{
  "id": "2009.12018",
  "version": "v1",
  "published": "2020-09-25T03:44:06.000Z",
  "updated": "2020-09-25T03:44:06.000Z",
  "title": "Stochastic Integrals and Two Filtrations",
  "authors": [
    "Rajeeva L. Karandikar",
    "B. V. Rao"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In the definition of the stochastic integral, apart from the integrand and the integrator, there is an underlying filtration that plays a role. Thus, it is natural to ask: {\\it Does the stochastic integral depend upon the filtration?} In other words, if we have two filtrations, $({\\mathcal F}_\\centerdot)$ and $({\\mathcal G}_\\centerdot)$, a process $X$ that is semimartingale under both the filtrations and a process $f$ that is predictable for both the filtrations, then are the two stochastic integrals - $Y=\\int f\\,dX$, with filtration $({\\mathcal F}_\\centerdot)$ and $Z=\\int f\\,dX$, with filtration $({\\mathcal G}_\\centerdot)$ the same? When $f$ is left continuous with right limits, then the answer is yes. When one filtration is an enlargement of the other, the two integrals are equal if $f$ is bounded but this may not be the case when $f$ is unbounded. We discuss this and give sufficient conditions under which the two integrals are equal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-25T03:44:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic integral",
      "filtration",
      "sufficient conditions",
      "right limits",
      "integrator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}