Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T06:42:49.015Z Has data issue: false hasContentIssue false
  • English
  • Français

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

Published online by Cambridge University Press:  17 April 2015

Attahiru Sule Alfa  and
Steve Drekic
Attahiru Sule Alfa
Affiliation:
Department of Electrical and Computer Engineering, University of Manitoba, 75A Chancellor’s Circle, Winnipeg, Manitoba, Canada R3T 5V6, E-mail: [email protected]
Steve Drekic
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, 200 University Ave. West, Waterloo, Ontario, Canada N2L 3G1, E-mail: [email protected]
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.

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

Keywords

Sparre Andersen modeldiscrete timematrix analytic methodstime of ruinsurplus immediately prior to ruindeficit at ruin
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 37 , Issue 2 , November 2007 , pp. 293 - 317
Copyright
Copyright © ASTIN Bulletin 2007

References

Alfa, A.S. (2004) “Markov chain representations of discrete distributions applied to queueing models”. Computers & Operations Research 31, 2365–2385.CrossRefGoogle Scholar
Alfa, A.S. and Neuts, M.F. (1995) “Modelling vehicular traffic using the discrete time Markovian arrival process”. Transportation Science 29, 109–117.CrossRefGoogle Scholar
Cardoso, R.M.R. and Egidio dos Reis, A.D. (2002) “Recursive calculation of time to ruin distributions”. Insurance: Mathematics & Economics 30, 219–230.Google Scholar
Cheng, S., Gerber, H.U. and Shiu, E.S.W. (2000) “Discounted probabilities and ruin theory in the compound binomial model”. Insurance: Mathematics & Economics 26, 239–250.Google Scholar
Cossette, H., Landriault, D. and Marceau, E. (2006) “Ruin probabilities in the discrete time renewal risk model”. Insurance: Mathematics & Economics 38, 309–323.Google Scholar
De Vylder, F.E. and Marceau, E. (1996) “Classical numerical ruin probabilities”. Scandinavian Actuarial Journal, 109–123.CrossRefGoogle Scholar
Dickson, D.C.M. (1994) “Some comments on the compound binomial model”. ASTIN Bulletin 24, 33–45.CrossRefGoogle Scholar
Dickson, D.C.M. (2005) Insurance Risk and Ruin, Cambridge University Press, Cambridge.CrossRefGoogle 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., Egidio dos Reis, A.D. and Waters, H.R. (1995) “Some stable algorithms in ruin theory and their applications”. ASTIN Bulletin 25, 153–175.CrossRefGoogle Scholar
Dickson, D.C.M. and Waters, H.R. (1991) “Recursive calculation of survival probabilities”. ASTIN Bulletin 21, 199–221.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
Gerber, H.U. (1988) “Mathematical fun with the compound binomial process”. ASTIN Bulletin 18, 161–168.CrossRefGoogle 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
Karlin, S. and Taylor, H.M. (1975) A First Course in Stochastic Processes, 2nd edition, Academic Press, New York.Google Scholar
Li, S. (2005a) “On a class of discrete time renewal risk models”. Scandinavian Actuarial Journal, 241–260.CrossRefGoogle Scholar
Li, S. (2005b) “Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models”. Scandinavian Actuarial Journal, 271–284.CrossRefGoogle Scholar
Li, S. and Garrido, J. (2002) “On the time value of ruin in the discrete time risk model”. Business Economics Series 12, Working Paper 02-18, Universidad Carlos III de Madrid, Spain, 28 pp.Google Scholar
Michel, R. (1989) “Representation of a time-discrete probability of eventual ruin”. Insurance: Mathematics & Economics 8, 149–152.Google Scholar
Pavlova, K.P. and Willmot, G.E. (2004) “The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function”. Insurance: Mathematics & Economics 35, 267–277.Google Scholar
Shiu, E.S.W. (1989) “Probability of eventual ruin in the compound binomial model”. ASTIN Bulletin 19, 179–190.CrossRefGoogle Scholar
Willmot, G.E. (1993) “Ruin probabilities in the compound binomial model”. Insurance: Mathematics & Economics 12, 133–142.Google Scholar