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MEASURABILITY OF OPTIMAL TRANSPORTATION AND STRONG COUPLING OF MARTINGALE MEASURES

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2010
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Fontbona Torres, Joaquín
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MEASURABILITY OF OPTIMAL TRANSPORTATION AND STRONG COUPLING OF MARTINGALE MEASURES
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Author
  • Fontbona Torres, Joaquín;
  • Guérin, Hélène;
  • Méléard, Sylvie;
Abstract
We consider the optimal mass transportation problem in Rd with measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.
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SUPPORTED BY FONDECYT PROYECT 1070743, ECOS-CONICYT C05E02 AND BASAL-CONICYT, SUPPORTED BY ECOS-CONICYT C05E02, SUPPORTED BY ECOS-CONICYT C05E02
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URI: https://repositorio.uchile.cl/handle/2250/125354
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Electronic Communications in Probability 15 (2010), 124–133
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