A Probabilistic View on the Adapted Wasserstein Distance

TL;DR

This paper provides a probabilistic framework for the adapted Wasserstein distance, including new representations and equivalences that connect it to classical probability theory and stochastic process concepts.

Contribution

It introduces probabilistic formulations of the adapted Wasserstein distance, including a Skorokhod representation and Markovian lift equivalences, bridging optimal transport and stochastic processes.

Findings

Proves a Skorokhod representation theorem for adapted weak convergence.

Reformulates stochastic process equivalences using Markovian lifts.

Provides an expression for the adapted Wasserstein distance on a common stochastic basis.

Abstract

Causal optimal transport and adapted Wasserstein distance have applications in different fields from optimization to mathematical finance and machine learning. The goal of this article is to provide equivalent formulations of these concepts in classic probabilistic language. In particular, we prove a Skorokhod representation theorem for adapted weak convergence, reformulate the equivalence of stochastic processes using Markovian lifts, and give an expression for the adapted Wasserstein distance based on representing processes on a common stochastic basis.