Wasserstein perturbations of Markovian transition semigroups - Sven Fuhrmann; Michael Kupper; Max Nendel (Bielefeld University, Center for Mathematical Economics (IMW), 2021)
In this paper, we deal with a class of time-homogeneous continuous-time Markov processes with transition probabilities bearing a nonparametric uncertainty. The uncertainty is modelled 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 satisfying a nonlinear PDE in a viscosity sense. A remarkable observation is that, in standard situations, the nonlinear transition operators arising from nonparametric uncertainty coincide with the ones related to parametric drift uncertainty. On the level of the generator, the uncertainty is reflected as an additive perturbation in terms of a convex functional of first order derivatives. We additionally provide sensitivity bounds for the convex semigroup relative to the reference model. The results are illustrated with Wasserstein perturbations of Levy processes, infinite-dimensional Ornstein-Uhlenbeck processes, geometric Brownian motions, and Koopman semigroups.
urn:nbn:de:0070-pub-29548628
Keywords: 330; Wasserstein distance; nonparametric uncertainty; convex semigroup; nonlinearPDE; viscosity solution