{
  "id": "2005.02276",
  "version": "v1",
  "published": "2020-05-05T15:15:16.000Z",
  "updated": "2020-05-05T15:15:16.000Z",
  "title": "On Absolute Continuity and Singularity of Multidimensional Diffusions",
  "authors": [
    "David Criens"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider two laws \\(P\\) and \\(Q\\) of multidimensional possibly explosive diffusions with common diffusion coefficient \\(\\mathfrak{a}\\) and drift coefficients \\(\\mathfrak{b}\\) and \\(\\mathfrak{b} + \\mathfrak{a} \\mathfrak{c}\\), respectively, and the law \\(P^\\circ\\) of an auxiliary diffusion with diffusion coefficient \\(\\langle \\mathfrak{c},\\mathfrak{a}\\mathfrak{c}\\rangle^{-1}\\mathfrak{a}\\) and drift coefficient \\(\\langle \\mathfrak{c}, \\mathfrak{a}\\mathfrak{c}\\rangle^{-1}\\mathfrak{b}\\). We show that \\(P \\ll Q\\) if and only if the auxiliary diffusion \\(P^\\circ\\) explodes almost surely and that \\(P\\perp Q\\) if and only if the auxiliary diffusion \\(P^\\circ\\) almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-05T15:15:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60J35",
      "60G44",
      "60H10"
    ],
    "keywords": [
      "absolute continuity",
      "multidimensional diffusions",
      "auxiliary diffusion",
      "singularity",
      "drift coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}