{
  "id": "1406.2351",
  "version": "v1",
  "published": "2014-06-09T20:49:34.000Z",
  "updated": "2014-06-09T20:49:34.000Z",
  "title": "Stochastic Calculus for Markov Processes Associated with Semi-Dirichlet Forms",
  "authors": [
    "Chuan-Zhong Chen",
    "Li Ma",
    "Wei Sun"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(\\mathcal{E},D(\\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\\geq0}$ be the associated Markov process. For $u\\in D(\\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\\tilde{u}(X_{t})-\\tilde{u}(X_{0})$ and $F^{[u]}_t:=\\sum_{0<s\\leq t}(\\tilde u(X_{s})-\\tilde u(X_{s-}))1_{\\{|\\tilde u(X_{s})-\\tilde u(X_{s-})|>1\\}}$, where $\\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^{[u]}+Z_t^{[u]}+F_t^{[u]}.$$ Further, we define the stochastic integral $\\int_0^t\\tilde v(X_{s-})dA_s^{[u]}$ for $v\\in D(\\mathcal{E})_{loc}$ and derive the related It\\^{o}'s formula.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-09T20:49:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "60J25"
    ],
    "keywords": [
      "semi-dirichlet form",
      "markov processes",
      "stochastic calculus",
      "locally square integrable martingale",
      "continuous local additive functional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.2351C"
    }
  }
}