{
  "id": "1107.3760",
  "version": "v1",
  "published": "2011-07-19T16:20:00.000Z",
  "updated": "2011-07-19T16:20:00.000Z",
  "title": "On the density of exponential functionals of Lévy processes",
  "authors": [
    "Juan Carlos Pardo",
    "Victor Rivero",
    "Kees van Schaik"
  ],
  "comment": "9 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the existence of the density associated to the exponential functional of the L\\'evy process $\\xi$, \\[ I_{\\ee_q}:=\\int_0^{\\ee_q} e^{\\xi_s} \\, \\mathrm{d}s, \\] where $\\ee_q$ is an independent exponential r.v. with parameter $q\\geq 0$. In the case when $\\xi$ is the negative of a subordinator, we prove that the density of $I_{\\ee_q}$, here denoted by $k$, satisfies an integral equation that generalizes the one found by Carmona et al. \\cite{Carmona97}. Finally when $q=0$, we describe explicitly the asymptotic behaviour at 0 of the density $k$ when $\\xi$ is the negative of a subordinator and at $\\infty$ when $\\xi$ is a spectrally positive L\\'evy process that drifts to $+\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-19T16:20:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51"
    ],
    "keywords": [
      "exponential functional",
      "lévy processes",
      "independent exponential",
      "subordinator",
      "integral equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.3760P"
    }
  }
}