Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T16:32:15.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Simulation of Ruin Probabilities for Subexponential Claims

Published online by Cambridge University Press:  29 August 2014

S. Asmussen  and
K. Binswanger
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.

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