{
  "id": "1206.2710",
  "version": "v2",
  "published": "2012-06-13T03:35:02.000Z",
  "updated": "2015-04-13T10:49:22.000Z",
  "title": "Stochastic differential equations driven by fractional Brownian motion and Poisson point process",
  "authors": [
    "Lihua Bai",
    "Jin Ma"
  ],
  "comment": "Published at http://dx.doi.org/10.3150/13-BEJ568 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)",
  "journal": "Bernoulli 2015, Vol. 21, No. 1, 303-334",
  "doi": "10.3150/13-BEJ568",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion (fBM) and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which the long term dependence between the claim payment and the past history of liability becomes the main focus. We establish some new fractional calculus on the fractional Wiener-Poisson space, from which we define the weak solution of the SDE and prove its existence and uniqueness. Using an extended form of Krylov-type estimate for the combined noise of fBM and compound Poisson, we prove the existence of the strong solution, along the lines of Gy\\\"{o}ngy and Pardoux (Probab. Theory Related Fields 94 (1993) 413-425). Our result in particular extends the one by Mishura and Nualart (Statist. Probab. Lett. 70 (2004) 253-261).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-13T03:35:02.000Z",
      "title": "Stochastic Differential Equations Driven by Fractional Brownian Motion and Poisson Point Process",
      "abstract": "In this paper we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which the long term dependence between the claim payment and the past history of liability becomes the main focus. We establish some new fractional calculus on the fractional Wiener-Poisson space, from which we define the weak solution of the SDE and prove its existence and uniqueness. Using a extended form of Krylov-type estimate for the combined noise of fBM and compound Poisson, we prove the existence of the strong solution, along the lines of Gy\\\"ongy and Pardoux (1993). Our result in particular extends a recent work of Mishura-Nualart (2004).",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-13T10:49:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential equations driven",
      "fractional brownian motion",
      "poisson point process",
      "general insurance model",
      "long term dependence"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.2710B"
    }
  }
}