{
  "id": "2003.06869",
  "version": "v1",
  "published": "2020-03-15T17:05:50.000Z",
  "updated": "2020-03-15T17:05:50.000Z",
  "title": "$L_p$ optimal prediction of the last zero of a spectrally negative Lévy process",
  "authors": [
    "Erik J. Baurdoux",
    "J. M. Pedraza"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Given a spectrally negative L\\'evy process $X$ drifting to infinity, we are interested in finding a stopping time which minimises the $L^p$ distance ($p>1$) with $g$, the last time $X$ is negative. The solution is substantially more difficult compared to the $p=1$ case for which it was shown in \\cite{baurdoux2018predicting} that it is optimal to stop as soon as $X$ exceeds a constant barrier. In the case of $p>1$ treated here, we prove that solving this optimal prediction problem is equivalent to solving an optimal stopping problem in terms of a two-dimensional strong Markov process which incorporates the length of the current excursion away from $0$. We show that an optimal stopping time is now given by the first time that $X$ exceeds a non-increasing and non-negative curve depending on the length of the current excursion away from $0$. We also show that smooth fit holds at the boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-15T17:05:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectrally negative lévy process",
      "current excursion away",
      "two-dimensional strong markov process",
      "optimal prediction problem",
      "stopping time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}