{
  "id": "1408.5267",
  "version": "v1",
  "published": "2014-08-22T11:22:05.000Z",
  "updated": "2014-08-22T11:22:05.000Z",
  "title": "An overview of Viscosity Solutions of Path-Dependent PDEs",
  "authors": [
    "Zhenjie Ren",
    "Nizar Touzi",
    "Jianfeng Zhang"
  ],
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-22T11:22:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "viscosity solutions",
      "path-dependent pdes",
      "path-dependent partial di erential equations",
      "barles-souganidis monotonic scheme approximation method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.5267R"
    }
  }
}