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We propose center-outward superquantile and expected shortfall functions,with applications to multivariate risk measurements, extending the standardnotion of value at risk and conditional value at risk from the real line to$\RR^d$. Our new concepts are built upon the recent definition ofMonge-Kantorovich quantiles based on the theory of optimal transport, and theyprovide a natural way to characterize multivariate tail probabilities andcentral areas of point clouds. They preserve the univariate interpretation of atypical observation that lies beyond or ahead a quantile, but in a meaningfulmultivariate way. We show that they characterize random vectors and theirconvergence in distribution, which underlines their importance. Our newconcepts are illustrated on both simulated and real datasets.

Monge-Kantorovich superquantiles and expected shortfalls with applications to multivariate risk measurements