On s-convex stochastic extrema for arithmetic risks

Recently, Denuit and Lefèvre (Insurance: Mathematics and Economics 20 (1997) 197–213) have introduced a class of discrete s-convex stochastic orderings for comparing arithmetic risks in actuarial sciences inter alia. The present paper is concerned with the construction of the extremal distributions with respect to these orderings. Firstly, the general problem of bounding such risks is studied in some details. Then, improved extrema are obtained for the case where the risks are known to have a decreasing density function. For illustration, the results are applied to derive bounds for the probability of ruin in the compound binomial risk model.