We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\mathbb{R}^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal transport, and they provide a natural way to characterize multivariate tail probabilities and central areas of point clouds. They preserve the univariate interpretation of a typical observation that lies beyond or ahead a quantile, but in a meaningful multivariate way. We show that they characterize random vectors and their convergence in distribution, which underlines their importance. Our new concepts are illustrated on both simulated and real datasets.

翻译：我们提出了中心外向超分位数和预期亏损函数，并将其应用于多元风险度量，将标准的风险价值与条件风险价值概念从实数轴扩展到$\mathbb{R}^d$。这些新概念建立在基于最优传输理论的Monge-Kantorovich分位数最新定义之上，为刻画多元分布的尾部概率和点云中心区域提供了自然方法。它们保持了单变量情形下观测值超出或领先于分位数的直观解释，但在多元意义下更具内涵。我们证明这些新概念能表征随机向量及其依分布收敛性，这凸显了其重要性。最后通过模拟数据和真实数据集对上述新概念进行了验证。