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.
Multicausal transport: barycenters and dynamic matching
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abstract
We introduce a multivariate version of causal transport, which we name multicausal transport, involving several filtered processes among which causality constraints are imposed. Subsequently, we consider the barycenter problem for stochastic processes with respect to causal and bicausal optimal transport, and study its connection to specific multicausal transport problems. Attainment and duality of the aforementioned problems are provided. As an application, we study a matching problem in a dynamic setting where agent types evolve over time. We link this to a causal barycenter problem and thereby show existence of equilibria.
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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.