Robust Λ$\Lambda$‐Quantiles and Extremal Distributions
研究了损失分布信息不完全时鲁棒Λ分位数的模型，发现其等于极值分布的Λ分位数，并应用于矩约束、Wasserstein距离约束和风险聚合边际约束下的不确定性集，得到显式表达式，用于模型不确定下的最优投资组合选择。
ABSTRACT In this paper, we investigate the robust models for ‐quantiles with partial information regarding the loss distribution, where ‐quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function . We find that, under some assumptions, the robust ‐quantiles equal the ‐quantiles of the extremal distributions. This finding allows us to obtain the robust ‐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 ‐quantiles by deriving the extremal distributions for each uncertainty set. These results are applied to optimal portfolio selection under model uncertainty.