Researchers from different areas have independently defined extensions of the usual weak topology between laws of stochastic processes. This includes Aldous' extended weak convergence, Hellwig's information topology and convergence in adapted distribution in the sense of Hoover-Keisler. In this talk, we show that on the set of continuous processes with canonical filtration these topologies coincide and are metrized by a suitable adapted Wasserstein distance. Moreover we show that the resulting topology is the weakest topology that guarantees continuity of optimal stopping.

While the set of canonical processes is not complete, we establish that its completion is the space of filtered processes. We also observe that this complete space is Polish, Martingales form a closed subset and approximation results like Donsker's theorem extend to the adapted Wasserstein distance. This talk is based on the joint work with Daniel Bartl, Mathias Beiglböck, Gudmund Pammer and Stefan Schrott.

Speaker(s): Zhang Xin (University of Vienna)