{
  "id": "1607.00616",
  "version": "v1",
  "published": "2016-07-03T08:54:47.000Z",
  "updated": "2016-07-03T08:54:47.000Z",
  "title": "Properties of $G$-martingales with finite variation and the application to $G$-Sobolev spaces",
  "authors": [
    "Yongsheng Song"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "As is known, a process of form $\\int_0^t\\eta_sd\\langle B\\rangle_s-\\int_0^t2G(\\eta_s)ds$, $\\eta\\in M^1_G(0,T)$, is a non-increasing $G$-martingale. In this paper, we shall show that a non-increasing $G$-martingale could not be form of $\\int_0^t\\eta_sds$ or $\\int_0^t\\gamma_sd\\langle B\\rangle_s$, $\\eta, \\gamma \\in M^1_G(0,T)$, which implies that the decomposition for generalized $G$-It\\^o processes is unique: For $\\zeta\\in H^1_G(0,T)$, $\\eta\\in M^1_G(0,T)$ and non-increasing $G$-martingales $K, L$, if \\[\\int_0^t\\zeta_s dB_s+\\int_0^t\\eta_sds+K_t=L_t,\\ t\\in[0,T],\\] then we have $\\eta\\equiv0$, $\\zeta\\equiv0$ and $K_t=L_t$. As an application, we give a characterization to the $G$-Sobolev spaces introduced in Peng and Song (2015).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-03T08:54:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G44",
      "60G45",
      "60G48"
    ],
    "keywords": [
      "sobolev spaces",
      "finite variation",
      "martingale",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}