The Wasserstein Space of Stochastic Processes in Continuous Time

TL;DR

The paper unifies multiple adapted topologies for laws of stochastic processes into a single canonical adapted weak topology, shown to be metrizable by the adapted Wasserstein distance $\mathcal{AW}_p$ and to admit the completion ${\rm FP}$ of processes with general filtrations. It proves that the Aldous–Meyer–Zheng, Hoover–Keisler, Hellwig, and optimal-stopping viewpoints coincide under this topology and that martingales form a closed subset, with Donsker-type discrete-to-continuous limits holding in $\mathcal{AW}$. By introducing filter-preserving canonical representatives and studying the stability of optimal stopping, the work delivers both qualitative and quantitative convergence results for processes and their discretizations. The findings establish a robust framework for convergence under information-flow constraints, enabling stable approximations of SDEs and Brownian motion and underpinning applications in stochastic control and finance. Overall, the unified theory provides a powerful, information-aware metric for continuous-time stochastic processes and their approximations.

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 $\mathcal{AW}$.

Figures (1)

• Figure 1: $\mathbb{P}$ on the left and $\mathbb{P}^\varepsilon$ on the right.

Theorems & Definitions (127)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.10