Aggregating risks with partial dependence information

We consider the problem of aggregating dependent risks in the presence of partial dependence information. More concretely, we assume that the risks involved belong to independent subgroups and the dependence structure within each group is unknown. We show that a sharp convex upper bound exists in this setting and that the constrained upper bound improves the existing, unconstrained, comonotonic up- per bound in convex order. Moreover, we characterize the constrained upper bound in terms of the distribution of its sum. Numerical illustrations are provided to show the improvement of the new upper bound.