{
  "id": "1307.3656",
  "version": "v2",
  "published": "2013-07-13T15:20:25.000Z",
  "updated": "2014-10-30T10:11:08.000Z",
  "title": "Optimal Transport and Skorokhod Embedding",
  "authors": [
    "Mathias Beiglboeck",
    "Martin Huesmann"
  ],
  "categories": [
    "math.PR",
    "math.OC"
  ],
  "abstract": "The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-13T15:20:25.000Z",
      "abstract": "It is well known that several solutions to the Skorokhod problem optimize certain \"cost\"- or \"payoff\"-functionals. We use the theory of Monge-Kantorovich transport to study the corresponding optimization problem. We formulate a dual problem and establish duality based on the duality theory of optimal transport. Notably the primal as well as the dual problem have a natural interpretation in terms of model-independent no arbitrage theory. In optimal transport the notion of c-monotonicity is used to characterize the geometry of optimal transport plans. We derive a similar optimality principle that provides a geometric characterization of optimal stopping times. We then use this principle to derive the Root- and Rost solutions to the Skorokhod embedding problem.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-30T10:11:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G42"
    ],
    "keywords": [
      "dual problem",
      "similar optimality principle",
      "optimal transport plans",
      "optimal stopping times",
      "geometric characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.3656B"
    }
  }
}