{
  "id": "1407.1876",
  "version": "v2",
  "published": "2014-07-07T20:43:49.000Z",
  "updated": "2015-05-09T09:25:46.000Z",
  "title": "Stochastic Variational Inequalities on Non-Convex Domains",
  "authors": [
    "Rainer Buckdahn",
    "Lucian Maticiuc",
    "Etienne Pardoux",
    "Aurel Răşcanu"
  ],
  "comment": "39 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "The objective of this work is to prove, in a first step, the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: $dx(t)+\\partial ^-\\varphi (x(t))(dt)\\ni dm(t),\\ t>0$, $x(0)=x_0$, where $m:\\mathbb{R}_+\\rightarrow\\mathbb{R}^d$ is a continuous function and $\\partial^-\\varphi$ is the Fr\\'{e}chet subdifferential of a semiconvex function $\\varphi$; the domain of $\\varphi$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\\mapsto x:C([0,T];\\mathbb{R}^{d})\\rightarrow C([0,T] ;\\mathbb{R}^{d})$, which associate the input function $m$ with the solution $x$ of the above equation, as well as tightness criteria allow to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: $X_t+K_t = \\xi+\\int_0^t F(s,X_{s})ds + \\int_0^t G(s,X_s) dB_s,\\; t\\geq0$, $\\;$ with $dK_{t}(\\omega)\\in\\partial^-\\varphi( X_t (\\omega))(dt)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-07T20:43:49.000Z",
      "abstract": "The objective of this work is to prove, in a first step, the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: $dx(t)+\\partial ^-\\varphi (x(t))(dt)\\ni dm(t),\\ t>0$, $x(0)=x_0$, where $m:\\mathbb{R}_+\\rightarrow\\mathbb{R}^d$ is a continuous function and $\\partial^-\\varphi$ is the Fr\\'{e}chet subdifferential of a semiconvex function $\\varphi$; the domain of $\\varphi$ can be non-convex, but some regularities of the boundary are required. The continuity of the map $m\\longmapsto x:C([0,T];\\mathbb{R}^{d})\\rightarrow C([0,T] ;\\mathbb{R}^{d})$, which associate the input function $m$ with the solution $x$ of the above equation, as well as tightness criteria allow to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: $X_t+K_t = \\xi+\\int_0^t F(s,X_{s})ds + \\int_0^t G(s,X_s) dB_s,\\quad t\\geq0$, $dK_{t}(\\omega)\\in\\partial^-\\varphi( X_t (\\omega))(dt)$.",
      "comment": "38 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-05-09T09:25:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60J60"
    ],
    "keywords": [
      "non-convex domains",
      "stochastic variational inequality driven",
      "multivalued deterministic differential equation",
      "multi-dimensional brownian motion",
      "semiconvex function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.1876B"
    }
  }
}