February 2022 Estimating processes in adapted Wasserstein distance
Julio Backhoff, Daniel Bartl, Mathias Beiglböck, Johannes Wiesel
Author Affiliations +
Ann. Appl. Probab. 32(1): 529-550 (February 2022). DOI: 10.1214/21-AAP1687


A number of researchers have independently introduced topologies on the set of laws of stochastic processes that extend the usual weak topology. Depending on the respective scientific background this was motivated by applications and connections to various areas (e.g., Plug–Pichler—stochastic programming, Hellwig—game theory, Aldous—stability of optimal stopping, Hoover–Keisler—model theory). Remarkably, all these seemingly independent approaches define the same adapted weak topology in finite discrete time. Our first main result is to construct an adapted variant of the empirical measure that consistently estimates the laws of stochastic processes in full generality.

A natural compatible metric for the adapted weak topology is the given by an adapted refinement of the Wasserstein distance, as established in the seminal works of Pflug–Pichler. Specifically, the adapted Wasserstein distance allows to control the error in stochastic optimization problems, pricing and hedging problems, optimal stopping problems, etcetera in a Lipschitz fashion. The second main result of this article yields quantitative bounds for the convergence of the adapted empirical measure with respect to adapted Wasserstein distance. Surprisingly, we obtain virtually the same optimal rates and concentration results that are known for the classical empirical measure wrt. Wasserstein distance.

Funding Statement

Daniel Bartl is grateful for financial support through the Vienna Science and Technology Fund (WWTF) project MA16-021 and the Austrian Science Fund (FWF) project P28661.
Mathias Beiglböck is grateful for financial support through the Austrian Science Fund (FWF) under project Y782.
Johannes Wiesel acknowledges support by the German National Academic Foundation.


Julio Backhoff is also affiliated with the University of Vienna.


Download Citation

Julio Backhoff. Daniel Bartl. Mathias Beiglböck. Johannes Wiesel. "Estimating processes in adapted Wasserstein distance." Ann. Appl. Probab. 32 (1) 529 - 550, February 2022. https://doi.org/10.1214/21-AAP1687


Received: 1 May 2020; Revised: 1 December 2020; Published: February 2022
First available in Project Euclid: 27 February 2022

MathSciNet: MR4386535
zbMATH: 1492.60104
Digital Object Identifier: 10.1214/21-AAP1687

Primary: 58E30 , 60G42 , 90C46

Keywords: adapted weak topology , empirical measure , nested distance , Wasserstein distance

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 1 • February 2022
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