{
  "id": "2205.14416",
  "version": "v1",
  "published": "2022-05-28T12:36:26.000Z",
  "updated": "2022-05-28T12:36:26.000Z",
  "title": "When is the convex hull of a Lévy process smooth?",
  "authors": [
    "David Bang",
    "Jorge Ignacio González Cázares",
    "Aleksandar Mijatović"
  ],
  "comment": "33 pages, 3 figures, YouTube video on: https://youtu.be/qPxBqaq2AsQ",
  "categories": [
    "math.PR"
  ],
  "abstract": "We characterise, in terms of their transition laws, the class of one-dimensional L\\'evy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation L\\'evy processes and depends subtly on the behaviour of the L\\'evy measure at zero. We introduce a class of strongly eroded L\\'evy processes, whose Dini derivatives vanish at every local minimum of the trajectory for all perturbations with a linear drift, and prove that these are precisely the processes with smooth convex hulls. We study how the smoothness of the convex hull can break and construct examples exhibiting a variety of smooth/non-smooth behaviours. Finally, we conjecture that an infinite variation L\\'evy process is either strongly eroded or abrupt, a claim implied by Vigon's point-hitting conjecture. In the finite variation case, we characterise the points of smoothness of the hull in terms of the L\\'evy measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-28T12:36:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51"
    ],
    "keywords": [
      "lévy process smooth",
      "infinite variation levy process",
      "levy measure",
      "one-dimensional levy processes",
      "finite variation case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}