{
  "id": "1106.2302",
  "version": "v2",
  "published": "2011-06-12T10:53:28.000Z",
  "updated": "2011-06-18T07:58:34.000Z",
  "title": "The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2",
  "authors": [
    "Litan Yan",
    "Chao Chen",
    "Junfeng Liu"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1/2$. In this paper we study the {\\it generalized quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t=\\lim_{\\epsilon\\downarrow 0}\\frac{2H}{\\epsilon^{2H}}\\int_0^t\\{f(B^{H}_{s+\\epsilon})-f(B^{H}_s)\\}(B^{H}_{s+\\epsilon}- B^{H}_s)s^{2H-1}ds, $$ where the limit is uniform in probability and $x\\mapsto f(x)$ is a deterministic function. We construct a Banach space ${\\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2$ and the Bouleau-Yor identity takes the form $$ [f(B^H),B^H]_t^{(W)}=-\\int_{\\mathbb {R}}f(x){\\mathscr L}^{H}(dx,t) $$ provided $f\\in {\\mathscr H}$, where ${\\mathscr L}^{H}(x,t)$ is the weighted local time of $B^H$. This allows us to write the fractional It\\^{o} formula for absolutely continuous functions with derivative belonging to ${\\mathscr H}$. These are also extended to the time-dependent case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-18T07:58:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60H05",
      "60H07"
    ],
    "keywords": [
      "generalized quadratic covariation",
      "fractional brownian motion",
      "hurst index",
      "deterministic function",
      "banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.2302Y"
    }
  }
}