{
  "id": "1107.4415",
  "version": "v1",
  "published": "2011-07-22T04:45:13.000Z",
  "updated": "2011-07-22T04:45:13.000Z",
  "title": "Asymptotic behaviour of first passage time distributions for Lévy processes",
  "authors": [
    "Ronald Doney",
    "Victor Rivero"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a real valued L\\'evy process that is in the domain of attraction of a stable law without centering with norming function $c.$ As an analogue of the random walk results in \\cite{vw} and \\cite{rad} we study the local behaviour of the distribution of the lifetime $\\zeta$ under the characteristic measure $\\underline{n}$ of excursions away from 0 of the process $X$ reflected in its past infimum, and of the first passage time of $X$ below $0,$ $T_{0}=\\inf \\{t>0:X_{t}<0\\},$ under $\\mathbb{P}_{x}(\\cdot),$ for $x>0,$ in two different regimes for $x,$ viz. $x=o(c(\\cdot))$ and $x>D c(\\cdot),$ for some $D>0.$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $T_{0}$ and discontinuous passage. In the way to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $X$ reflected in its past infimum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-22T04:45:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "G52",
      "60F99"
    ],
    "keywords": [
      "first passage time distributions",
      "lévy processes",
      "asymptotic behaviour",
      "past infimum",
      "real valued levy process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.4415D"
    }
  }
}