×

Practical approximations for multivariate characteristics of risk processes. (English) Zbl 0946.91026

Summary: The applicability aspects of power series expansions with respect to the arrival intensity, based on recursive algorithms, when approximating multivariate finite time ruin probabilities will be substantially enhanced using double Laplace transforms. We will also prove that power series methodology may be considered as an outstanding complement to diffusion approximations when the time horizon considered is not large.

MSC:

91B30 Risk theory, insurance (MSC2010)

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Asmussen, S., 1984. Approximations for the probability of ruin within finite time. Scandinavian Actuarial Journal, 31-57.; Asmussen, S., 1984. Approximations for the probability of ruin within finite time. Scandinavian Actuarial Journal, 31-57. · Zbl 0568.62092
[2] Blanc, J. P.C., The power-series algorithm applied to cyclic polling systems, Stochastic models, 7, 527-545 (1991) · Zbl 0749.60087
[3] Bohman, H., 1971. Ruin probabilities. Skandinavisk Aktuarietidskrift, 159-163.; Bohman, H., 1971. Ruin probabilities. Skandinavisk Aktuarietidskrift, 159-163. · Zbl 0253.60091
[4] Bohman, H., 1974. Fourier inversion-distribution functions-long tails. Scandinavian Actuarial Journal, 43-45.; Bohman, H., 1974. Fourier inversion-distribution functions-long tails. Scandinavian Actuarial Journal, 43-45. · Zbl 0279.60011
[5] Bohman, H., 1975. Numerical inversion of characteristic functions. Scandinavian Actuarial Journal, 121-124.; Bohman, H., 1975. Numerical inversion of characteristic functions. Scandinavian Actuarial Journal, 121-124. · Zbl 0313.60014
[6] Char, B. et al., 1991. Maple V Library Reference Manual. Springer, New York.; Char, B. et al., 1991. Maple V Library Reference Manual. Springer, New York. · Zbl 0763.68046
[7] Cramèr, H., 1955. Collective Risk Theory. Jubille Volume of F. Skandia.; Cramèr, H., 1955. Collective Risk Theory. Jubille Volume of F. Skandia.
[8] Davies, B., Martin, B., 1979. Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of Computational Physics 33.; Davies, B., Martin, B., 1979. Numerical inversion of the Laplace transform: a survey and comparison of methods. Journal of Computational Physics 33. · Zbl 0416.65077
[9] Dickson, C., 1989. Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics 8.; Dickson, C., 1989. Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics 8. · Zbl 0682.62083
[10] Dickson, C., Waters, H., 1992. The probability and severity of ruin in finite and infinite time. ASTIN Bulletin, 22, 2.; Dickson, C., Waters, H., 1992. The probability and severity of ruin in finite and infinite time. ASTIN Bulletin, 22, 2.
[11] Dickson, C., 1993. On the distribution of the claim causing ruin. Insurance: Mathematics and Economics 12.; Dickson, C., 1993. On the distribution of the claim causing ruin. Insurance: Mathematics and Economics 12. · Zbl 0783.62083
[12] Dufresne, F., Gerber, H., 1988. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7.; Dufresne, F., Gerber, H., 1988. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7. · Zbl 0674.62072
[13] Frey, A., Schmidt, V., 1996. Taylor Series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 18.; Frey, A., Schmidt, V., 1996. Taylor Series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 18. · Zbl 0855.62094
[14] Gaver, D. P., Operational Research, 14, 444-459 (1966)
[15] Gerber, H., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin, 172.; Gerber, H., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin, 172.
[16] Gradshteyn, I.S., Ryzhik, I.M., 1994. Table of Integrals, Series and Products, 5th ed. Academic Press, New York.; Gradshteyn, I.S., Ryzhik, I.M., 1994. Table of Integrals, Series and Products, 5th ed. Academic Press, New York. · Zbl 0918.65002
[17] Grandell, J., 1977. A class of approximations of ruin probabilities. Scandinavian Actuarial Journal, 37-52.; Grandell, J., 1977. A class of approximations of ruin probabilities. Scandinavian Actuarial Journal, 37-52. · Zbl 0384.60057
[18] Hooghiemstra, G.; Keane, M.; van de Ree, S., Power series for the stationary distributions of coupled processor models, SIAM Journal of Applied mathematics, 48, 1159-1166 (1988) · Zbl 0652.60097
[19] Iglehart, D. L., Diffusion approximation in collective risk theory, Journal of Applied Probability, 6, 285-292 (1969) · Zbl 0191.51202
[20] Kroese, D.P., Schmidt, V., 1995. Light-traffic analysis for queues with spatially distributed arrivals. Mathematics of Operations Research. 21(1) 135-157.; Kroese, D.P., Schmidt, V., 1995. Light-traffic analysis for queues with spatially distributed arrivals. Mathematics of Operations Research. 21(1) 135-157. · Zbl 0848.60087
[21] Panjer, H. H., Recursive calculation of a family of compound distributions, ASTIN Bulletin, 12, 22-26 (1981)
[22] Piessens, R., New quadrature formulas for the numerical inversion of Laplace transforms, BIT, 9, 351-361 (1969) · Zbl 0194.47105
[23] Rieman, M.; Simon, B., Light traffic limits of sojourn time distributions in Markovian queuing networks, Stochastic models, 4, 191-233 (1988) · Zbl 0651.60090
[24] Schmidli, H.P., 1992. A general insurance risk model. Doctoral Thesis, ETH, Zürich.; Schmidli, H.P., 1992. A general insurance risk model. Doctoral Thesis, ETH, Zürich.
[25] Seal, H., Numerical calculation of the Bohman-Esscher family of convolution-mixed negative binomial distribution functions, Mitt. Verein. schweiz. Versich.-Mathr., 71, 71-94 (1971) · Zbl 0217.51902
[26] Seal, H., 1974. The numerical calculation of \(Uwt\); Seal, H., 1974. The numerical calculation of \(Uwt\) · Zbl 0288.60088
[27] Seal, H., 1977. Numerical inversion of characteristic functions. Scandinavian Actuarial Journal, 48-53.; Seal, H., 1977. Numerical inversion of characteristic functions. Scandinavian Actuarial Journal, 48-53. · Zbl 0365.65075
[28] Simon, B., Calculating light traffic limits for sojourn time distributions in open Markovian queuing systems, Stochastic models, 9, 213-231 (1993) · Zbl 0771.60084
[29] Stehfest, H., 1970. Numerical inversion of Laplace transform. Communications of the ACM, 19 (1).; Stehfest, H., 1970. Numerical inversion of Laplace transform. Communications of the ACM, 19 (1).
[30] Thorin, O., 1970. Some remarks on the ruin problem in case the epochs of claims form a renewal process. Skandinavisk Aktuarietidskrift, 29-50.; Thorin, O., 1970. Some remarks on the ruin problem in case the epochs of claims form a renewal process. Skandinavisk Aktuarietidskrift, 29-50. · Zbl 0218.60082
[31] Thorin, O., Further remarks on the ruin problem in case the epochs of claims form a renewal process, Skandinavisk Aktuarietidskrift, 14-38, 121-142 (1971) · Zbl 0283.62096
[32] Thorin, O., 1973. The ruin problem in case the tail of a distribution is completely monotone. Skandinavisk Aktuarietidskrift, 100-119.; Thorin, O., 1973. The ruin problem in case the tail of a distribution is completely monotone. Skandinavisk Aktuarietidskrift, 100-119. · Zbl 0283.60098
[33] Thorin, O., 1977. Ruin probabilities prepared for numerical calculation. Scandinavian Actuarial Journal.; Thorin, O., 1977. Ruin probabilities prepared for numerical calculation. Scandinavian Actuarial Journal. · Zbl 0368.62101
[34] Thorin, O.; Wikstad, N., Numerical evaluation of ruin probabilities for a finite period, ASTIN Bulletin, VII, 2, 138-153 (1973)
[35] Usábel, M. A., 1999. A note on the Taylor-series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 25, 133-142.; Usábel, M. A., 1999. A note on the Taylor-series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 25, 133-142.
[36] Wikstad, N., 1971. Exemplifications of ruin probabilities. ASTIN Bulletin VI, Part 2.; Wikstad, N., 1971. Exemplifications of ruin probabilities. ASTIN Bulletin VI, Part 2.
[37] Wikstad, N., 1977. How to calculate ruin probabilities according to the classical risk theory. Scandinavian Actuarial Journal.; Wikstad, N., 1977. How to calculate ruin probabilities according to the classical risk theory. Scandinavian Actuarial Journal. · Zbl 0361.62097
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.