{
  "id": "1306.1458",
  "version": "v2",
  "published": "2013-06-06T16:24:41.000Z",
  "updated": "2017-03-04T04:45:13.000Z",
  "title": "Modified Euler approximation scheme for stochastic differential equations driven by fractional Brownian motions",
  "authors": [
    "Yaozhong Hu",
    "Yanghui Liu",
    "David Nualart"
  ],
  "comment": "This paper has been withdrawn. It has been replaced an updated version of the paper: arXiv:1408.6471",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter $H> \\frac12$ it is known that the classical Euler scheme has the rate of convergence $2H-1$. In this paper we introduce a new numerical scheme which is closer to the classical Euler scheme for diffusion processes, in the sense that it has the rate of convergence $2H-\\frac12$. In particular, the rate of convergence becomes $\\frac 12$ when $H$ is formally set to $\\frac 12$ (the rate of Euler scheme for classical Brownian motion). The rate of weak convergence is also deduced for this scheme. The main tools are fractional calculus and Malliavin calculus. We also apply our approach to the classical Euler scheme.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-06-06T16:24:41.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-03-04T04:45:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential equations driven",
      "fractional brownian motion",
      "modified euler approximation scheme",
      "classical euler scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.1458H"
    }
  }
}