probability under the compound binomial risk model. European Actuarial Journal, 3, 249–271.Google Scholar
Nie, C., Dickson, D.C.M. & Li, S. (2015). The finite time ruin probability in a risk model with capital injections. Scandinavian Actuarial Journal, 2015, 301–318.Google 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
Zhao, C. & Zhang, C. (2013). Joint density of the number of claims until ruin and the time to ruin in the delayed renewal risk model with Erlang(n) claims. Journal of Computational and Applied Mathematics, 244, 102–114.Google Scholar