{
  "id": "2305.09766",
  "version": "v1",
  "published": "2023-05-16T19:36:02.000Z",
  "updated": "2023-05-16T19:36:02.000Z",
  "title": "Stopping Times of Boundaries: Relaxation and Continuity",
  "authors": [
    "H. Mete Soner",
    "Valentin Tissot-Daguette"
  ],
  "comment": "22 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study boundaries and their hitting times in the context of optimal stopping and Bermudan-style options. We prove the continuity of the map linking the boundary to the value of its associated stopping policy. While the supremum norm is used to compare continuous boundaries, a weaker topology induced by the so-called relaxed $L^{\\infty}$ distance is used in the semicontinuous case. Relaxed stopping rules and their properties are also discussed. Incidentally, we provide a convergence analysis for neural stopping boundaries [Reppen, Soner, and Tissot-Daguette, Neural Optimal Stopping Boundary, 2022] entailing the universal approximation capability of neural networks and the notion of inf/sup convolution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-16T19:36:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "91G20",
      "60G57",
      "68T07"
    ],
    "keywords": [
      "stopping times",
      "continuity",
      "relaxation",
      "neural optimal stopping boundary",
      "universal approximation capability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}