Multivariate copula-dependent distortion risk measures

In this talk, we will introduce two new classes of multivariate risk measures, which are referred to as multivariate copula-dependent distortion risk measures. We define and axiomatically characterize the class of multivariate scalar copula-dependent distortion risk measures through the tool of multivariate Choquet integral. As a by-product, this characterization can also be regarded as a multivariate extension of the univariate Greco's Representation Theorem. Furthermore, based on the representations for the multivariate scalar copula-dependent distortion risk measures, we will introduce the class of multivariate vector-valued copula-dependent distortion risk measures, and their properties of copula-dependent monotonicity, translation invariance, positive homogeneity and pi-comonotone additivity are shown. Finally, we present several examples, among which one example introduces a new class of vector-valued risk measures, while the others demonstrate the comparisons of the introduced multivariate vector-valued distortion risk measures with those vector-valued risk measures known as in the literature. This talk is based on a joint work with Suo Gong and Linxiao Wei.