Robust Λ-Quantiles and Extremal Distributions

In this paper, we investigate the robust models for Λ-quantiles with partial information regarding the loss distribution, where Λ-quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function Λ. We find that, under some assumptions, the robust Λ-quantiles equal the Λ-quantiles of the extreme probabilities. This finding allows us to obtain the robustΛ-quantiles by applying the results of robust quantiles in the literature. Our results are applied to uncertainty sets characterized by three different constraints respectively: moment constraints, probability distance constraints via Wasserstein metric, and marginal constraints in risk aggregation. We obtain some explicit expressions for robust Λ-quantiles by deriving the extreme probabilities for each uncertainty set. Those results are applied to optimal portfolio selection under model uncertainty.