{
  "id": "1309.4935",
  "version": "v2",
  "published": "2013-09-19T11:27:42.000Z",
  "updated": "2015-02-10T14:53:21.000Z",
  "title": "On the continuity of the probabilistic representation of a semilinear Neumann--Dirichlet problem",
  "authors": [
    "Lucian Maticiuc",
    "Aurel Răşcanu"
  ],
  "comment": "A slight change to the title has been made. Also we changed the structure of the article by creating Sections. Some new proofs have been added",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article we prove the continuity of the deterministic function $u:[0,T]\\times \\mathcal{\\bar{D}}\\rightarrow \\mathbb{R}$, defined by $u(t,x):=Y_{t}^{t,x}$, where the process $(Y_{s}^{t,x})_{s\\in[t,T]}$ is given by the generalized multivalued backward stochastic differential equation: \\begin{equation*} \\left\\{ \\begin{array}{l} -dY_{s}^{t,x}+\\partial \\varphi(Y_{s}^{t,x})ds+\\partial\\psi(Y_{s}^{t,x})dA_{s}^{t,x}\\ni f(s,X_{s}^{t,x},Y_{s}^{t,x})ds \\\\ \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+g(s,X_{s}^{t,x},Y_{s}^{t,x})dA_{s}^{t,x}-Z_{s}^{t,x}dW_{s}~,\\;t\\leq s < T, \\\\ {Y_{T}=h(X_{T}^{t,x}).} \\end{array} \\right. \\end{equation*} Here, the process $(X_{s}^{t,x},A_{s}^{t,x})_{s\\geq t}$ is the solution of a stochastic differential equation with reflecting boundary conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-19T11:27:42.000Z",
      "title": "On the continuity of $u(t,x)=Y_{t}^{t,x}$ from Feynman-Kac formula for a Neumann-Dirichlet problem",
      "abstract": "In this note we prove the continuity of the deterministic function $u:[0,T] \\times \\mathcal{\\bar{D}}\\rightarrow \\mathbb{R}$, $u(t,x)=Y_{t}^{t,x}$, where process $(Y_{s}^{t,x})_{s\\in [t,T]}$ is defined by the generalized multivalued backward stochastic differential equation: \\begin{equation*} \\left\\{\\begin{array}{r} dY_{s}^{t,x}{+}F(s,X_{s}^{t,x},Y_{s}^{t,x})ds{+} G(s,X_{s}^{t,x},Y_{s}^{t,x})dA_{s}^{t,x}\\in \\partial \\varphi (Y_{s}^{t,x})ds\\medskip \\\\ {+}\\partial \\psi (Y_{s}^{t,x})dA_{s}^{t,x}{+}Z_{s}^{t,x}dW_{s},\\;t\\leq s\\leq T,\\medskip \\\\ \\multicolumn{1}{l}{Y_{T}=\\xi .} \\end{array} \\right. \\end{equation*} The process $(X_{s}^{t,x},A_{s}^{t,x})_{s\\geq t}$ is given by a stochastic differential equation with reflecting boundary conditions.",
      "comment": "17 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-10T14:53:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "feynman-kac formula",
      "neumann-dirichlet problem",
      "continuity",
      "multivalued backward stochastic differential equation",
      "deterministic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.4935M"
    }
  }
}