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

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  • Max Nendel
  • Alessandro Sgarabottolo

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.

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  • Max Nendel & Alessandro Sgarabottolo, 2022. "A parametric approach to the estimation of convex risk functionals based on Wasserstein distance," Papers 2210.14340, arXiv.org, revised Aug 2024.
    • Handle: RePEc:arx:papers:2210.14340
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    References listed on IDEAS

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    Cited by:

    1. Michael Kupper & Max Nendel & Alessandro Sgarabottolo, 2023. "Risk measures based on weak optimal transport," Papers 2312.05973, arXiv.org.
    2. Ariel Neufeld & Matthew Ng Cheng En & Ying Zhang, 2024. "Robust SGLD algorithm for solving non-convex distributionally robust optimisation problems," Papers 2403.09532, arXiv.org.
    3. Daniel Bartl & Ariel Neufeld & Kyunghyun Park, 2023. "Sensitivity of robust optimization problems under drift and volatility uncertainty," Papers 2311.11248, arXiv.org.
    4. Daniel Bartl & Johannes Wiesel, 2022. "Sensitivity of multiperiod optimization problems in adapted Wasserstein distance," Papers 2208.05656, arXiv.org, revised Jun 2023.

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