Robust Lambda-quantiles and extremal distributions

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