A note on the adapted weak topology in discrete time

TL;DR

This paper investigates the adapted weak topology in discrete time, showing all such topologies coincide and providing new characterizations, including on Markov processes and measures, using topological and optimal transport methods.

Contribution

It recovers a key result that all adapted topologies in discrete time are the same and introduces new characterizations using topological and optimal transport techniques.

Findings

All adapted topologies in discrete time coincide.

New characterizations of the adapted weak topology on Markov processes.

A description of the classical weak topology via a weak Wasserstein metric.

Abstract

The adapted weak topology is an extension of the weak topology for stochastic processes designed to adequately capture properties of underlying filtrations. With the recent work of Bart--Beiglb\"ock-P. as starting point, the purpose of this note is to recover with topological arguments the intriguing result by Backhoff-Bartl-Beiglb\"ock-Eder that all adapted topologies in discrete time coincide. We also derive new characterizations of this topology including descriptions of its trace on the sets of Markov processes and processes equipped with their natural filtration. To emphasize the generality of the argument, we also describe the classical weak topology for measures on $\mathbb R^d$ by a weak Wasserstein metric based on the theory of weak optimal transport initiated by Gozlan-Roberto-Samson-Tetali.