Multi-Period Multivariate Risk Measures with Extensions to Multivariate Risk Statistics

In this paper, we investigate multi-period multivariate risk measures as well as multivariate risk statistics. To be precise, by proposing new sets of axioms, we introduce two new classes of multi-period coherent and convex multivariate risk measures first, and then we provide dual representations for them. Equivalent conditions for dual representation are also given. Time consistency is also discussed. Moreover, we axiomatically introduce multi-period coherent and convex multivariate risk statistics. Dual representations for these multi-period multivariate risk statistics are derived. This paper provides some relevant theoretical results about the evaluation of multivariate risks in the multi-period setting, and provides a non-trivial generalization of several known achievements in the literature.