{
  "id": "1804.10737",
  "version": "v1",
  "published": "2018-04-28T03:54:40.000Z",
  "updated": "2018-04-28T03:54:40.000Z",
  "title": "An Iterative Approximation of the Sublinear Expectation of an Arbitrary Function of $G$-normal Distribution and the Solution to the Corresponding $G$-heat Equation",
  "authors": [
    "Yifan Li",
    "Reg Kulperger"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "It has been a well-known problem in the $G$-framework that it is hard to compute the sublinear expectation of the $G$-normal distribution $\\hat{\\mathbb{E}}[\\varphi(X)]$ when $\\varphi$ is neither convex nor concave, if not involving any PDE techniques to solve the corresponding $G$-heat equation. Recently, we have established an efficient iterative method able to compute the sublinear expectation of \\emph{arbitrary} functions of the $G$-normal distribution, which directly applies the \\emph{Nonlinear Central Limit Theorem} in the $G$-framework to a sequence of variance-uncertain random variables following the \\emph{Semi-$G$-normal Distribution}, a newly defined concept with a nice \\emph{Integral Representation}, behaving like a ladder in both theory and intuition, helping us climb from the ground of classical normal distribution to approach the peak of $G$-normal distribution through the \\emph{iteratively maximizing} steps. The series of iteration functions actually produce the whole \\emph{solution surface} of the $G$-heat equation on a given time grid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-28T03:54:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sublinear expectation",
      "heat equation",
      "arbitrary function",
      "iterative approximation",
      "corresponding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}