Adapted optimal transport between Gaussian processes in discrete time

TL;DR

This paper introduces an explicit formulation of the adapted 2-Wasserstein distance for Gaussian distributions, providing a new bicausal coupling characterization and an adapted Bures-Wasserstein distance for positive definite matrices.

Contribution

It derives an explicit form of the adapted 2-Wasserstein distance between Gaussian distributions and characterizes the optimal bicausal couplings, extending the Bures-Wasserstein distance.

Findings

Explicit formula for adapted 2-Wasserstein distance between Gaussians

Characterization of optimal bicausal couplings

Extension to adapted Bures-Wasserstein distance

Abstract

We derive explicitly the adapted $2$-Wasserstein distance between non-degenerate Gaussian distributions on $\mathbb{R}^N$ and characterize the optimal bicausal coupling(s). This leads to an adapted version of the Bures-Wasserstein distance on the space of positive definite matrices.