{
  "id": "math/0606783",
  "version": "v1",
  "published": "2006-06-30T07:45:30.000Z",
  "updated": "2006-06-30T07:45:30.000Z",
  "title": "On the absolute continuity of Lévy processes with drift",
  "authors": [
    "Ivan Nourdin",
    "Thomas Simon"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117905000000620 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 3, 1035-1051",
  "doi": "10.1214/009117905000000620",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the problem of absolute continuity for the one-dimensional SDE \\[X_t=x+\\int_0^ta(X_s) ds+Z_t,\\] where $Z$ is a real L\\'{e}vy process without Brownian part and $a$ a function of class $\\mathcal{C}^1$ with bounded derivative. Using an elementary stratification method, we show that if the drift $a$ is monotonous at the initial point $x$, then $X_t$ is absolutely continuous for every $t>0$ if and only if $Z$ jumps infinitely often. This means that the drift term has a regularizing effect, since $Z_t$ itself may not have a density. We also prove that when $Z_t$ is absolutely continuous, then the same holds for $X_t$, in full generality on $a$ and at every fixed time $t$. These results are then extended to a larger class of elliptic jump processes, yielding an optimal criterion on the driving Poisson measure for their absolute continuity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-30T07:45:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51",
      "60H10"
    ],
    "keywords": [
      "absolute continuity",
      "lévy processes",
      "elementary stratification method",
      "elliptic jump processes",
      "one-dimensional sde"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6783N"
    }
  }
}