A parametric approach to the estimation of convex risk functionals based on Wasserstein distance

In this paper, we explore a static setting for the assessment of risk in the context of mathematical finance and actuarial science that takes into account model uncertainty in the distribution of a possibly infinite-dimensional risk factor. We study convex risk functionals that incorporate a safety margin with respect to nonparametric uncertainty by penalizing perturbations from a given baseline model using Wasserstein distance. We investigate to which extent this form of probabilistic imprecision can be approximated by restricting to a parametric family of models. The particular form of the parametrization allows to develop numerical methods based on neural networks, which give both the value of the risk functional and the worst-case perturbation of the reference measure. Moreover, we consider additional constraints on the perturbations, namely, mean and martingale constraints. We show that, in both cases, under suitable conditions on the loss function, it is still possible to estimate the risk functional by passing to a parametric family of perturbed models, which again allows for numerical approximations via neural networks.