×

zbMATH — the first resource for mathematics

Simple approximations of ruin probabilities. (English) Zbl 0986.62086
Summary: A “simple approximation” of a ruin probability is an approximation using only some moments of the claim distribution and not the detailed tail behaviour of that distribution. Such approximations may be based on limit theorems or on more or less ad hoc arguments. The most successful simple approximation is certainly the de Vylder approximation [F. de Vylder, Scand. Actuar. J. 1978, 114-119 (1978)] which is based on the idea to replace the risk process with a risk process with exponentially distributed claims such that the three first moments coincide. That approximation is known to work extremely well for “kind” claim distributions.
The main purpose of this paper is to analyse the de Vylder approximation and other simple approximations from a more mathematical point of view and to give a possible explanation why the de Vylder approximation is so good.

MSC:
62P05 Applications of statistics to actuarial sciences and financial mathematics
62E17 Approximations to statistical distributions (nonasymptotic)
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Asmussen, S., Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue, Journal of applied probability, 14, 143-170, (1982) · Zbl 0501.60076
[2] Asmussen, S., 2000. Ruin Probability. World Scientific, Singapore, in press.
[3] Asmussen, S.; Klüppelberg, C., Large deviation results for subexponential tails, with applications to insurance risk, Stochastic processes and their applications, 64, 103-125, (1996) · Zbl 0879.60020
[4] Beekman, J., 1969. A ruin function approximation. Transactions of the Society of Actuaries 21, 41-48, 275-279.
[5] Benckert, L.-G.; Jung, J., Statistical models of claim distribution in fire insurance, Astin bulletin, 7, 1-25, (1974)
[6] Cramér, H., 1930. On the Mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm. In: Martin-Löf, A. (Ed.), Harald Cramér Collected Works, Vol. I. Springer, Berlin, 1994, pp. 601-678.
[7] Cramér, H., 1945. Mathematical Methods of Statistics. Almqvist & Wiksell/Princeton University Press, Stockholm/Princeton.
[8] Cramér, H., 1955. Collective Risk Theory. Skandia Jubilee Volume, Stockholm. In: Martin-Löf, A. (Ed.), Harald Cramér Collected Works, Vol. II. Springer, Berlin, 1994, pp. 1028-1115.
[9] De Vylder, F.E., 1978. A practical solution to the problem of ultimate ruin probability. Scandinavian Actuarial Journal, 114-119.
[10] De Vylder, F.E., 1996. Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Université de Bruxelles and Swiss Association of Actuaries.
[11] Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[12] Embrechts, P.; Veraverbeke, N., Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance: mathematics and economics, 1, 55-72, (1982) · Zbl 0518.62083
[13] Grandell, J., 1977. A class of approximations of ruin probabilities. Scandinavian Actuarial Journal (Suppl.), 38-52. · Zbl 0384.60057
[14] Grandell, J., 1978. A remark on ‘A class of approximations of ruin probabilities’. Scandinavian Actuarial Journal, 77-78. · Zbl 0389.62082
[15] Grandell, J., Approximate waiting times in thinned point processes, Liet. matem. rink. 20 nr., 4, 29-47, (1980) · Zbl 0461.60069
[16] Grandell, J., 1991. Aspects of Risk Theory. Springer, New York. · Zbl 0717.62100
[17] Grandell, J., 1997. Mixed Poisson Processes. Chapman & Hall, London. · Zbl 0922.60005
[18] Grandell, J., Segerdahl, C.-O., 1971. A comparison of some approximations of ruin probabilities, Skand. AktuarTidskr., 144-158. · Zbl 0246.62096
[19] Hadwiger, H., Über die wahrscheinlichkeit des ruins bei einer grossen zahl von geschäften, Arkiv für mathematische wirtschaft-und sozialforschung, 6, 131-135, (1940) · JFM 66.0605.01
[20] Kalashnikov, V., 1997. Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic Publishers, Dordrecht. · Zbl 0881.60043
[21] Lundberg, F., 1926. Försäkringsteknisk Riskutjämning. F. Englunds boktryckeri A.B., Stockholm.
[22] Lundberg, O., 1964. On Random Processes and their Application to Sickness and Accident Statistics, 1st Edition. Almqvist & Wiksell, Uppsala. · JFM 66.0678.01
[23] Pakes, A.G., On the tails of waiting-time distributions, Journal of applied probability, 12, 555-564, (1975) · Zbl 0314.60072
[24] Rényi, A., 1956. A Poisson-folyamat egy jellemzíse, Magyar Tud. Akad., Mat. Kutató, Int. Közl. 1, 519-527. English translation: A characterization of Poisson processes. In: Turán, P. (Ed.), Selected Papers of Alfred Rényi, Vol. 1. Akadémiai Kiadó, Budapest, 1976, pp. 622-628.
[25] Thorin, O.; Wikstad, N., Calculation of ruin probabilities when the claim distribution is lognormal, Astin bulletin, 9, 231-246, (1977)
[26] Wikstad, N., Exemplification of ruin probabilities, Astin bulletin, 6, 147-152, (1971)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.