The Wasserstein space of stochastic processes

Abstract

Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of classical transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate probabilistic variant, the adapted Wasserstein distance $\mathcal{A}\mathcal{W}$, can play a similar role for the class $\mathrm{FP}$ of filtered processes, i.e., stochastic processes together with a filtration. In contrast to other topologies for stochastic processes, probabilistic operations such as the Doob decomposition, optimal stopping and stochastic control are continuous with respect to $\mathcal{A}\mathcal{W}$. We also show that $(\mathrm{FP},\mathcal{A}\mathcal{W})$ is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace.