{
  "id": "1810.07504",
  "version": "v1",
  "published": "2018-10-17T12:49:23.000Z",
  "updated": "2018-10-17T12:49:23.000Z",
  "title": "Existence of densities for stochastic differential equations driven by Lévy processes with anisotropic jumps",
  "authors": [
    "Martin Friesen",
    "Peng Jin",
    "Barbara Rüdiger"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We study existence of densities for solutions to stochastic differential equations with H\\\"older continuous coefficients and driven by a $d$-dimensional L\\'evy process $Z=(Z_{t})_{t\\geq 0}$, where, for $t>0$, the density function $f_{t}$ of $Z_{t}$ exists and satisfies, for some $(\\alpha_{i})_{i=1,\\dots,d}\\subset(0,2)$ and $C>0$, \\begin{align*} \\limsup\\limits _{t \\to 0}t^{1/\\alpha_{i}}\\int\\limits _{\\mathbb{R}^{d}}|f_{t}(z+e_{i}h)-f_{t}(z)|dz\\leq C|h|,\\ \\ h\\in \\mathbb{R},\\ \\ i=1,\\dots,d. \\end{align*} Here $e_{1},\\dots,e_{d}$ denote the canonical basis vectors in $\\mathbb{R}^{d}$. The latter condition covers anisotropic $(\\alpha_{1},\\dots,\\alpha_{d})$-stable laws but also particular cases of subordinate Brownian motion. To prove our result we use some ideas taken from \\citep{DF13}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-17T12:49:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60E07",
      "60G30"
    ],
    "keywords": [
      "stochastic differential equations driven",
      "lévy processes",
      "anisotropic jumps",
      "dimensional levy process",
      "condition covers anisotropic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}