{
  "id": "1210.4572",
  "version": "v1",
  "published": "2012-10-16T20:44:33.000Z",
  "updated": "2012-10-16T20:44:33.000Z",
  "title": "Fractional smoothness of functionals of diffusion processes under a change of measure",
  "authors": [
    "Stefan Geiss",
    "Emmanuel Gobet"
  ],
  "categories": [
    "math.PR",
    "math.AP",
    "math.FA"
  ],
  "abstract": "Let $v:[0,T]\\times \\R^d \\to \\R$ be the solution of the parabolic backward equation $ \\partial_t v + (1/2) \\sum_{i,l} [\\sigma \\sigma^\\perp]_{il} \\partial_{x_i \\partial_{x_l} v + \\sum_{i} b_i \\partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t)_{t\\in [0,T]}$ be the associated $\\R^d$-valued diffusion process on some appropriate $(\\Omega,\\cF,\\Q)$. For $p\\in [2,\\infty)$ and a measure $d\\P=\\lambda_T d\\Q$, where $\\lambda_T$ satisfies the Muckenhoupt condition $A_\\alpha$ for $\\alpha \\in (1,p)$, we relate the behavior of $\\|g(X_T)-\\ept g(X_T) \\|_{L_p(\\P)}$, $\\|\\nabla v(t,X_t) \\|_{L_p(\\P)}$ and $\\|D^2 v(t,X_t) \\|_{L_p(\\P)}$ to each other, where $D^2v:=(\\partial_{x_i \\partial_{x_l}v)_{i,l}$ is the Hessian matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-10-16T20:44:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H30",
      "46B70",
      "35K10",
      "35Bxx"
    ],
    "keywords": [
      "diffusion processes",
      "fractional smoothness",
      "functionals",
      "hessian matrix",
      "parabolic backward equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.4572G"
    }
  }
}