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The Schmitter Problem and a Related Problem: A Partial Solution

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  • Kaas, R.

Abstract

At the 1990 ASTIN-colloquium, Schmitter posed the problem of finding the extreme values of the ultimate ruin probability ψ(u) in a risk process with initial capital u, fixed safety margin θ, and mean μ and variance σ2 of the individual claims. This note aims to give some more insight into this problem. Schmitter's conjecture that the maximizing individual claims distribution is always diatomic is disproved by a counterexample. It is shown that if one uses the distribution maximizing the upper bound e−Ru to find a ‘large’ ruin probability among risks with range [0, b], incorrect results are found if b is large or u small. The related problem of finding extreme values of stop-loss premiums for a compound Poisson (λ) distribution with identical restrictions on the individual claims is analyzed by the same methods. The results obtained are very similar.

Suggested Citation

  • Kaas, R., 1991. "The Schmitter Problem and a Related Problem: A Partial Solution," ASTIN Bulletin, Cambridge University Press, vol. 21(1), pages 133-146, April.
    • Handle: RePEc:cup:astinb:v:21:y:1991:i:01:p:133-146_00
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    Cited by:

    1. De Vylder, F. & Marceau, E., 1996. "The numerical solution of the Schmitter problems: Theory," Insurance: Mathematics and Economics, Elsevier, vol. 19(1), pages 1-18, December.
    2. Hansjörg Albrecher & José Carlos Araujo-Acuna, 2022. "On The Randomized Schmitter Problem," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 515-535, June.
    3. De Vylder, F. & Goovaerts, M. & Marceau, E., 1997. "The solution of Schmitter's simple problem: Numerical illustration," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 43-58, June.
    4. De Vylder, F. & Goovaerts, M. & Marceau, E., 1997. "The bi-atomic uniform minimal solution of Schmitter's problem," Insurance: Mathematics and Economics, Elsevier, vol. 20(1), pages 59-78, June.

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