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Recursive Methods for Computing Finite-Time Ruin Probabilities for Phase-Distributed Claim Sizes

Published online by Cambridge University Press:  29 August 2014

D.A. Stanford  and
K.J. Stroiński
D.A. Stanford
Affiliation:
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada, N6A 5B7
K.J. Stroiński
Affiliation:
Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, Canada, N6A 5B7
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Abstract

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Finite time ruin methods typically rely on diffusion approximations or discretization. We propose a new method by looking at the surplus process embedded at claim instants and develop a recursive scheme for calculating ruin probabilities. It is assumed that claim sizes follow a phase-type distribution. The proposed method is exact. The application of the method reveals where in the future the relative vulnerability to the company lies.

Keywords

Finite-time ruin probabilityRecursive methodsPhase-type distributionsRisk theory
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 24 , Issue 2 , November 1994 , pp. 235 - 254
Copyright
Copyright © International Actuarial Association 1994

References

Arfwedson, G. (1950) Some problems in the collective theory of risk. Skand. Aktuartidskr. 1–38.Google Scholar
Asmussen, S. and Bladt, M. (1992) Phase-type distributions and risk processes with state-dependent premiums. Technical Report R92-2022, The University of Aalborg.Google Scholar
Asmussen, S. and Rolski, T. (1991) Computational methods in risk theory: a matrix-algorithmic approach. Technical Report R91-13, The University of Aalborg.Google Scholar
Beekman, J. A. (1966) Research on the collective risk stochastic process. Skand. Aktuartidskr. 65–77.Google Scholar
Bowers, N.L., Gerber, H.V., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Itasca, Illinois.Google Scholar
Cramér, M. (1955) Collective Risk Theory. Jubilee Volume of Försäkringsaktiebogalet Skandia.Google Scholar
De Vylder, F. and Goovaerts, M.J. (1988) Recursive calculations of finite-time ruin probabilities. Insurance: Mathematics and Economics 7, 1–8.Google 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 Waters, H.R. (1992) The probability of severity of ruin in finite and infinite time. ASTIN Bulletin 22, 177–190.CrossRefGoogle Scholar
Dufresne, F. and Gerber, H. U. (1988) The probability and severity of ruin for combinations of exponential claim amount distributions and their Translations. Insurance: Mathematics and Economics 7, 75–80.Google Scholar
Garrido, J. (1988) Diffusion premiums for claim severities subject to inflation. Insurance: Mathematics and Economics 7, 123–129.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Monograph No. 8, S.S. Huebuer Foundation, Distributed by R. Irwin, Homewood, IL.Google Scholar
Gerber, H.U., Goovaerts, M.J. and Kaas, R. (1987) On the probability and severity of ruin. ASTIN Bulletin 17, 151–163.CrossRefGoogle Scholar
Janssen, J. (1980) Some transient results on the M/SM/1 special semi-Markov model in risk and queueing theories. ASTIN Bulletin 11, 41–51.CrossRefGoogle Scholar
Janssen, J. (1982) On the interaction between risk and queueing theories. Blätter 15, 383–395.CrossRefGoogle Scholar
Janssen, J. and Deilfosse, P. (1982) Some numerical aspects in transient risk theory. ASTIN Bulletin 13, 99–113.CrossRefGoogle Scholar
Neuts, M. F. (1981) Matrix-geometric Solutions in Stochastic Models. John Hopkins University Press, Baltimore.Google Scholar
Panjer, H.H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 22–26.CrossRefGoogle Scholar
Prabhu, N.U. (1961) On the ruin problem of collective risk theory. Annals of Math. Statistics 32, 757–764.CrossRefGoogle Scholar
Prabhu, N.U. (1965) Queues and Inventories, John Wiley & Sons, New York.Google Scholar
Shiu, E.S.W. (1988) Calculation of the probability of eventual ruin by Beekman's convolution series. Insurance: Mathematics and Economics 7, 41–47.Google Scholar
Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes, John Wiley & Sons, New York.Google Scholar
Taylor, G.C. (1978) Representation and explicit calculation of finite time ruin probabilities. Scan. Actuarial Journal 78 (1), 1–18.Google Scholar
Taylor, G.C. (1985) A heuristic review of some ruin theory results. ASTIN Bulletin 15, 73–88.CrossRefGoogle Scholar
Thorin, O. (1968) An identity in the collective risk theory with some applications. Skand. Aktuartidskr. 26–44.Google Scholar
Willmot, G. (1990) A queueing theoretic approach to the analysis of the claims payment process. Transactions of the Society of Actuaries, XLII, 447–497.Google Scholar