{
  "id": "2112.13332",
  "version": "v2",
  "published": "2021-12-26T08:07:08.000Z",
  "updated": "2022-01-26T13:58:37.000Z",
  "title": "Drift estimation for a multi-dimensional diffusion process using deep neural networks",
  "authors": [
    "Akihiro Oga",
    "Yuta Koike"
  ],
  "comment": "32 pages. The order of Lemmas 4.6 and 4.7 has been reversed to avoid an integrability issue",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Recently, many studies have shed light on the high adaptivity of deep neural network methods in nonparametric regression models, and their superior performance has been established for various function classes. Motivated by this development, we study a deep neural network method to estimate the drift coefficient of a multi-dimensional diffusion process from discrete observations. We derive generalization error bounds for least squares estimates based on deep neural networks and show that they achieve the minimax rate of convergence up to a logarithmic factor when the drift function has a compositional structure.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-01-26T13:58:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62M45",
      "62G05"
    ],
    "keywords": [
      "multi-dimensional diffusion process",
      "deep neural network method",
      "drift estimation",
      "nonparametric regression models",
      "derive generalization error bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}