{
  "id": "0906.5083",
  "version": "v1",
  "published": "2009-06-27T14:51:34.000Z",
  "updated": "2009-06-27T14:51:34.000Z",
  "title": "Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes",
  "authors": [
    "Mark S. Veillette",
    "Murad S. Taqqu"
  ],
  "comment": "13 pages, one figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{D(s), s \\geq 0 \\}$ be a L\\'evy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that $D(0) = 0$. We study the first-hitting time of the process $D$, namely, the process $E(t) = \\inf \\{s: D(s) > t \\}$, $t \\geq 0$. The process $E$ is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the $n$-time tail distribution function $P[E(t_1) > s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution given natural boundary conditions. This PDE can be used to derive all $n$-time moments of the process $E$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-27T14:51:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "60G51",
      "60J75",
      "60E07"
    ],
    "keywords": [
      "increasing levy processes",
      "first-passage times",
      "joint moments",
      "time tail distribution function",
      "partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.5083V"
    }
  }
}