Adapted Wasserstein distance between the laws of SDEs

TL;DR

This paper introduces a novel optimal transport approach for laws of scalar SDEs, establishing the optimality of synchronous coupling and revealing connections to discrete rearrangements, with implications for path-dependent and multidimensional cases.

Contribution

It proves the optimality of synchronous coupling for laws of scalar SDEs and links it to the Knothe--Rosenblatt rearrangement, extending understanding of optimal transport in stochastic processes.

Findings

Optimality of synchronous coupling for scalar SDEs laws

Connection between synchronous coupling and Knothe--Rosenblatt rearrangement

Examples showing coupling variations in path-dependent and multidimensional cases

Abstract

We consider the bicausal optimal transport problem between the laws of scalar time-homogeneous stochastic differential equations, and we establish the optimality of the synchronous coupling between these laws. The proof of this result is based on time-discretisation and reveals a novel connection between the synchronous coupling and the celebrated discrete-time Knothe--Rosenblatt rearrangement. We also prove a result on equality of topologies restricted to a certain subset of laws of continuous-time processes. We complement our main results with examples showing how the optimal coupling may change in path-dependent and multidimensional settings.