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The bi-atomic uniform minimal solution of Schmitter’s problem. (English) Zbl 0906.62107

Summary: The problem posed by H. Schmitter [see Mitt. Ver. Schweiz. Versicherungsmath. 1984, 89-103 (1984; Zbl 0568.62089)] was to maximize the ruin probability when mean and variance of the claim size distribution are given. We prove that the minimal ruin probability is given by the bi-atomic distribution with the maximal possible claim size as one of its mass points. A by-product is a lower bound \(ce^{-\rho u}\) for the ruin probability \(\psi (u)\), where \(\rho\) is the adjustment coefficient, and \(c\) a constant not depending on the allowed claim size distributions.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
60F99 Limit theorems in probability theory
60K05 Renewal theory
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References:

[1] Brockett, P.; Goovaerts, M.; Taylor, G., The Schmitter problem, Astin Bulletin, 21, 1, 129-132 (1991)
[2] De Vylder, F., Advanced Risk Theory (1996), Editions de l’Université de Bruxelles: Editions de l’Université de Bruxelles Brussels
[3] Feller, W., (An Introduction to Probability Theory and Its applications, Vol. 2 (1966), Wiley: Wiley New York) · Zbl 0138.10207
[4] Kaas, R., The Schmitter problem and a related problem: A partial solution, Astin Bulletin, 21, 1, 133-146 (1991)
[5] Kaas, R.; Vanneste, M.; Goovaerts, M., Maximizing compound poisson stop-loss premiums numerically with given mean and variance, Astin Bulletin, 22, 2, 225-234 (1992)
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