In continuous time, the Aldous, Hoover-Keisler, Hellwig, and optimal-stopping topologies on naturally filtered processes coincide and are metrized by an adapted Wasserstein distance, whose completion is the space of general filtered processes.
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The Wasserstein Space of Stochastic Processes in Continuous Time
In continuous time, the Aldous, Hoover-Keisler, Hellwig, and optimal-stopping topologies on naturally filtered processes coincide and are metrized by an adapted Wasserstein distance, whose completion is the space of general filtered processes.