Robust Lambda-quantiles and extremal distributions

Abstract

In this paper, we investigate the robust models for $\hat{I}›$-quantiles with partial information regarding the loss distribution, where $\hat{I}›$-quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function $\hat{I}›$. We find that, under some assumptions, the robust $\hat{I}›$-quantiles equal the $\hat{I}›$-quantiles of the extremal distributions. This finding allows us to obtain the robust $\hat{I}›$-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 $\hat{I}›$-quantiles by deriving the extremal distributions for each uncertainty set. These results are applied to optimal portfolio selection under model uncertainty.

33 pages