{
  "id": "2102.06541",
  "version": "v1",
  "published": "2021-02-12T14:09:47.000Z",
  "updated": "2021-02-12T14:09:47.000Z",
  "title": "Upper functions for sample paths of Lévy(-type) processes",
  "authors": [
    "Franziska Kühn"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the small-time asymptotics of sample paths of L\\'evy processes and L\\'evy-type processes. Namely, we investigate under which conditions the limit $$\\limsup_{t \\to 0} \\frac{1}{f(t)} |X_t-X_0|$$ is finite resp.\\ infinite with probability $1$. We establish integral criteria in terms of the infinitesimal characteristics and the symbol of the process. Our results apply to a wide class of processes, including solutions to L\\'evy-driven SDEs and stable-like processes. For the particular case of L\\'evy processes, we recover and extend earlier results from the literature. Moreover, we present a new maximal inequality for L\\'evy-type processes, which is of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-12T14:09:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G17",
      "60G51",
      "60G53",
      "60J76",
      "47G20"
    ],
    "keywords": [
      "sample paths",
      "upper functions",
      "levy processes",
      "levy-type processes",
      "extend earlier results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}