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The Schmitter Problem and a Related Problem: A Partial Solution

Published online by Cambridge University Press:  29 August 2014

R. Kaas
R. Kaas*
Affiliation:
University of Amsterdam, the Netherlands
*
Institute for Actuarial Science and Econometrics, University of Amsterdam, Jodenbreestraat 23, NL-1011 NH Amsterdam, the Netherlands.
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Abstract

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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.

Type
Discussion Papers
Information
ASTIN Bulletin: The Journal of the IAA , Volume 21 , Issue 1 , April 1991 , pp. 133 - 146
Copyright
Copyright © International Actuarial Association 1991

References

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