Max Nendel

Abstract:

In mathematical finance and actuarial science, the assessment of risk is closely related to matching the distribution of the underlying risk factors in an accurate way; a major issue in this process is the so-called model uncertainty or epistemic uncertainty. The latter refer to the impossibility of perfectly capturing the randomness of future states in a single stochastic framework. We first explore a static setting that takes into account model uncertainty in the distribution of a risk factor by allowing for perturbations around a baseline model, measured via Wasserstein distances. We try to understand to which extent this nonparametric form of probabilistic imprecision can be parametrized. In a second step, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is again modeled by considering perturbations of the transition probabilities within a proximity in Wasserstein distance. As a limit over progressively finer time periods, on which the level of uncertainty scales proportionally, we obtain a convex semigroup, which solves a Hamilton-Jacobi-Bellman equation in a viscosity sense. We show that, in standard situations, the nonlinear transition operators arising from Wasserstein uncertainty coincide with the value function of an optimal control problem and provide sensitivity bounds for the convex semigroup relative to the reference model. The talk is based on joint works with Sven Fuhrmann, Michael Kupper, and Alessandro Sgarabottolo.