{
  "id": "2205.00203",
  "version": "v1",
  "published": "2022-04-30T08:36:08.000Z",
  "updated": "2022-04-30T08:36:08.000Z",
  "title": "A universal robust limit theorem for nonlinear Lévy processes under sublinear expectation",
  "authors": [
    "Mingshang Hu",
    "Lianzi Jiang",
    "Gechun Liang",
    "Shige Peng"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "This article establishes a universal robust limit theorem under a sublinear expectation framework. Under moment and consistency conditions, we show that, for $\\alpha \\in (1,2)$, the i.i.d. sequence \\begin{equation*} \\left\\{ \\left( \\frac{1}{\\sqrt{n}}\\sum_{i=1}^{n}X_{i},\\frac{1}{n} \\sum_{i=1}^{n}Y_{i},\\frac{1}{\\sqrt[\\alpha ]{n}}\\sum_{i=1}^{n}Z_{i}\\right) \\right\\} _{n=1}^{\\infty } \\end{equation*} converges in distribution to $\\tilde{L}_{1}$, where $\\tilde{L}_{t}=(\\tilde{\\xi}_{t},\\tilde{\\eta}_{t},\\tilde{\\zeta}_{t})$, $t\\in \\lbrack 0,1]$, is a multidimensional nonlinear L\\'{e}vy process with an uncertainty set $\\Theta$ as a set of L\\'{e}vy triplets. This nonlinear L\\'{e}vy process is characterized by a fully nonlinear degenerate partial integro-differential equation (PIDE) \\begin{equation*} \\left\\{ \\begin{array}{l} \\displaystyle\\partial _{t}u(t,x,y,z)-\\sup\\limits_{(F_{\\mu },q,Q)\\in \\Theta }\\left\\{ \\int_{\\mathbb{R}^{d}}\\delta _{\\lambda }u(t,x,y,z)F_{\\mu }(d\\lambda )\\right. \\\\ \\displaystyle\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\left. +\\langle D_{y}u(t,x,y,z),q\\rangle +\\frac{1}{2}tr[D_{x}^{2}u(t,x,y,z)Q]\\right\\} =0, \\\\ \\displaystyle u(0,x,y,z)=\\phi (x,y,z),\\ \\ \\forall (t,x,y,z)\\in \\lbrack 0,1]\\times \\mathbb{R}^{3d}, \\end{array} \\right. \\end{equation*} with $\\delta _{\\lambda }u(t,x,y,z):=u(t,x,y,z+\\lambda )-u(t,x,y,z)-\\langle D_{z}u(t,x,y,z),\\lambda \\rangle $. To construct the limit process $(\\tilde{L}_{t})_{t\\in \\lbrack 0,1]}$, we develop a novel weak convergence approach based on the notion of tightness and weak compactness on a sublinear expectation space. We further prove a new type of L\\'{e}vy-Khintchine representation formula to characterize $(\\tilde{L}_{t})_{t\\in \\lbrack 0,1]}$. As a byproduct, we also provide a probabilistic approach to prove the existence of the above fully nonlinear degenerate PIDE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-30T08:36:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G51",
      "60G52",
      "60G65",
      "45K05"
    ],
    "keywords": [
      "universal robust limit theorem",
      "sublinear expectation",
      "nonlinear lévy processes",
      "nonlinear degenerate partial integro-differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}