{
  "id": "2008.01385",
  "version": "v1",
  "published": "2020-08-04T07:49:14.000Z",
  "updated": "2020-08-04T07:49:14.000Z",
  "title": "The Multiplicative Chaos of $H=0$ Fractional Brownian Fields",
  "authors": [
    "Paul Hager",
    "Eyal Neuman"
  ],
  "comment": "48 pages, 1 figure",
  "categories": [
    "math.PR",
    "q-fin.MF"
  ],
  "abstract": "We consider a family of fractional Brownian fields $\\{B^{H}\\}_{H\\in (0,1)}$ on $\\mathbb{R}^{d}$, where $H$ denotes their Hurst parameter. We first define a rich class of normalizing kernels $\\psi$ such that the covariance of $$ X^{H}(x) = \\Gamma(H)^{\\frac{1}{2}} \\left( B^{H}(x) - \\int_{\\mathbb{R}^{d}} B^{H}(u) \\psi(u, x)du\\right), $$ converges to the covariance of a log-correlated Gaussian field when $H \\downarrow 0$. We then use Berestycki's ``good points'' approach in order to derive the limiting measure of the so-called multiplicative chaos of the fractional Brownian field $$ M^{H}_\\gamma(dx) = e^{\\gamma X^{H}(x) - \\frac{\\gamma^{2}}{2} E[X^{H}(x)^{2}] }dx, $$ as $H\\downarrow 0$ for all $\\gamma \\in (0,\\gamma^{*}(d)]$, where $\\gamma^{*}(d)>\\sqrt{\\frac{7}{4}d}$. As a corollary we establish the $L^{2}$ convergence of $M^{H}_\\gamma$ over the sets of ``good points'', where the field $X^H$ has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of ``good points'' with probability close to $1$. Moreover, on these sets the volatility converges in $L^2$ to the volatility of multifractal random walks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-04T07:49:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G57",
      "60G60",
      "60G18"
    ],
    "keywords": [
      "fractional brownian field",
      "multiplicative chaos",
      "log-normal rough volatility models",
      "small hurst parameter",
      "multifractal random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}