{
  "id": "0711.1313",
  "version": "v2",
  "published": "2007-11-08T16:36:29.000Z",
  "updated": "2009-12-09T10:15:41.000Z",
  "title": "Fractional martingales and characterization of the fractional Brownian motion",
  "authors": [
    "Yaozhong Hu",
    "David Nualart",
    "Jian Song"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/09-AOP464 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2009, Vol. 37, No. 6, 2404-2430",
  "doi": "10.1214/09-AOP464",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\\alpha$ of a continuous local martingale, where $\\alpha\\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order $\\frac{2}{1+2\\alpha}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of L\\'evy's characterization theorem for the fractional Brownian motion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-12-09T10:15:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G44",
      "60J65",
      "60G15",
      "26A45"
    ],
    "keywords": [
      "fractional brownian motion",
      "fractional martingale",
      "nonzero finite variation",
      "levys characterization theorem",
      "continuous local martingale"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.1313H"
    }
  }
}