{
  "id": "1208.1100",
  "version": "v1",
  "published": "2012-08-06T07:50:03.000Z",
  "updated": "2012-08-06T07:50:03.000Z",
  "title": "Hölder regularity and series representation of a class of stochastic volatility models",
  "authors": [
    "Antoine Ayache",
    "Qidi Peng"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\Phi:\\R\\rightarrow\\R$ be an arbitrary continuously differentiable deterministic function such that $|\\Phi|+|\\Phi'|$ is bounded by a polynomial. In this article we consider the class of stochastic volatility models in which ${Z(t)}_{t\\in [0,1]}$, the logarithm of the price process, is of the form $Z(t)=\\int_{0}^t \\Phi(X(s)) dW(s)$, where ${X(s)}_{s\\in[0,1]}$ denotes an arbitrary centered Gaussian process whose trajectories are, with probability 1, H\\\"older continuous functions of an arbitrary order $\\alpha\\in (1/2,1]$, and where ${W(s)}_{s\\in[0,1]}$ is a standard Brownian motion independent on ${X(s)}_{s\\in [0,1]}$. First we show that the critical H\\\"older regularity of a typical trajectory of ${Z(t)}_{t\\in[0,1]}$ is equal to 1/2. Next we provide for such a trajectory an expression as a random series which converges at a geometric rate in any H\\\"older space of an arbitrary order $\\gamma<1/2$; this expression is obtained through the expansion of the random function $s\\mapsto \\Phi(X(s))$ on the Haar basis. Finally, thanks to it, we give an efficient iterative simulation method for ${Z(t)}_{t\\in[0,1]}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-06T07:50:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic volatility models",
      "hölder regularity",
      "series representation",
      "arbitrary order",
      "standard brownian motion independent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1100A"
    }
  }
}