2025/01/23 by Daniel Bartl, Bartl, Daniel, Mathias Beiglböck +7
Economics, Econometrics and Finance · #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2501.14135
openalex publication_date 2025/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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, AW) exhibits several desirable properties. Specifically, it is Polish, martingales form a closed subset and approximation results such as Donsker's theorem extend to AW.