{
  "id": "1304.1377",
  "version": "v2",
  "published": "2013-04-04T14:29:17.000Z",
  "updated": "2013-10-20T19:08:13.000Z",
  "title": "On the loss of the semimartingale property at the hitting time of a level",
  "authors": [
    "Aleksandar Mijatović",
    "Mikhail Urusov"
  ],
  "comment": "introduction revised, 27 pages, to appear in Journal of Theoretical Probability",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper studies the loss of the semimartingale property of the process $g(Y)$ at the time a one-dimensional diffusion $Y$ hits a level, where $g$ is a difference of two convex functions. We show that the process $g(Y)$ can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the \\textit{first} and \\textit{second kind}. We give a deterministic if and only if condition (in terms of $g$ and the coefficients of $Y$) for $g(Y)$ to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion $Y$ on $[0,\\infty)$ and a \\emph{predictable} finite stopping time $\\zeta$, such that $Y$ is a semimartingale on the stochastic interval $[0,\\zeta)$, continuous at $\\zeta$ and constant after $\\zeta$, but is \\emph{not} a semimartingale on $[0,\\infty)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-20T19:08:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "60J60",
      "60J55"
    ],
    "keywords": [
      "semimartingale property",
      "hitting time",
      "paper studies",
      "one-dimensional diffusion",
      "convex functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1377M"
    }
  }
}