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

Abstract

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized 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. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely 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. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

Uncertainty orders on the sublinear expectation space

“…We then introduce the recent results of uncertainty orders in the capacity space introduced by [14,15]. We also establish some characterizations by distortion functions.…”

“…Some characterizations about these uncertainty orders were considered in [14,15], see Propositions 2.6-2.8 and Proposition 3.1 in [15]. …”

“…We give the following definitions, which generalize the definitions of VaR and AVaR under a priori probability measure. These definitions can also be found in [14,15], where Grigorova used the different notations but the economic implications are the same. Definition 3.6.…”

“…On the other hand, we study the uncertainty orders in the sublinear expectation space by a family of probability measures P induced by the sublinear expectation E. We can define the capacity space from P, then we get some characterizations with the help of the recent results about capacity orders obtained by [14,15] in the capacity space. Besides, we also give the characterization of uncertainty orders by distortion functions.…”

Stochastic dominance with respect to a capacity and risk measures

“…In our previous paper [16], motivated by the Choquet expected utility theory, we have generalized some "classical" results on the increasing convex stochastic dominance to the case where the measurable space (Ω, F) is endowed with a given capacity which is not necessarily a probability measure. We have established in particular (cf.…”