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The martingale part in the semimartingale decomposition of a Brownian motionwith respect to an enlargement of its filtration, is an anticipative mapping ofthe given Brownian motion. In analogy to optimal transport theory, we definecausal transport plans in the context of enlargement of filtrations, as theKantorovich counterparts of the aforementioned non-adapted mappings. We providea necessary and sufficient condition for a Brownian motion to remain asemimartingale in an enlarged filtration, in terms of certain minimizationproblems over sets of causal transport plans. The latter are also used in orderto give robust transport-based estimates for the value of having additionalinformation, as well as model sensitivity with respect to the referencemeasure, for the classical stochastic optimization problems of utilitymaximization and optimal stopping. Our results have natural extensions to thecase of general multidimensional continuous semimartingales.

Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization