A transfer principle for computing the adapted Wasserstein distance between stochastic processes

TL;DR

This paper introduces a transfer principle for the adapted Wasserstein distance between stochastic processes, providing explicit formulas and characterizations for Gaussian and fractional processes, extending the analysis beyond Markov or semi-martingale frameworks.

Contribution

It develops an explicit formula for the adapted Wasserstein distance between Gaussian processes using causal factorization and links it to process representations, covering a broad class of non-Markovian processes.

Findings

Explicit formula for Gaussian processes' distance via causal factorization

Characterization of Gaussian Volterra processes through filtrations

Synchronous coupling attains the distance for fractional SDEs under conditions

Abstract

We propose a transfer principle to study the adapted 2-Wasserstein distance between stochastic processes. First, we obtain an explicit formula for the distance between real-valued mean-square continuous Gaussian processes by introducing the causal factorization as an infinite-dimensional analogue of the Cholesky decomposition for operators on Hilbert spaces. We discuss the existence and uniqueness of this causal factorization and link it to the canonical representation of Gaussian processes. As a byproduct, we characterize mean-square continuous Gaussian Volterra processes in terms of their natural filtrations. Moreover, for real-valued fractional stochastic differential equations, we show that the synchronous coupling between the driving fractional noises attains the adapted Wasserstein distance under some monotonicity conditions. Our results cover a wide class of stochastic processes which are neither Markov processes nor semi-martingales, including fractional Brownian motions and fractional Ornstein--Uhlenbeck processes.