{
  "id": "math/0606084",
  "version": "v1",
  "published": "2006-06-04T07:12:51.000Z",
  "updated": "2006-06-04T07:12:51.000Z",
  "title": "Some properties of exponential integrals of Lévy processes and examples",
  "authors": [
    "Hitoshi Kondo",
    "Makoto Maejima",
    "Ken-iti Sato"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The improper stochastic integral $Z=\\int_0^{\\infty-}\\exp(-X_{s-})dY_s$ is studied, where $\\{(X_t, Y_t), t \\geqslant 0 \\}$ is a L\\'evy process on $\\mathbb R ^{1+d}$ with $\\{X_t \\}$ and $\\{Y_t \\}$ being $\\mathbb R$-valued and $\\mathbb R ^d$-valued, respectively. The condition for existence and finiteness of $Z$ is given and then the law $\\mathcal L(Z)$ of $Z$ is considered. Some sufficient conditions for $\\mathcal L(Z)$ to be selfdecomposable and some sufficient conditions for $\\mathcal L(Z)$ to be non-selfdecomposable but semi-selfdecomposable are given. Attention is paid to the case where $d=1$, $\\{X_t\\}$ is a Poisson process, and $\\{X_t\\}$ and $\\{Y_t\\}$ are independent. An example of $Z$ of type $G$ with selfdecomposable mixing distribution is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-04T07:12:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E07",
      "60G51",
      "60H05"
    ],
    "keywords": [
      "exponential integrals",
      "lévy processes",
      "properties",
      "sufficient conditions",
      "improper stochastic integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6084K"
    }
  }
}