{
  "id": "0710.2977",
  "version": "v1",
  "published": "2007-10-16T08:22:52.000Z",
  "updated": "2007-10-16T08:22:52.000Z",
  "title": "An explicit formula for the Skorokhod map on $[0,a]$",
  "authors": [
    "Lukasz Kruk",
    "John Lehoczky",
    "Kavita Ramanan",
    "Steven Shreve"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/009117906000000890 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 5, 1740-1768",
  "doi": "10.1214/009117906000000890",
  "categories": [
    "math.PR"
  ],
  "abstract": "The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map $\\Gamma_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $\\mathcal{D}[0,\\infty)$ of right-continuous functions with left limits taking values in $\\mathbb{R}$, $\\Gamma_{0,a}=\\Lambda_a\\circ \\Gamma_0$, where $\\Lambda_a:\\mathcal{D}[0,\\infty)\\to\\mathcal{D}[0,\\infty)$ is defined by \\[\\Lambda_a(\\phi)(t)=\\phi(t)-\\sup_{s\\in[0,t]}\\biggl[\\bigl(\\ phi(s)-a\\bigr)^+\\wedge\\inf_{u\\in[s,t]}\\phi(u)\\biggr]\\] and $\\Gamma_0:\\mathcal{D}[0,\\infty)\\to\\mathcal{D}[0,\\infty)$ is the Skorokhod map on $[0,\\infty)$, which is given explicitly by \\[\\Gamma_0(\\psi)(t)=\\psi(t)+\\sup_{s\\in[0,t]}[-\\psi(s)]^+.\\] In addition, properties of $\\Lambda_a$ are developed and comparison properties of $\\Gamma_{0,a}$ are established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-16T08:22:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G05",
      "60G17",
      "60J60",
      "90B05",
      "90B22"
    ],
    "keywords": [
      "skorokhod map",
      "explicit formula",
      "stochastic differential equations",
      "convenient tool",
      "left limits"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.2977K"
    }
  }
}