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.
Adapted optimal transport between Gaussian processes in discrete time
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We derive explicitly the adapted $2$-Wasserstein distance between non-degenerate Gaussian distributions on $\mathbb{R}^N$ and characterize the optimal bicausal coupling(s). This leads to an adapted version of the Bures-Wasserstein distance on the space of positive definite matrices.
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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.