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Abstract: We consider the optimal mass transportation problem in $\RR^d$ withmeasurably parameterized marginals, for general cost functions and underconditions ensuring the existence of a unique optimal transport map. We prove ajoint measurability result for this map, with respect to the space variable andto the parameter. The proof needs to establish the measurability of someset-valued mappings, related to the support of the optimal transference plans,which we use to perform a suitable discrete approximation procedure. Amotivation is the construction of a strong coupling between orthogonalmartingale measures. By this we mean that, given a martingale measure, weconstruct in the same probability space a second one with specified covariancemeasure. This is done by pushing forward one martingale measure through apredictable version of the optimal transport map between the covariancemeasures. This coupling allows us to obtain quantitative estimates in terms ofthe Wasserstein distance between those covariance measures.



Author: Joaquin Fontbona, Helene Guerin, Sylvie Meleard

Source: https://arxiv.org/







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