{
  "id": "1304.2466",
  "version": "v5",
  "published": "2013-04-09T06:34:22.000Z",
  "updated": "2014-09-11T13:50:18.000Z",
  "title": "Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind",
  "authors": [
    "Ehsan Azmoodeh",
    "Lauri Viitasaari"
  ],
  "comment": "Modified version. arXiv admin note: text overlap with arXiv:1302.6047",
  "categories": [
    "math.PR"
  ],
  "abstract": "Fractional Ornstein-Uhlenbeck process of the second kind $(\\text{fOU}_{2})$ is solution of the Langevin equation $\\mathrm{d}X_t = -\\theta X_t\\,\\mathrm{d}t+\\mathrm{d}Y_t^{(1)}, \\ \\theta >0$ with Gaussian driving noise $ Y_t^{(1)} := \\int^t_0 e^{-s} \\,\\mathrm{d}B_{a_s}$, where $ a_t= H e^{\\frac{t}{H}}$ and $B$ is a fractional Brownian motion with Hurst parameter $H \\in (0,1)$. In this article, we consider the case $H>\\frac{1}{2}$. Then using the ergodicity of $\\text{fOU}_{2}$ process, we construct consistent estimators of drift parameter $\\theta$ based on discrete observations in two possible cases: $(i)$ the Hurst parameter $H$ is known and $(ii)$ the Hurst parameter $H$ is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range $H \\in (\\frac{1}{2},1)$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-01-13T10:55:11.000Z",
      "abstract": "Fractional Ornstein-Uhlenbeck process of the second kind $(\\text{fOU}_{2})$ is solution of the Langevin equation $\\mathrm{d}X_t = -\\theta X_t\\,\\mathrm{d}t+\\mathrm{d}Y_t^{(1)}, \\ \\theta >0$ with driving noise $ Y_t^{(1)} := \\int^t_0 e^{-s} \\,\\mathrm{d}B_{a_s}; \\ a_t= H e^{{t}{H}}$ where $B$ is a fractional Brownian motion with Hurst parameter $H \\in (0,1)$. In this article, we consider the case $H>{1}{2}$. Then using ergodicity of $\\text{fOU}_{2}$, we construct consistent estimators of drift parameter $\\theta$ based on discrete observations in both cases: $(i)$ the Hurst parameter $H$ is known and $(ii)$ the Hurst parameter $H$ is unknown. Moreover, using Malliavin calculus technique, we prove central limit theorems for our estimators which is valid for the whole range $H \\in (\\frac{1}{2},1)$.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-09-11T13:50:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G22",
      "60H07",
      "62F99"
    ],
    "keywords": [
      "fractional ornstein-uhlenbeck process",
      "discrete observations",
      "second kind",
      "parameter estimation",
      "hurst parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.2466A"
    }
  }
}