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Ultimate Ruin Probabilities for Generalized Gamma-Convolutions Claim Sizes

Published online by Cambridge University Press:  29 August 2014

M. Usábel
M. Usábel*
Affiliation:
Universidad Carlos III de Madrid
*
Universidad Carlos III de Madrid, Avda. Universidad Carlos, 22, Colmenarejo 28270 (Madrid), Spain E-mail: [email protected]
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Abstract

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A method of inverting the Laplace transform based on the integration between zeros technique and a simple acceleration algorithm is presented. This approach was designed to approximate ultimate ruin probabilities for Γ-convolutions claim sizes, but it can be also used with other distributions. The stable algorithm obtained yields interval approximations (lower and upper bounds) to any desired degree of accuracy even for very large values of u (1,000,000), initial reserves, without increasing the number of computations. This last fact can be considered an interesting property compared with other recursive methods previously used in actuarial literature or other methods of inverting Laplace transforms.

Keywords

Ultimate ruin probabilityupper and lower boundsstable recursive algorithmsnumerical inversion of the Laplace transform
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 31 , Issue 1 , May 2001 , pp. 59 - 79
Copyright
Copyright © International Actuarial Association 2001

References

Abate, J. and Whitt, W. (1992) The Fourier-series method for inverting transforms of probability distributions. Queueing Systems 10, 5–88.CrossRefGoogle Scholar
Abate, , and Whitt, W. (1995) Numerical inversion of Laplace transforms of probability distributions. ORSA Journal on Computing 7, No. 1, 36–43.CrossRefGoogle Scholar
Abate, , and Whitt, W. (1995) An operational calculus for probability distributions via Laplace Transforms. Advances in Applied Probability, 28, 75–113.CrossRefGoogle Scholar
Abate, J., Choudhury, G. and Whitt, W. (1994) Waiting-time tail probabilities in queues with long-tail service-time distribution. Queueing Systems, 16, 311–338.CrossRefGoogle Scholar
Abate, J., Choudhury, G. and Whitt, W. (1995) Calculating the M/G/1 busy period density and LIFO waiting-time distribution by direct numerical transform inversion. Operations Research Letters 18, 113–119.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. Ninth Edition. New York, N.Y. Dover Publications.Google Scholar
Berg, C. (1981) The Pareto distribution is a generalized Γ-convolution – a new proof. Scan. Act. Journal, 117–119.CrossRefGoogle Scholar
Boas, R. (1987) Invitation to complex analysis. The Random House/Birkhäuser mathematics series. New York.Google Scholar
Bohman, H. (1971) Ruin probabilities. Skandinavisk Aktuarietidskrift, 159–163.Google Scholar
Bohman, H. (1974) Fourier Inversion-Distribution functions-Long tails. Scand. Actuarial Journal, 43–45.CrossRefGoogle Scholar
Bohman, H. (1975) Numerical inversion of characteristic functions. Scan. Act. Journal, 121–124.CrossRefGoogle Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbith, C.J. (1997) Actuarial Mathematics. Society of Actuaries.Google Scholar
Cai, J. and Garrido, J. (1998) Aging properties and bounds for ruin probabilities and stop-loss premiums. Insurance: Mathematics and Economics 23, Number 1, 33–44.Google Scholar
Choudhury, M.L., Gupt, U.C., Agarwal, M. (1992) Exact and approximate numerical solutions to steady-state single-server queues: M/G/1 – a unified approach. Queueing Systems 10, 351–380.CrossRefGoogle Scholar
Coudhury, M.L. and Whitt, W. (1997) Scaling for Numerical Transform Inversion to analyze Teletraffic Models. ITC 15, 933–942.Google Scholar
Conway, J. (1978) Functions of one complex variable I. Second edition. Springer-Verlag. New York.CrossRefGoogle Scholar
Cramèr, H. (1955) Collective Risk Theory. Jubille Volume of F. Skandia.Google Scholar
Davies, B. and Martin, B. (1979) Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of computational physics 33, 1–32.CrossRefGoogle Scholar
Davies, J.D. and Rabinowitz, P. (1984) Methods of numerical Integration. Second edition. Academic Press Inc. Boston.Google Scholar
Dickson, D.C.M. (1989) Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics 8, 145–148.Google Scholar
Dickson, D.C.M., Egidio dos Reis, A.D., Waters, H.R. (1995) Some Stable Algorithms in Ruin Theory and their Application. 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
Embrechts, P. and Veraverbeke, N. (1982) Estimates for Probability of Ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1, 55–72.Google Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. Huebner Foundation Monograph 8. Philadelphia, Pa., University of Pennsylvania.Google Scholar
Gerber, H.Goovaerts, M., Kaas, R. (1987) On the probability and severity of ruin. ASTIN Bulletin 17, n. 2, 151–163.CrossRefGoogle Scholar
Glynn, P. and Whitt, W. (1995) Heavy-traffic extreme-value limits for queues. Operations Research Letters 18, 107–111.CrossRefGoogle Scholar
Goovaerts, M.D'Hooge, L., De Pril, N. (1977) On a class of generalized Γ-convolutions. Scan. Act. Journal, 21–30.CrossRefGoogle Scholar
Goovaerts, M. and de Vylder, F. (1984) A Stable Recursive Algorithm for Evaluation of Ultimate Ruin Probabilities. ASTIN Bulletin 14, 53–59.CrossRefGoogle Scholar
Gradshteyn, I.S. and Ryzhik, I.M. (1994) Table of Integrals, Series and Products. Fifth edition. Academic Press.Google Scholar
Heckman, P. and Meyers, G. (1983) The calculation of aggregate loss distributions from claim severity and claim count distributions. Proceedings of the Casuality Actuarial Society LXX, 22–61.Google Scholar
Hosono, T. (1984) Fast Inversion of the Laplace Transform by Basic. Kyoritsu Publishers, Japan.Google Scholar
Joyce, D.C. (1971) Survey of extraolation processes in numerical analysis. SI AM Rev. 13, 435–490.CrossRefGoogle Scholar
Kendall, M. and Stuart, A. (1977) The Advanced Theory of Statistics. Vol. I, 4th Edition. MacMillan.Google Scholar
Levin, D. (1973) Development of non-linear transformations for improving convergence of sequences. Internat. J. Comput. Math. 3, 371–388.CrossRefGoogle Scholar
Longman, I.M. (1956) Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Philos. Soc. 52, 764–768.CrossRefGoogle Scholar
Panjer, H.H. (1986) Direct Calculation of Ruin Probabilities. Journal of Risk and Insurance 53, 521–529.CrossRefGoogle Scholar
Panjer, H.H., Wang, S. (1993) On the Stability of recursive Formulas. ASTIN Bulletin 23, 227–258.CrossRefGoogle Scholar
Panjer, H.H., Willmot, G.E. (1992) Insurance risk Models. Society of Actuaries, Schaumburg.Google Scholar
Piessens, R. (1969) New quadrature formulas for the numerical inversion of Laplace transforms. BIT 9, 351–361.CrossRefGoogle Scholar
Piessens, R. (1971) Gaussian Quadrature Formulas for the Numerical Integration of Bromwich's integral and the Inversion of the Laplace Transform. Journal of Engineering Math. 5, 1–5.CrossRefGoogle Scholar
Piessens, R. and Branders, M. (1971) Numerical Inversion of the Laplace Transform using generalised Laguerre Polynomials. Proceedings of the IEE 118, No. 10.Google Scholar
Ramsay, C.M. (1992a) A Practical Algorithm for Approximating the Probability of Ruin. Transactions of the Society of Actuaries XLIV, 443–59.Google Scholar
Ramsay, C.M. (1992b) Improving Goovaerts' and De Vylder's Stable Recursive Algorithm. ASTIN Bulletin 22, 51–59.CrossRefGoogle Scholar
Ramsay, C.M. and Usäbel, M.A. (1997). Calculating Ruin probabilities via Product integration. ASTIN Bulletin 27, n. 2, 263–271.CrossRefGoogle Scholar
Seal, H. (1971) Numerical calculation of the Bohman-Esscher family of convolution-mixed negative binomial distribution functions. Mitt. Verein. Schweiz. Versich.-Mathr. 71, 71–94.Google Scholar
Seal, H. (1974) The numerical calculation of U(w, t), the probability of non-ruin in an interval (0, t). Scan. Act. Journal, 121–139.CrossRefGoogle Scholar
Seal, H.L. (1975) A note on the use of Laguerre polynomials in the inversion of Laplace transforms. Blätter der DAVM 12, 131–134.CrossRefGoogle Scholar
Seal, H. (1977) Numerical inversion of characteristic functions. Scan. Act. Journal, 48–53.CrossRefGoogle Scholar
Thorin, O. (1970) Some remarks on the ruin problem in case the epochs of claims form a renewal process. Skandinavisk Aktuarietidskrift, 29–50.Google Scholar
Thorin, O. (1971) Further remarks on the ruin problem in case the epochs of claims form a renewal process. Skandinavisk Aktuarietidskrift, 14–38, 121–142.Google Scholar
Thorin, O. (1973) The ruin problem in case the tail of a distribution is completely monotone. Skandinavisk Aktuarietidskrift, 100–119.Google Scholar
Thorin, O. (1977) Ruin probabilities prepared for numerical calculation. Scandinavian Actuarial Journal.CrossRefGoogle Scholar
Thorin, O. (1977) On the infinite divisibility of the Pareto distribution. Scan. Act. Journal, 31–40.CrossRefGoogle Scholar
Thorin, O. (1977) On the infinite divisibility of the Lognormal distribution. Scan. Act. Journal, 121–148.CrossRefGoogle Scholar
Thorin, O. (1978) An extension of the notion of a generalized Γ-convolution. Scan. Act. Journal, 141–149.CrossRefGoogle Scholar
Thorin, O. and Wikstad, N. (1973) Numerical evaluation of ruin probabilities for a finite period. ASTIN Bulletin VII:2, 138–153.Google Scholar
Usábel, M. (1999) Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique. IME 25, 133–142.Google Scholar
van de Vooren, A.I. and van Linde, H.J. (1966) Numerical calculation of integrals with strongly oscillating integrand. Math. Comp., 232–245.Google Scholar
Wikstad, N. (1971) Exemplifications of ruin probabilities. ASTIN Bulletin 6, part 2.CrossRefGoogle Scholar
Wikstad, N. (1977) How to calculate Ruin probabilities according to the classical Risk Theory. Scand. Actuarial Journal.CrossRefGoogle Scholar
Willekens, E. and Teugels, J.L. (1992) Asymptotic expansions for waiting time probabilities in a M/G/1 with long-tailed service time. Queueing Systems 10, 295–312.CrossRefGoogle Scholar