Bounds on integrals with respect to multivariate copulas

In this paper, we present a method to obtain upper and lower bounds on integrals with respect to copulas by solving the corresponding assignment problems (AP's). In their 2014 paper, Hofer and Iaco proposed this approach for two dimensions and stated the generalization to arbitrary dimensons as an open problem. We will clarify the connection between copulas and AP's and thus find an extension to the multidimensional case. Furthermore, we provide convergence statements and, as applications, we consider three dimensional dependence measures as well as an example from finance.