{
  "id": "1702.03361",
  "version": "v1",
  "published": "2017-02-11T00:34:36.000Z",
  "updated": "2017-02-11T00:34:36.000Z",
  "title": "Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube",
  "authors": [
    "Zhijian He"
  ],
  "categories": [
    "math.NA"
  ],
  "abstract": "This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $[0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only $o(n^{-1/2})$ for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of $O(n^{-1/2-1/(4d-2)+\\epsilon})$ for arbitrarily small $\\epsilon>0$. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains $O(n^{-1+\\epsilon})$ if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-11T00:34:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit cube",
      "discontinuous integrands",
      "singularities",
      "root mean square error",
      "discontinuity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}