Bounding adapted Wasserstein metrics

TL;DR

This paper establishes explicit upper bounds relating the adapted Wasserstein distance to the classical Wasserstein distance, revealing how smoothing influences the topology and providing new insights into metrics for stochastic processes.

Contribution

It provides the first explicit bounds of adapted Wasserstein metrics in terms of Wasserstein metrics, including a novel inequality for Lipschitz kernels and a characterization of the smoothed topology.

Findings

Upper bounds of adapted Wasserstein in terms of Wasserstein distance are derived.

The inequality $ ext{AW}_1 \\le C \\sqrt{\text{W}_1}$ is established for measures with Lipschitz kernels.

The topology induced by the smooth adapted Wasserstein distance interpolates between adapted weak and weak topologies.

Abstract

The Wasserstein distance $\mathcal{W}_p$ is an important instance of an optimal transport cost. Its numerous mathematical properties as well as applications to various fields such as mathematical finance and statistics have been well studied in recent years. The adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p$ extends this theory to laws of discrete time stochastic processes in their natural filtrations, making it particularly well suited for analyzing time-dependent stochastic optimization problems. While the topological differences between $\mathcal{A}\mathcal{W}_p$ and $\mathcal{W}_p$ are well understood, their differences as metrics remain largely unexplored beyond the trivial bound $\mathcal{W}_p\lesssim \mathcal{A}\mathcal{W}_p$. This paper closes this gap by providing upper bounds of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$ through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of $\mathcal{W}_p$, Eder's modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on $\mathcal{W}_p$ automatically hold for $\mathcal{AW}_p$ under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the set of measures that have Lipschitz kernels. Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.