{
  "id": "1904.06162",
  "version": "v1",
  "published": "2019-04-12T11:38:12.000Z",
  "updated": "2019-04-12T11:38:12.000Z",
  "title": "Zooming-in on a Lévy process: Failure to observe threshold exceedance over a dense grid",
  "authors": [
    "Krzysztof Bisewski",
    "Jevgenijs Ivanovs"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For a L\\'evy process $X$ on a finite time interval consider the probability that it exceeds some fixed threshold $x>0$ while staying below $x$ at the points of a regular grid. We establish exact asymptotic behavior of this probability as the number of grid points tends to infinity. We assume that $X$ has a zooming-in limit, which necessarily is $1/\\alpha$-self-similar L\\'evy process with $\\alpha\\in(0,2]$, and restrict to $\\alpha>1$. Moreover, the moments of the difference of the supremum and the maximum over the grid points are analyzed and their asymptotic behavior is derived. It is also shown that the zooming-in assumption implies certain regularity properties of the ladder process, and the decay rate of the left tail of the supremum distribution is determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-12T11:38:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51"
    ],
    "keywords": [
      "lévy process",
      "threshold exceedance",
      "dense grid",
      "establish exact asymptotic behavior",
      "zooming-in assumption implies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}