{
  "id": "1310.1587",
  "version": "v1",
  "published": "2013-10-06T14:08:31.000Z",
  "updated": "2013-10-06T14:08:31.000Z",
  "title": "The asymptotic behavior of the density of the supremum of Lévy processes",
  "authors": [
    "Loïc Chaumont",
    "Jacek Malecki"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let us consider a real L\\'evy process X whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0,\\infty). We show that its density f_t(x) is continuous on (0,\\infty) if and only if the potential density h' of the upward ladder height process is continuous on (0,\\infty). Then we prove that f_t behaves at 0 as h'. We also describe the asymptotic behaviour of f_t, when t tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the L\\'evy process conditioned to stay positive.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-06T14:08:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60J75"
    ],
    "keywords": [
      "asymptotic behavior",
      "lévy processes",
      "real levy process",
      "upward ladder height process",
      "asymptotic behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1587C"
    }
  }
}