Adapted Wasserstein distance between the laws of SDEs

Understand

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.