{
  "id": "1310.3364",
  "version": "v2",
  "published": "2013-10-12T12:09:26.000Z",
  "updated": "2015-11-17T09:34:18.000Z",
  "title": "Capacities, Measurable Selection and Dynamic Programming Part II: Application in Stochastic Control Problems",
  "authors": [
    "Nicole El Karoui",
    "Xiaolu Tan"
  ],
  "comment": "38 pages",
  "categories": [
    "math.OC"
  ],
  "abstract": "We aim to give an overview on how to derive the dynamic programming principle for a general stochastic control/stopping problem, using measurable selection techniques. By considering their martingale problem formulation, we show how to check the required measurability conditions for different versions of control/stopping problem, including in particular the controlled/stopped diffusion processes problem. Further, we also study the stability of the control problem, i.e. the approximation of the control process by piecewise constant control problems. It induces in particular an equivalence result for different versions of the controlled/stopped diffusion processes problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-12T12:09:26.000Z",
      "abstract": "Under the framework in our previous paper, we derive a dynamic programming principle (DPP) for a general controlled/stopped martingale problem. Next, we restrict to the diffusion process case, where the martingale problem gives a weak and a relaxed formulation of the control problem. We then study its strong formulation, including its DPP and also its equivalence to other formulations under some regularity conditions. Finally, for the controlled Markov diffusion-jump processes problem, we present a Markovian selection approach.",
      "comment": "30 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-11-17T09:34:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28B20",
      "49L20",
      "93E20",
      "60H30"
    ],
    "keywords": [
      "stochastic control problems",
      "dynamic programming part",
      "measurable selection",
      "capacities",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.3364E"
    }
  }
}