{
  "id": "0803.3665",
  "version": "v2",
  "published": "2008-03-26T04:52:25.000Z",
  "updated": "2008-12-04T12:30:53.000Z",
  "title": "Integration with respect to fractional local times with Hurst index $H$ greater than 1/2",
  "authors": [
    "Litan Yan",
    "Junfeng Liu",
    "Xiangfeng Yang"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let ${\\mathscr L}^H(x,t)=2H\\int_0^t\\delta(B^H_s-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$. In this paper, we use Young integration to study the integral of determinate functions $\\int_{\\mathbb R}f(x){\\mathscr L}^H(dx,t)$. As an application, we investigate the {\\it weighted quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t:=\\lim_{n\\to \\infty}2H\\sum_{k=0}^{n-1} k^{2H-1}\\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\\}(B^H_{t_{k+1}}-B^H_{t_{k}}), $$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$ [f(B^H),B^H]^{(W)}_t=-\\int_{\\mathbb R}f(x){\\mathscr L}^H(dx,t), $$ provided $f$ is of bounded $p$-variation with $1\\leq p<\\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional It\\^{o} formula for new classes of functions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-12-04T12:30:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60H05",
      "60H07"
    ],
    "keywords": [
      "fractional local times",
      "hurst index",
      "fractional brownian motion",
      "weighted local time",
      "young integration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0803.3665Y"
    }
  }
}