{
  "id": "0807.4415",
  "version": "v2",
  "published": "2008-07-28T11:39:11.000Z",
  "updated": "2010-02-23T13:03:12.000Z",
  "title": "Viscosity solutions for systems of parabolic variational inequalities",
  "authors": [
    "Lucian Maticiuc",
    "Etienne Pardoux",
    "Aurel Răşcanu",
    "Adrian Zălinescu"
  ],
  "comment": "Published in at http://dx.doi.org/10.3150/09-BEJ204 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)",
  "journal": "Bernoulli 2010, Vol. 16, No. 1, 258-273",
  "doi": "10.3150/09-BEJ204",
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\\[\\{[c]{l}\\dfrac{\\partial u}{\\partial t}(t,x)+\\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\\in\\partial\\phi (u(t,x)),\\quad t\\in[0,T),x\\in\\mathbb{R}^d, u(T,x)=h(x),\\quad x\\in\\mathbb{R}^d,\\] where $\\partial\\phi$ is the subdifferential operator of the proper convex lower semicontinuous function $\\phi:\\mathbb{R}^k\\to (-\\infty,+\\infty]$ and $\\mathcal{L}_t$ is a second differential operator given by $\\mathcal{L}_tv_i(x)={1/2}\\operatorname {Tr}[\\sigma(t,x)\\sigma^*(t,x)\\mathrm{D}^2v_i(x)]+< b(t,x),\\nabla v_i(x)>$, $i\\in\\bar{1,k}$. We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution $u:[0,T]\\times\\mathbb{R}^d\\to\\mathbb{R}^k$ of the above parabolic variational inequality.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-02-23T13:03:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic variational inequality",
      "viscosity solution",
      "proper convex lower semicontinuous function",
      "second differential operator",
      "partial differential equations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.4415M"
    }
  }
}