{
  "id": "1001.3972",
  "version": "v1",
  "published": "2010-01-22T12:17:30.000Z",
  "updated": "2010-01-22T12:17:30.000Z",
  "title": "Martingale representation for Poisson processes with applications to minimal variance hedging",
  "authors": [
    "Guenter Last",
    "Mathew D. Penrose"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR",
    "q-fin.CP"
  ],
  "abstract": "We consider a Poisson process $\\eta$ on a measurable space $(\\BY,\\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $\\lambda$ of $\\eta$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $\\eta$), which was previously known only in the special case, when $\\lambda$ is the product of Lebesgue measure on $\\R_+$ and a $\\sigma$-finite measure on another space $\\BX$. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of It\\^o of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-22T12:17:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60G44",
      "60G51"
    ],
    "keywords": [
      "poisson processes",
      "minimal variance hedging",
      "martingale representation",
      "independent random measure",
      "quite general financial market driven"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}