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

TL;DR

This paper introduces a neural network-based method for estimating convex risk functionals that account for model uncertainty using Wasserstein distance, with applications in finance and actuarial science.

Contribution

It develops a parametric framework for risk assessment under Wasserstein perturbations, enabling numerical approximation and incorporating additional constraints.

Findings

Neural network algorithms effectively estimate risk functionals.

Parametric models capture model uncertainty with Wasserstein distance.

Method accommodates mean and martingale constraints.

Abstract

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 allow for perturbations around a baseline model, measured via Wasserstein distance, and we investigate to which extent this form of probabilistic imprecision can be parametrized. The aim is to come up with a convex risk functional that incorporates a sefety margin with respect to nonparametric uncertainty and still can be approximated through parametrized models. The particular form of the parametrization allows us to develop a numerical method, based on neural networks, which gives both the value of the risk functional and the optimal perturbation of the reference measure. Moreover, we study the problem under additional constraints on the perturbations, namely, a mean and a martingale constraint. 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 a numerical approximation via neural networks.