{
  "id": "1704.08053",
  "version": "v1",
  "published": "2017-04-26T10:46:10.000Z",
  "updated": "2017-04-26T10:46:10.000Z",
  "title": "Canonical RDEs and general semimartingales as rough paths",
  "authors": [
    "Ilya Chevyrev",
    "Peter K. Friz"
  ],
  "comment": "38 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of L\\'epingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-26T10:46:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H99",
      "60H10"
    ],
    "keywords": [
      "general semimartingales",
      "canonical rdes",
      "defined rough path lifts",
      "multidimensional semimartingales admit",
      "canonically defined rough path"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}