Open Access
2024 A note on the adapted weak topology in discrete time
Gudmund Pammer
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Electron. Commun. Probab. 29: 1-13 (2024). DOI: 10.1214/23-ECP572


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–Beiglböck–P. [7] as starting point, the purpose of this note is to recover with topological arguments the intriguing result by Backhoff–Bartl–Beiglböck–Eder [3] 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 Rd by a weak Wasserstein metric based on the theory of weak optimal transport that was initiated by Gozlan–Roberto–Samson–Tetali [11].


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Gudmund Pammer. "A note on the adapted weak topology in discrete time." Electron. Commun. Probab. 29 1 - 13, 2024.


Received: 11 May 2022; Accepted: 26 December 2023; Published: 2024
First available in Project Euclid: 8 January 2024

Digital Object Identifier: 10.1214/23-ECP572

Primary: 49Q22 , 60B10 , 60G05

Keywords: adapted weak topology , Stochastic processes

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