{
  "id": "math/0212281",
  "version": "v1",
  "published": "2002-12-19T21:36:27.000Z",
  "updated": "2002-12-19T21:36:27.000Z",
  "title": "Small values of the maximum for the integral of fractional Brownian motion",
  "authors": [
    "G. M. Molchan",
    "A. V. Khokhlov"
  ],
  "comment": "23 pages,3 figures, TeX/LaTeX 3.14159",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the integral of fractional Brownian motion (IFBM) and its functionals $\\xi_T$ on the intervals $(0,T)$ and $(-T,T)$ of the following types: the maximum $M_T$, the position of the maximum, the occupation time above zero etc. We show how the asymptotics of $P(\\xi_T<1)=p_T, T\\to \\infty$, is related to the Hausdorff dimension of Lagrangian regular points for the inviscid Burgers equation with FBM initial velocity. We produce computational evidence in favor of a power asymptotics for $p_T$. The data do not reject the hypothesis that the exponent $\\theta$ of the power law is related to the similarity parameter $H$ of fractional Brownian motion as follows: $\\theta =-(1-H)$ for the interval $(-T,T)$ and $\\theta =-H(1-H)$ for $(0,T)$. The point 0 is special in that IFBM and its derivative both vanish there.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-12-19T21:36:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J25",
      "60G15"
    ],
    "keywords": [
      "fractional brownian motion",
      "small values",
      "produce computational evidence",
      "fbm initial velocity",
      "inviscid burgers equation"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....12281M"
    }
  }
}