{
  "id": "2206.01049",
  "version": "v1",
  "published": "2022-06-02T14:01:02.000Z",
  "updated": "2022-06-02T14:01:02.000Z",
  "title": "A path-dependent stochastic Gronwall inequality and strong convergence rate for stochastic functional differential equations",
  "authors": [
    "Martin Hutzenthaler",
    "Tuan Anh Nguyen"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive a stochastic Gronwall lemma with suprema over the paths in the upper bound of the assumed affine-linear growth assumption. This allows applications to It\\^o processes with coefficients which depend on earlier time points such as stochastic delay equations or Euler-type approximations of stochastic differential equations. We apply our stochastic Gronwall lemma with path-suprema to stochastic functional differential equations and prove a strong convergence rate for coefficient functions which depend on path-suprema.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-02T14:01:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "65C30",
      "34K50"
    ],
    "keywords": [
      "stochastic functional differential equations",
      "path-dependent stochastic gronwall inequality",
      "strong convergence rate",
      "stochastic gronwall lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}