Robust Λ$\Lambda$‐Quantiles and Extremal Distributions

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