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

TL;DR

This paper introduces multivariate superquantiles and expected shortfalls based on optimal transport theory, extending univariate risk measures to higher dimensions for better multivariate risk assessment.

Contribution

It develops a novel framework for multivariate risk measures using Monge-Kantorovich quantiles, providing a natural extension of univariate concepts to multivariate settings.

Findings

Characterize multivariate tail probabilities and central regions.

Ensure convergence in distribution of multivariate risks.

Demonstrate applicability on simulated and real data.

Abstract

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.