{
  "id": "0809.1111",
  "version": "v1",
  "published": "2008-09-05T21:51:34.000Z",
  "updated": "2008-09-05T21:51:34.000Z",
  "title": "Measurability of optimal transportation and strong coupling of martingale measures",
  "authors": [
    "Joaquin Fontbona",
    "Helene Guerin",
    "Sylvie Meleard"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the optimal mass transportation problem in $\\RR^d$ with measurably parameterized marginals, for general cost functions and 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 specified covariance measure. This is done by pushing forward one 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-09-05T21:51:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q20",
      "60G57"
    ],
    "keywords": [
      "martingale measure",
      "optimal transportation",
      "strong coupling",
      "measurability",
      "covariance measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.1111F"
    }
  }
}