{
  "id": "1705.08046",
  "version": "v1",
  "published": "2017-05-23T00:00:32.000Z",
  "updated": "2017-05-23T00:00:32.000Z",
  "title": "An Elementary Proof for the Structure of Derivatives in Probability Measures",
  "authors": [
    "Cong Wu",
    "Jianfeng Zhang"
  ],
  "comment": "3 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $F: \\mathbb{L}^2(\\Omega, \\mathbb{R}) \\to \\mathbb{R}$ be a law invariant and continuously Fr\\'echet differentiable mapping. Based on Lions \\cite{Lions}, Cardaliaguet \\cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \\bea \\label{Derivative} D F (\\xi) = g(\\xi), \\eea where $g: \\mathbb{R} \\to \\mathbb{R}$ is a deterministic function which depends only on the law of $\\xi$. See also Carmona \\& Delarue \\cite{CD} Section 5.2. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \\cite{WZ}, which deals with a more general situation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-23T00:00:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elementary proof",
      "probability measures",
      "derivatives",
      "law invariant",
      "general situation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}