{
  "id": "1311.6910",
  "version": "v2",
  "published": "2013-11-27T09:47:47.000Z",
  "updated": "2016-04-18T20:23:44.000Z",
  "title": "Minimal Supersolutions of Convex BSDEs under Constraints",
  "authors": [
    "Gregor Heyne",
    "Michael Kupper",
    "Christoph Mainberger",
    "Ludovic Tangpi"
  ],
  "comment": "23 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study supersolutions of a backward stochastic differential equation, the control processes of which are constrained to be continuous semimartingales of the form $dZ = {\\Delta}dt + {\\Gamma}dW$. The generator may depend on the decomposition $({\\Delta},{\\Gamma})$ and is assumed to be positive, jointly convex and lower semicontinuous, and to satisfy a superquadratic growth condition in ${\\Delta}$ and ${\\Gamma}$. We prove the existence of a supersolution that is minimal at time zero and derive stability properties of the non-linear operator that maps terminal conditions to the time zero value of this minimal supersolution such as monotone convergence, Fatou's lemma and $L^1$-lower semicontinuity. Furthermore, we provide duality results within the present framework and thereby give conditions for the existence of solutions under constraints.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-27T09:47:47.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-04-18T20:23:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H20",
      "60H30"
    ],
    "keywords": [
      "minimal supersolution",
      "convex bsdes",
      "constraints",
      "maps terminal conditions",
      "superquadratic growth condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.6910H"
    }
  }
}