{
  "id": "1608.00511",
  "version": "v1",
  "published": "2016-08-01T18:09:26.000Z",
  "updated": "2016-08-01T18:09:26.000Z",
  "title": "On Finite difference schemes for partial integro-differential equations of Lévy type",
  "authors": [
    "Konstantinos Dareiotis"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NA"
  ],
  "abstract": "In this article we introduce a finite difference approximation for integro-differential operators of L\\'evy type. We approximate solutions of integro-differential equations, where the second order operator is allowed to degenerate. In the existing literature, the L\\'evy operator is treated as a zero/first order operator outside of a centered ball of radius $\\delta$, leading to error estimates of order $\\xi (\\delta)+N(\\delta)(h+\\sqrt{\\tau})$, where $h$ and $\\tau$ are the spatial and temporal discretization parameters respectively. In these estimates $\\xi (\\delta) \\downarrow 0$, but $N(\\delta )\\uparrow \\infty$ as $\\delta \\downarrow 0$. In contrast, we treat the integro-differential operator as a second order operator on the whole unit ball. By this method we obtain error estimates of order $(h+\\tau^k)$ for $k\\in \\{1/2,1\\}$, eliminating the additional errors and the blowing up constants. Moreover, we do not pose any conditions on the L\\'evy measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-01T18:09:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R09",
      "65M06",
      "60G51"
    ],
    "keywords": [
      "finite difference schemes",
      "partial integro-differential equations",
      "lévy type",
      "second order operator",
      "integro-differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}