{
  "id": "math/0310413",
  "version": "v1",
  "published": "2003-10-26T17:42:40.000Z",
  "updated": "2003-10-26T17:42:40.000Z",
  "title": "Unilateral Small Deviations for the Integral of Fractional Brownian Motion",
  "authors": [
    "G. Molchan",
    "A. Khokhlov"
  ],
  "comment": "15 pages, 4 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at $\\infty$ with exponent $\\theta$. In applications this parameter can provide information on fractal properties of processes that are subordinate to $x(\\cdot)$. For this reason the estimation of $\\theta$ is an important theoretical problem. Here, we consider the process $x(t)$ whose derivative is fractional Brownian motion with self-similarity parameter $0<H<1$. For this case we produce new computational evidence in favor of the relations $\\log p(T)=-\\theta \\log T(1+o(1))$ and $\\theta =H(1-H)$. The estimates of $\\theta$ are to within 0.01 in the range $0.1\\le H\\le 0.9$. An analytical result for the problem in hand is known for the markovian case alone, i.e., for $H=1/2$. We point out other statistics of $x(t)$ whose small values have probabilities of the same order as $p(T)$ in the $\\log$ scale.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-10-26T17:42:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G18"
    ],
    "keywords": [
      "fractional brownian motion",
      "unilateral small deviations",
      "gaussian random process",
      "large time interval",
      "small values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....10413M"
    }
  }
}