{
  "id": "1105.0062",
  "version": "v2",
  "published": "2011-04-30T08:42:45.000Z",
  "updated": "2012-02-07T21:30:10.000Z",
  "title": "A Wiener-Hopf Type Factorization for the Exponential Functional of Levy Processes",
  "authors": [
    "Pierre Patie",
    "Juan Carlos Pardo Milan",
    "Mladen Savov"
  ],
  "doi": "10.1112/jlms/jds028",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a L\\'evy process $\\xi=(\\xi_t)_{t\\geq0}$ drifting to $-\\infty$, we define the so-called exponential functional as follows \\[{\\rm{I}}_{\\xi}=\\int_0^{\\infty}e^{\\xi_t} dt.\\] Under mild conditions on $\\xi$, we show that the following factorization of exponential functionals \\[{\\rm{I}}_{\\xi}\\stackrel{d}={\\rm{I}}_{H^-} \\times {\\rm{I}}_{Y}\\] holds, where, $\\times $ stands for the product of independent random variables, $H^-$ is the descending ladder height process of $\\xi$ and $Y$ is a spectrally positive L\\'evy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of ${\\rm{I}}_{\\xi}$ for a large class of L\\'evy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein-Uhlenbeck processes which is of independent interest on its own. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-02-07T21:30:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60J25",
      "47A68",
      "60E07"
    ],
    "keywords": [
      "exponential functional",
      "wiener-hopf type factorization",
      "levy processes",
      "ascending ladder height process",
      "fine markovian study"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.0062P"
    }
  }
}