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Simulation of Ruin Probabilities for Subexponential Claims

Published online by Cambridge University Press:  29 August 2014

S. Asmussen  and
K. Binswanger
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Abstract

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We consider the classical risk model with subexponential claim size distribution. Three methods are presented to simulate the probability of ultimate ruin and we investigate their asymptotic efficiency. One, based upon a conditional Monte Carlo idea involving the order statistics, is shown to be asymptotically efficient in a certain sense. We use the simulation methods to study the accuracy of the standard Embrechts-Veraverbeke [16] approximation for the ruin probability and also suggest a new one based upon ideas of Hogan [21].

Keywords

Conditional Monte Carlocorrected diffusion approximationladder heightsorder statisticsPollaczeck-Khinchine formulaprobability of ultimate ruinrare eventsregular variationsubexponential distribution
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 27 , Issue 2 , November 1997 , pp. 297 - 318
Copyright
Copyright © International Actuarial Association 1997

References

REFERENCES

[1]Abate, J., Choudhury, G. L., and Whitt, W.Waiting-time tail probabilities in queues with long-tailed service-time distributions. Queueing Systems 16 (1994), 311–338.CrossRefGoogle Scholar
[2]Abate, J., and Whitt, W.The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10 (1992), 5–88.CrossRefGoogle Scholar
[3]Arnold, B. C., Balakrishnan, N., and Nagaraja, H. N.A First Course in Order Statistics. Wiley, New York, 1992.Google Scholar
[4]Asmussen, S.Conjugate processes and the simulation of ruin problems. Stochastic Processes and their Applications 20 (1985), 213–229.CrossRefGoogle Scholar
[5]Asmussein, S.Applied Probability and Queues. Wiley, Chichester, 1987.Google Scholar
[6]Asmussen, S.Approximations for the probability of ruin within finite time. Scandinavian Actuarial Journal p 31–57, 1984, (p:64, 1985).Google Scholar
[7]Asmussen, S., and Rubinstein, R.Steady-state Rare Events Simulation in Queueing Models and its Complexity Properties, Advances in Queueing (J. Dhashalou, ed), 79-102. CRC Press, Boca Raton, Florida, 1995.Google Scholar
[8]Binswanger, K.Rare Events and Insurance. ETH Zürich, PhD-thesis No. 12233.Google Scholar
[9]Buchwald, M., Chevallier, E., and Klüppelberg, C.Approximation methods for the total claimsize distribution – an algorithmic and graphical presentation. Mitteilungen der Schweizerischen Vereinigung der Versicherungsmathematiker 2 (1993), 187–226.Google Scholar
[10]De Vylder, F., and Goovaerts, M.Bounds for classical ruin probabilities. Insurance: Mathematics and Economics 3 (1984), 121–131.Google Scholar
[11]Devroye, L.Non-Uniform Random Variate Generation. Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
[12]Dickson, D. C. M.An upper bound for the probability of ultimate ruin. Scandinavian Actuarial Journal 2 (1994), 131–138.CrossRefGoogle Scholar
[13]Dickson, D. C. M.A review of Panjer's recursion formula and its applications. British Actuarial Journal 1 (1995), 107–124.CrossRefGoogle Scholar
[14]Embrechts, P., Grübel, R., and Pitts, S.Some applications of the fast Fourier transform algorithm in insurance mathematics. Statistica Neerlandica 47 (1993), 59–75.CrossRefGoogle Scholar
[15]Embrechts, P., and Klüppelberg, C.Some aspects of insurance mathematics. Theory of Probability and its Applications 39 (1994), 262–295.CrossRefGoogle Scholar
[16]Embrechts, P., and Veraverbeke, N.Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1 (1982), 55–72.Google Scholar
[17]Feilmeier, M., and Bertram, J.Anwendungen numerischer Methoden in der Risikotheorie. Verlag Versicherungswirtschaft E. V., Karlsruhe, 1987.Google Scholar
[18]Glynn, P. W., and Iglehart, D. L.Importance sampling for stochastic simulations. Management Science 35 (1989), 1367–1392.CrossRefGoogle Scholar
[19]Grandell, J.A class of approximations of ruin probabilities. Scandinavian Actuarial Journal Suppl. (1977; 1978: 77-78, 1978), 1977: 37–52.Google Scholar
[20]Heidelberger, P.Fast simulation of rare events in queueing and reliability models. ACM Trans actions on Computer Modeling and Simulation 5 (1995), 43–85.Google Scholar
[21]Hogan, M. L.Comment on corrected diffusion approximations in certain random walk problems. Journal of Applied Probability 23 (1986), 89–96.CrossRefGoogle Scholar
[22]Iglehart, D. L.Diffusion approximations in collective risk theory. Journal of Applied Probability 6 (1969), 285–292.CrossRefGoogle Scholar
[23]Klüppelberg, C.Estimation of ruin probabilities by means of hazard rates. Insurance: Mathematics and Economics 8 (1989), 279–285.Google Scholar
[24]Omey, E., and Willekens, E.Second order behaviour of the tail of a subordinated probability distribution. Stochastic Processes and their Applications 21 (1986), 339–353.CrossRefGoogle Scholar
[25]Omey, E., and Willekens, E.Second order behaviour of distributions subordinate to a distribution with finite mean. Communications in Statistics: Stochastic Models 3(3) (1987), 311–342.Google Scholar
[26]Panjer, H., and Doray, L.Further results on the probability of ruin with an absorbing upper barrier. Transactions of the 23rd International Congress of Actuaries Helsinki (1988), 159–177.Google Scholar
[27]Panjer, H. H., and Willmot, G. E.Insurance Risk Models. Society of Actuaries, 1992.Google Scholar
[28]Rubinstein, R. Y.Simulation and the Monte Carlo Method. Wiley, New York, 1981.CrossRefGoogle Scholar
[29]Siegmund, D.Importance sampling in the Monte Carlo study of sequential tests. The Annals of Statistics 4 (1976), 673–684.CrossRefGoogle Scholar
[30]Siegmund, D.Corrected diffusion approximations in certain random walk problems. Advances in Applied Probability 11 (1979), 701–719.CrossRefGoogle Scholar
[31]Thorin, O., and Wikstad, N.Calculation of the ruin probabilities when the claim distribution is lognormal. ASTIN Bulletin 9 (1979), 231–246.CrossRefGoogle Scholar