The bi-atomic uniform minimal solution of Schmitterʹs problem

The problem posed by Schmitter was to maximize the ruin probability when mean and variance of the claim size distribution are given. In this note 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 c e−pu for the ruin probability ψ(u), where p is the adjustment coefficient, and c a constant not depending on the allowed claim size distributions.