Mathias Beiglböck: Adapted optimal transport

Abstract

Adapted transport theory and adapted Wasserstein distance aim to extend classical transport theory for probability measures to the case of stochastic processes. A fundamental difference is that it takes the temporal flow of information into account.

In contrast to other topologies for stochastic processes, one obtains that probabilistic operations such as the Doob-decomposition, optimal stopping, and stochastic control are continuous with respect to the adapted Wasserstein distance. Moreover, adapted transport enjoys desirable properties similar to classical theory. E.g. it turns the set of stochastic processes into a geodesic space, isometric to a classical Wasserstein space and allows for Talagrand-type inequalities as recently established by Föllmer.