Bounds on integrals with respect to multivariate copulas

Abstract

Finding upper and lower bounds to integrals with respect to copulas is a quite prominent problem in applied probability. In their 2014 paper [9], Hofer and Iacó showed how particular two dimensional copulas are related to optimal solutions of the two dimensional assignment problem. Using this, they managed to approximate integrals with respect to two dimensional copulas. In this paper, we will further illuminate this connection, extend it to d-dimensional copulas and therefore generalize the method from [9] to arbitrary dimensions. We also provide convergence statements. As an example, we consider three dimensional dependence measures.

On Majorization in Dependence Modeling

“…This procedure yields rigorous bounds but is computationally more costful and thus usually cannot provide the accuracy of the RA. For more details on the LP method and a comparison of the two approaches see [6] and the references therein.…”