Before revision:  First we focus on the case where the set of possible distributions of a financial loss is given by the neighborhood of a given baseline distribution in the Wasserstein distance, or more generally, an optimal-transport distance.
After revision:  First we focus on the case where the uncertainty is modeled by a nonlinear expectation penalizing distributions that are "far" in terms of optimal-transport distance (Wasserstein distance for instance) from a given baseline distribution in the Wasserstein distance, or more generally, an optimal-transport distance.