On a version of a multivariate integration by parts formula for Lebesgue integrals

Abstract

Multidimensional integration by parts formulas apply under the standard assumption that one of the functions is continuous and the other has bounded Hardy-Krause variation. Motivated by recently developed results in the probabilistic context of price and risk bounds, this paper provides a version of an integration by parts formula for the Lebesgue integral of measure-inducing functions which may both be discontinuous and may have infinite Hardy-Krause variation. To this end, we give a general definition of measure-inducing functions and establish various of their properties, such as a characterization in terms of Delta-monotone functions. As a consequence of the integration by parts formula, several convergence results are provided, allowing an extension of the Lebesgue integral of a measure-inducing function to the case where one integrates with respect to a continuous semi-copula. The latter class of aggregation functions includes quasi-copulas which serve as bounds for the dependence structure in many applications.