We introduce a new algorithm, called the swapping algorithm, to approximate numerically the minimal and maximal expected inner product of two random vectors with given marginal distributions. As a direct application, the algorithm computes an approximation of the L2-Wasserstein distance between...

We introduce the concepts of φ-complete mixability and φ-joint mixability and we investigate some necessary and sufficient conditions to the φ-mixability of a set of distribution functions for some supermodular functions φ. We give examples and numerical verifications which confirm our findings

We introduce the Mixability Detection Procedure (MDP) to check whether a set of d distribution functions is jointly mixable at a given confidence level. The procedure is based on newly established results regarding the convergence rate of the minimal variance problem within the class of joint...

The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges; recent developments in extremal dependence concepts have drawn a lot of attention in probability and...

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...

Most banks use the top-down approach to aggregate their risk types when computing total economic capital. Following this approach, marginal distributions for each risk type are first independently estimated and then merged into a joint model using a copula function. Due to lack of reliable data,...

We introduce a new scalar coefficient to measure linear correlation between random vectors which preserves all the relevant properties of Pearson's correlation in arbitrary dimensions. The new measure and its bounds are derived from a mass transportation approach in which the expected inner...

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...