{
  "id": "1803.04859",
  "version": "v1",
  "published": "2018-03-13T14:57:08.000Z",
  "updated": "2018-03-13T14:57:08.000Z",
  "title": "On moments of exponential functionals of additive processes",
  "authors": [
    "Paavo Salminenåbo",
    "Lioudmila Vostrikova"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let X = (X t) t$\\ge$0 be a real-valued additive process, i.e., a process with independent increments. In this paper we study the exponential integral functionals of X, namely, the functionals of the form I s,t = t s exp(--X u)du, 0 $\\le$ s < t $\\le$ $\\infty$. Our main interest is focused on the moments of I s,t of order $\\alpha$ $\\ge$ 0. In the case when the Laplace exponent of X t is explicitly known, we derive a recursive (in $\\alpha$) integral equation for the moments. This yields a multiple integral formula for the entire positive moments of I s,t. From these results emerges an easy-to-apply sufficient condition for the finiteness of all the entire moments of I $\\infty$ := I 0,$\\infty$. The corresponding formulas for L{\\'e}vy processes are also presented. As examples we discuss the finiteness of the moments of I $\\infty$ when X is the first hit process associated with a diffusion. In particular, we discuss the exponential functionals related with Bessel processes and geometric Brownian motions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-13T14:57:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exponential functionals",
      "additive process",
      "exponential integral functionals",
      "first hit process",
      "multiple integral formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}