{
  "id": "math/0602384",
  "version": "v2",
  "published": "2006-02-17T17:03:02.000Z",
  "updated": "2007-03-27T12:40:53.000Z",
  "title": "Nonsemimartingales: Stochastic differential equations and weak Dirichlet processes",
  "authors": [
    "Rosanna Coviello",
    "Francesco Russo"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117906000000566 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2007, Vol. 35, No. 1, 255-308",
  "doi": "10.1214/009117906000000566",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we discuss existence and uniqueness for a one-dimensional time inhomogeneous stochastic differential equation directed by an $\\mathbb{F}$-semimartingale $M$ and a finite cubic variation process $\\xi$ which has the structure $Q+R$, where $Q$ is a finite quadratic variation process and $R$ is strongly predictable in some technical sense: that condition implies, in particular, that $R$ is weak Dirichlet, and it is fulfilled, for instance, when $R$ is independent of $M$. The method is based on a transformation which reduces the diffusion coefficient multiplying $\\xi$ to 1. We use generalized It\\^{o} and It\\^{o}--Wentzell type formulae. A similar method allows us to discuss existence and uniqueness theorem when $\\xi$ is a H\\\"{o}lder continuous process and $\\sigma$ is only H\\\"{o}lder in space. Using an It\\^{o} formula for reversible semimartingales, we also show existence of a solution when $\\xi$ is a Brownian motion and $\\sigma$ is only continuous.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-03-27T12:40:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H05",
      "60H10",
      "60G18",
      "60G20"
    ],
    "keywords": [
      "weak dirichlet processes",
      "time inhomogeneous stochastic differential equation",
      "one-dimensional time inhomogeneous stochastic differential",
      "nonsemimartingales"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2384C"
    }
  }
}