{
  "id": "2008.03148",
  "version": "v1",
  "published": "2020-08-06T09:48:50.000Z",
  "updated": "2020-08-06T09:48:50.000Z",
  "title": "A note on the asymptotic stability of the Semi-Discrete method for Stochastic Differential Equations",
  "authors": [
    "Nikolaos Halidias",
    "Ioannis S. Stamatiou"
  ],
  "comment": "18 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:2001.07483",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of $\\mathcal L^2$-convergence of the truncated SD method and showed that it can be arbitrarily close to $1/2,$ see \\textit{Stamatiou, Halidias (2019), Convergence rates of the Semi-Discrete method for stochastic differential equations, Theory of Stochastic Processes, 24(40)}. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE, which we call Lamperti semi-discrete (LSD). Numerical simulations support our theoretical findings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-06T09:48:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60H35",
      "65C20",
      "65C30",
      "65J15",
      "65L20"
    ],
    "keywords": [
      "stochastic differential equations",
      "asymptotic stability",
      "semi-discrete method",
      "truncated sd method",
      "stochastic processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}