{
  "id": "2005.09354",
  "version": "v1",
  "published": "2020-05-19T10:35:54.000Z",
  "updated": "2020-05-19T10:35:54.000Z",
  "title": "Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise",
  "authors": [
    "Oumaima Bencheikh",
    "Benjamin Jourdain"
  ],
  "comment": "37 pages, 6 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $\\rho$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $\\rho \\ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-19T10:35:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H35",
      "60H10",
      "65C30",
      "65C05"
    ],
    "keywords": [
      "measurable drift coefficient",
      "euler-maruyama scheme",
      "diffusion processes",
      "additive noise",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}