{
  "id": "1401.6198",
  "version": "v2",
  "published": "2014-01-23T21:47:33.000Z",
  "updated": "2014-11-06T05:04:04.000Z",
  "title": "On a Class of Stochastic Differential Equations With Jumps and Its Properties",
  "authors": [
    "Ari Arapostathis",
    "Anup Biswas",
    "Luis Caffarelli"
  ],
  "comment": "46 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differential operator. One of the key results is the regularity of solutions of the Dirichlet problem for a class of operators with locally weakly H\\\"older continuous kernels.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-23T21:47:33.000Z",
      "title": "On a class of SDE with jumps and its properties",
      "abstract": "We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for a class of these operators. We also establish key connections between the recurrence properties of these jump processes and the non-local partial differential operator.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-06T05:04:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60J75",
      "58F11"
    ],
    "keywords": [
      "properties",
      "non-local partial differential operator",
      "study stochastic differential equations",
      "corresponding non-local partial differential equations",
      "basic stochastic characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.6198A"
    }
  }
}