{
  "id": "1501.06986",
  "version": "v1",
  "published": "2015-01-28T04:46:55.000Z",
  "updated": "2015-01-28T04:46:55.000Z",
  "title": "On the $\\frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter $H < 1/2$",
  "authors": [
    "El Hassan Essaky",
    "David Nualart"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the $\\frac{1}{H}$-variation of stochastic divergence integrals $X_t = \\int_0^t u_s {\\delta}B_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < \\frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $\\frac{1}{H}$-variation of $X$ exists in $L^1({\\Omega})$ and is equal to $e_H \\int_0^T|u_s|^H ds$, where $e_H = \\mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $\\|B_t\\|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < \\frac{1}{2}$. Using a multidimensional version of the result on the $\\frac{1}{H}$-variation of divergence integrals, we prove that if $2dH^2 > 1$, then the divergence integral in the integral representation of the fractional Bessel process has a $\\frac{1}{H}$-variation equals to a multiple of the Lebesgue measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-28T04:46:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H05",
      "60H07",
      "60G18"
    ],
    "keywords": [
      "hurst parameter",
      "fractional bessel process",
      "dimensional fractional brownian motion",
      "integral representation",
      "stochastic divergence integrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150106986H"
    }
  }
}