{
  "id": "2409.13615",
  "version": "v1",
  "published": "2024-09-20T16:23:10.000Z",
  "updated": "2024-09-20T16:23:10.000Z",
  "title": "Sharp supremum and Hölder bounds for stochastic integrals indexed by a parameter",
  "authors": [
    "Sonja Cox",
    "Joris van Winden"
  ],
  "comment": "29 pages",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We provide sharp bounds for the supremum of countably many stochastic convolutions taking values in a 2-smooth Banach space. As a consequence, we obtain sharp bounds on the modulus of continuity of a family of stochastic integrals indexed by parameter $x\\in M$, where $M$ is a metric space with finite doubling dimension. In particular, we obtain a theory of stochastic integration in H\\\"older spaces on arbitrary bounded subsets of $\\mathbb{R}^d$. This is done by relating the (generalized) H\\\"older-seminorm associated with a modulus of continuity to a supremum over countably many variables, using a Kolmogorov-type chaining argument. We provide two applications of our results: first, we show long-term bounds for Ornstein-Uhlenbeck processes, and second, we derive novel results regarding the modulus of continuity of the parabolic Anderson model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-20T16:23:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H05",
      "60G60",
      "60H15",
      "60G42"
    ],
    "keywords": [
      "stochastic integrals",
      "hölder bounds",
      "sharp supremum",
      "sharp bounds",
      "continuity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}