Seminar by Max Nendel

On Model Uncertainty and Risk Measures with Applications in Finance and Actuarial Science

Abstract. In mathematical finance and actuarial science, model uncertainty is an al-most omnipresent phenomenon, which appears, for example, due to imperfect knowledge about certain aspects of a random factor or insuÿcient data to perform reliable statis-tical estimation methods for the parameters of an underlying distribution. In this talk, we first give an overview over several topics related to the mathematical treatment and axiomatization of model uncertainty, e.g., in the context of insurance premia, default risk, and asset pricing. We then explore in more detail computational aspects related to the assessment of risk, taking into account model uncertainty in the form of pertur-bations around a baseline model. Such perturbations are described as models that are suÿciently close to a reference model with respect to an optimal transport distance. We investigate to which extent these forms of probabilistic imprecision can be described by parametric families of probability measures or nonlinear transforms. These descriptions form the basis for numerical methods, based on neural networks, which allow to com-pute both the value of the related risk measure and the worst-case perturbation of the reference measure. In a last step, we consider time-dependent families of risk measures, whose penalty functions are assumed to satisfy a certain growth condition in terms of an optimal transport problem around a given reference dynamics. After passing to a suitable scaling limit, we show that the related dynamic risk measure collapses to the value func-tion of a drift or volatility control problem. The talk is based on joint work with Jonas Blessing, Robert Denk, Sven Fuhrmann, Michael Kupper, Frank Riedel, Maren-Diane Schmeck, Alessandro Sgarabottolo, and Jan Streicher.