{
  "id": "1106.1144",
  "version": "v2",
  "published": "2011-06-06T18:31:00.000Z",
  "updated": "2012-02-19T16:15:23.000Z",
  "title": "Note on Viscosity Solution of Path-Dependent PDE and G-Martingales",
  "authors": [
    "Shige Peng"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In the 2nd version of this note we introduce the notion of viscosity solution for a type of fully nonlinear parabolic path-dependent partial differential equations (P-PDE). We then prove the comparison theorem (or maximum principle) of this new type of equation which is the key property of this framework. To overcome the well-known difficulty of non-compactness of the space of paths for the maximization, we have introduced a new approach, called left frozen maximization approach which permits us to obtain the comparison principle for smooth as well as viscosity solutions of path-dependent PDE. A solution of a backward stochastic differential equation and a G-martingale under a G-expectation are typical examples of such type of solutions of P-PDE. The maximum principle for viscosity solutions of classical PDE, called state dependent PDE, is a special case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-02-19T16:15:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "viscosity solution",
      "path-dependent pde",
      "g-martingale",
      "nonlinear parabolic path-dependent partial differential",
      "parabolic path-dependent partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.1144P"
    }
  }
}