{
  "id": "1310.1737",
  "version": "v3",
  "published": "2013-10-07T11:43:50.000Z",
  "updated": "2014-09-11T08:29:26.000Z",
  "title": "Markov chain approximations to scale functions of Lévy processes",
  "authors": [
    "Aleksandar Mijatović",
    "Matija Vidmar",
    "Saul Jacka"
  ],
  "comment": "45 pages, 4 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce a general algorithm for the computation of the scale functions of a spectrally negative L\\'evy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with, what are nonnegative, coefficients given explicitly in terms of the L\\'evy triplet of $X$. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of $X$ and its scale functions, not unlike the one-dimensional It\\^o diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-21T11:45:00.000Z",
      "abstract": "We introduce a general algorithm for the computation of the scale functions of a spectrally negative L\\'evy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with coefficients given explicitly in terms of the L\\'evy triplet of $X$. It is easy to implement and fast to execute. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of $X$ and its scale functions, not unlike the one-dimensional It\\^o diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.",
      "comment": "Reference to Matlab code for the algorithm (available online) added, Introduction revised, 43 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-11T08:29:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G51"
    ],
    "keywords": [
      "scale functions",
      "markov chain approximations",
      "lévy processes",
      "main result establishes sharp rates",
      "upwards skip-free continuous-time markov chains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1737M"
    }
  }
}