{
  "id": "2003.06871",
  "version": "v1",
  "published": "2020-03-15T17:10:49.000Z",
  "updated": "2020-03-15T17:10:49.000Z",
  "title": "On the last zero process of a spectrally negative Lévy process",
  "authors": [
    "Erik J. Baurdoux",
    "J. M. Pedraza"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X$ be a spectrally negative L\\'evy process and consider $g_t$ the last time $X$ is below the level zero before time $t\\geq 0$. We derive an It\\^o formula for the three dimensional process $\\{(g_t,t,X_t), t\\geq 0 \\}$ and its infinitesimal generator using a perturbation method for L\\'evy processes. We also find an explicit formula for calculating functionals that include the whole path of the length of current positive excursion at time $t\\geq 0$,q $U_t:=t-g_t$. These results are applied to optimal prediction problems for the last zero $g:=\\lim_{t \\rightarrow \\infty} g_t$, when $X$ drifts to infinity. Moreover, the joint Laplace transform of $(U_{e_q},X_{e_q})$ where $e_q$ is an independent exponential time is found and a formula for a density of the $q$-potential measure of the process $\\{(U_t,X_t,),t\\geq 0 \\}$ is derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-15T17:10:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectrally negative lévy process",
      "zero process",
      "levy process",
      "independent exponential time",
      "joint laplace transform"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}