Minimizing the Expected Absolute Difference with a Fixed Continuous Copula

GivenBaz, J. Díaz, I. Montes, S.  a fixed random variable X and a fixed continuous copula C, the existence of a random variable Y such that (X, Y) has copula C and the quantity \(E[|X-Y|]\) is minimum is proved. The proof, based on the Berge Maximum Theorem, also gives the expression of the distribution function of Y and implies a monotonicity property. In addition, the explicit solution for some of the most relevant copulas is stated.