  We consider settings in which the distribution of a multivariate random
variable is partly ambiguous. We assume the ambiguity lies on the level of the
dependence structure, and that the marginal distributions are known.
Furthermore, a current best guess for the distribution, called reference
measure, is available. We work with the set of distributions that are both
close to the given reference measure in a transportation distance (e.g. the
Wasserstein distance), and additionally have the correct marginal structure.
The goal is to find upper and lower bounds for integrals of interest with
respect to distributions in this set. The described problem appears naturally
in the context of risk aggregation. When aggregating different risks, the
marginal distributions of these risks are known and the task is to quantify
their joint effect on a given system. This is typically done by applying a
meaningful risk measure to the sum of the individual risks. For this purpose,
the stochastic interdependencies between the risks need to be specified. In
practice the models of this dependence structure are however subject to
relatively high model ambiguity. The contribution of this paper is twofold:
Firstly, we derive a dual representation of the considered problem and prove
that strong duality holds. Secondly, we propose a generally applicable and
computationally feasible method, which relies on neural networks, in order to
numerically solve the derived dual problem. The latter method is tested on a
number of toy examples, before it is finally applied to perform robust risk
aggregation in a real world instance.
