{
  "id": "2306.13405",
  "version": "v1",
  "published": "2023-06-23T09:40:03.000Z",
  "updated": "2023-06-23T09:40:03.000Z",
  "title": "New asymptotic expansion formula via Malliavin calculus and its application to rough differential equation driven by fractional Brownian motion",
  "authors": [
    "Akihiko Takahashi",
    "Toshihiro Yamada"
  ],
  "comment": "16 pages, 1 figure",
  "categories": [
    "math.PR",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index $H<1/2$, without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-23T09:40:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G22",
      "60H07",
      "60L20"
    ],
    "keywords": [
      "rough differential equation driven",
      "fractional brownian motion",
      "malliavin calculus",
      "generic asymptotic expansion formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}