{
  "id": "1712.03157",
  "version": "v1",
  "published": "2017-12-08T16:25:15.000Z",
  "updated": "2017-12-08T16:25:15.000Z",
  "title": "On solutions for stochastic differential equations with Hölder coefficients",
  "authors": [
    "Rongrong Tian",
    "Liang Ding",
    "Jinlong Wei"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the following stochastic differential equation (SDE for short) $$X_t=x+\\int\\limits_0^tb(s,X_s)ds+\\int\\limits_0^t\\sigma(s,X_s)dW_s,\\quad t>0, \\, x\\in\\mathbb{R}^d,$$ where $\\{W_s\\}_{s\\geq 0}$ is a $d$-dimensional standard Wiener process, $b\\in L^q_{loc}(\\mathbb{R}_+;\\mathcal{C}_b^\\alpha(\\mathbb{R}^d))$, $\\sigma\\in \\mathcal{C}(\\mathbb{R}_+;\\mathcal{C}_b^\\alpha(\\mathbb{R}^d))$ with $\\alpha\\in (0,1), q\\in [1,2]$. Suppose that $1+\\alpha-2/q>0$, and $\\sigma\\sigma^\\top$ meets uniformly elliptic condition, then there exits a weak solution to the above equation. Furthermore, if $q=2$, the weak solution is unique, the Markov semi-group has the strong Feller property and there is a density associated with the above SDE. Moreover, if $|\\nabla\\sigma|\\in L^2_{loc}(\\mathbb{R}_+;L^\\infty(\\mathbb{R}^d))$ in addition, the path-wise uniqueness holds and the unique solution $X$ lies in $L^p(\\Omega;\\mathcal{C}([0,T];\\mathcal{C}_{b}^\\beta(B_R)))$ for every $p\\geq 1$, every $R>0$, every $T>0$ and every $\\beta\\in (0,1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-08T16:25:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "34F05"
    ],
    "keywords": [
      "stochastic differential equation",
      "hölder coefficients",
      "dimensional standard wiener process",
      "weak solution",
      "meets uniformly elliptic condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}