{
  "id": "2011.13030",
  "version": "v1",
  "published": "2020-11-25T21:24:19.000Z",
  "updated": "2020-11-25T21:24:19.000Z",
  "title": "A weak law of large numbers for realised covariation in a Hilbert space setting",
  "authors": [
    "Fred Espen Benth",
    "Dennis Schroers",
    "Almut E. D. Veraart"
  ],
  "comment": "34 pages",
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert-Schmidt norm. In addition, we show that the conditions on the volatility process are valid for most common stochastic volatility models in Hilbert spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-25T21:24:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F99",
      "62M99"
    ],
    "keywords": [
      "weak law",
      "large numbers",
      "hilbert space setting",
      "realised covariation",
      "space-valued stochastic evolution equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}