{
  "id": "1410.7281",
  "version": "v1",
  "published": "2014-10-27T15:48:08.000Z",
  "updated": "2014-10-27T15:48:08.000Z",
  "title": "Comparison of Viscosity Solutions of Semi-linear Path-Dependent PDEs",
  "authors": [
    "Zhenjie Ren",
    "Nizar Touzi",
    "Jianfeng Zhang"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in \\cite{EKTZ} which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions, and reduces to the notion of stochastic viscosity solutions analyzed in \\cite{BayraktarSirbu1,BayraktarSirbu2}. Our main result takes advantage of this enlargement of the test functions, and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions \\cite{CaffarelliCabre}, and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in \\cite{EKTZ}. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-27T15:48:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D40",
      "35K10",
      "60H10",
      "60H30"
    ],
    "keywords": [
      "semi-linear path-dependent pdes",
      "test functions",
      "linear path-dependent heat equation",
      "path-dependent semilinear pdes",
      "aleksandroff smoothness result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.7281R"
    }
  }
}