{
  "id": "1709.09370",
  "version": "v1",
  "published": "2017-09-27T07:46:28.000Z",
  "updated": "2017-09-27T07:46:28.000Z",
  "title": "Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs",
  "authors": [
    "Charles-Edouard Bréhier"
  ],
  "categories": [
    "math.PR",
    "math.NA"
  ],
  "abstract": "This article investigates the role of the regularity of the test function when considering the weak error for standard discretizations of SPDEs of the form $dX(t)=AX(t)dt+F(X(t))dt+dW(t)$, driven by space-time white noise. In previous results, test functions are assumed (at least) of class $\\mathcal{C}^2$ with bounded derivatives, and the weak order is twice the strong order. We prove, in the case $F=0$, that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even when the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by $1$, does not converge to $0$ as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by $2$, i.e. it is not better than the strong order. This is in contrast with the finite dimensional case, where the Euler-Maruyama discretization of elliptic SDEs $dY(t)=f(Y(t))dt+dB_t$ has weak order of convergence $1$ even for bounded continuous functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-27T07:46:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H35",
      "65C30"
    ],
    "keywords": [
      "weak convergence",
      "numerical discretization",
      "weak order",
      "regularity",
      "bounded continuous functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}