Absolutely Continuous Curves of Stochastic Processes

TL;DR

This paper extends classical optimal transport theory to the adapted setting of stochastic processes, providing new representations, characterizations of geodesics, and an energy minimization framework in the adapted Wasserstein space.

Contribution

It introduces a probabilistic representation of absolutely continuous curves in the adapted Wasserstein space and derives an adapted Benamou--Brenier formula, extending classical results.

Findings

Characterization of geodesics in the adapted Wasserstein space

Probabilistic representation of absolutely continuous curves

An adapted Benamou--Brenier-type energy minimization formula

Abstract

We study absolutely continuous curves in the adapted Wasserstein space of filtered processes. We provide a probabilistic representation of such curves as flows of adapted processes on a common filtered probability space, extending classical results to the adapted setting. Moreover, we characterize geodesics in this space and derive an adapted Benamou--Brenier-type formula by reformulating adapted optimal transport as an energy minimization problem. As an application, we obtain a Skorokhod-type representation for sequences of filtered processes under the adapted weak topology.