{
  "id": "1904.07784",
  "version": "v1",
  "published": "2019-04-16T16:08:04.000Z",
  "updated": "2019-04-16T16:08:04.000Z",
  "title": "The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem",
  "authors": [
    "Andreas Neuenkirch",
    "Michaela Szölgyenyi"
  ],
  "categories": [
    "math.PR",
    "math.NA"
  ],
  "abstract": "We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a novel framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $\\kappa \\in (0,1)$ for the drift, our analysis of the quadrature problem yields the convergence order $\\min\\{3/4,(1+\\kappa)/2\\}-\\epsilon$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $\\epsilon>0$). The cut-off of the convergence order at $\\kappa=3/4$ can be overcome by using a suitable non-equidistant discretization, which yields the strong convergence order of $(1+\\kappa)/2-\\epsilon$ for the corresponding Euler-Maruyama scheme.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-16T16:08:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60H35",
      "65C30"
    ],
    "keywords": [
      "irregular drift",
      "convergence rates",
      "strong convergence order",
      "scalar stochastic differential equations",
      "equidistant euler-maruyama scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}