The Wasserstein space of stochastic processes

2021/04/29 by Daniel Bartl, Bartl, Daniel, Mathias Beiglböck +3 · 9 citations
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2104.14245

openalex publication_date 2021/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 AW, can play a similar role for the class 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 w.r.t. AW. We also show that (FP,AW) is a geodesic space, isometric to a classical Wasserstein space, and that martingales form a closed geodesically convex subspace.

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