Estimating processes in adapted Wasserstein distance

By Julio Backhoff, Daniel Bartl, Mathias Beiglbock, Johannes Wiesel. The Annals of Applied Probability, 2020

54 citations recorded by the source corpus.

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 weak adapted 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, etc. 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.

Subfields: Applied Mathematics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/454f27bc889eda433030d5edbe440ab4fc4aa0e5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Daniel Bartl
2. Mathias Beiglbock
3. Gudmund Pammer
4. Julio Backhoff-Veraguas
5. Nizar Touzi

In Finance

1. Gudmund Pammer
2. Daniel Bartl
3. Mathias Beiglbock
4. Nizar Touzi
5. Marcel Nutz
6. Julio Backhoff-Veraguas

In Management Science and Operations Research

1. Gudmund Pammer
2. Daniel Bartl
3. Mathias Beiglbock

Citations in the corpus

• The Wasserstein space of stochastic processes (2021, background): “While the weak adapted topology / AWp appear canonical and have recently seen a burst of applications (see [28, 64, 65, 35, 8, 2, 76, 66, 50, 58, 68, 67, 77, 9, 46] among others) we also highlight two limitations:”