The Wasserstein barycenter is an important notion in the analysis of high dimensional data with a broad range of applications in applied probability, economics, statistics, and in particular to clustering and image processing. We state a general version of the equivalence of the Wasserstein...

Based on a novel extension of classical Hoeffding-Fréchet bounds, we provide an upper VaR bound for joint risk portfolios with fixed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a...

Optimal transportation w.r.t. the Kantorovich metric l1 (resp. the Wasser- stein metric W1) between two absolutely continuous measures is known since the basic papers of Kantorovich and Rubinstein (1957) and Sudakov (1979) to occur on rays induced by a decomposition of the basic space, which is...

We show that the rearrangement algorithm introduced in Puccetti and Rüschendorf (2012) to compute distributional bounds can be used also to compute sharp lower and upper bounds on the expected value of a supermodular function of d random variables having fixed marginal distributions. Compared...

The problem of the fair allocation of indivisible items is a relevant and challenging economic problem with several applications. For small dimensional frameworks, the problem can be solved exactly by full enumeration of all the possible allocations of the items. For higher dimensional...

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence...

We derive lower and upper bounds for the Value-at-Risk of a portfolio of losses when the marginal distributions are known and independence among (some) subgroups of the marginal components is assumed. We provide several actuarial examples showing that the newly proposed bounds strongly improve...