{
  "id": "2106.00401",
  "version": "v1",
  "published": "2021-06-01T11:23:18.000Z",
  "updated": "2021-06-01T11:23:18.000Z",
  "title": "On moments of downwards passage times for spectrally negative Lévy processes",
  "authors": [
    "Anita Behme",
    "Philipp Lukas Strietzel"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The existence of moments of first downwards passage times of a spectrally negative L\\'evy process is governed by the general dynamics of the L\\'evy process, i.e. whether it is drifting to $+\\infty$, $-\\infty$ or oscillates. Whenever the L\\'evy process drifts to $+\\infty$, we prove that the $\\kappa$-th moment of the first passage time (conditioned to be finite) exists if and only if the $(\\kappa+1)$-th moment of the L\\'evy jump measure exists, thus generalizing a result shown earlier by Delbaen for Cram\\'er-Lundberg risk processes \\cite{Delbaen1990}. Whenever the L\\'evy process drifts to $-\\infty$ we prove that all moments of the passage time exist, while for an oscillating L\\'evy process we derive conditions for non-existence of the moments and in particular we show that no integer moments exist. Moreover we provide general formulae for integer moments of the first passage time (whenever they exist) in terms of the scale function of the L\\'evy process and its derivatives and antiderivatives.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-01T11:23:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60G40",
      "91G05"
    ],
    "keywords": [
      "spectrally negative lévy processes",
      "downwards passage times",
      "first passage time",
      "levy process drifts",
      "th moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}