Stochastic orderings with respect to a capacity and an application to a financial optimization problem

In an analogous way to the classical case of a probability measure, we extend the notion of an increasing convex (concave) stochastic dominance relation to the case of a normalised monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. We state a new version of the classical upper (resp. lower) Hardy-Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. We apply our results to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity.