The Wasserstein Space of Stochastic Processes in Continuous Time

TL;DR

This paper introduces a canonical adapted weak topology for continuous stochastic processes, metrized by an adapted Wasserstein distance, and explores its properties and completion, extending classical results to a broader process space.

Contribution

It defines a unified adapted weak topology on continuous processes, proves its metrizability by an adapted Wasserstein distance, and characterizes the process space completion.

Findings

The adapted weak topology is metrized by the adapted Wasserstein distance.

The process space with this topology is Polish.

Donsker's theorem extends to this setting.

Abstract

Researchers from different areas have independently defined extensions of the usual weak convergence of laws of stochastic processes with the goal of adequately accounting for the flow of information. Natural approaches are convergence of the Aldous--Knight prediction process, Hellwig's information topology, convergence in adapted distribution in the sense of Hoover--Keisler and the weak topology induced by optimal stopping problems. The first main contribution of this article is that on continuous processes with natural filtrations there exists a canonical adapted weak topology which can be defined by all of these approaches; moreover, the adapted weak topology is metrized by a suitable adapted Wasserstein distance $\mathcal{AW}$. While the set of processes with natural filtrations is not complete, we establish that its completion consists precisely of the space ${\rm FP}$ of stochastic processes with general filtrations. We also show that $({\rm FP}, \mathcal{AW})$ exhibits several desirable properties. Specifically, it is Polish, martingales form a closed subset and approximation results such as Donsker's theorem extend to $\mathcal{AW}$.