On the existence of a trivariate copula with given values of a trivariate quasi-copula at several points

Abstract

The existence of a (multivariate) copula with a given value of a (multivariate) quasi-copula at a single point is known. In the bivariate case, the existence of a copula with given values of a quasi-copula at two or three arbitrary points is also known. In this paper, we give an alternative proof of the existence of a trivariate copula with a given value of a trivariate quasi-copula at a single point. This proof relies on the reformulation of the existence problem as a linear programming minimization problem and its solution by means of the simplex algorithm. The same method is then used to prove the existence of a trivariate copula with given values of a trivariate quasi-copula at two arbitrary points in the unit cube. We furthermore establish a counter-example showing that the existence for given values at three points is not guaranteed. This completes the analysis of the trivariate case.