Stochastic Orders Under Uncertainty

We study stochastic order relations under uncertainty. This topic has hardly been considered in the literature so far. It is shown that we cannot expect any reasonable robustness of classical first order stochastic dominance, whereas such properties hold for some versions of almost stochastic dominance, as almost stochastic dominance is naturally related to robustness with respect to the Wasserstein distance. We also review other recent results on almost stochastic dominance under uncertainty where the uncertainty sets are either described by knowing only mean and variance or by knowing only the marginal distribution in a multivariate context.