{
  "id": "1904.03168",
  "version": "v1",
  "published": "2019-04-05T17:17:51.000Z",
  "updated": "2019-04-05T17:17:51.000Z",
  "title": "First passage times over stochastic boundaries for subdiffusive processes",
  "authors": [
    "C. Constantinescu",
    "R. Loeffen",
    "P. Patie"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\mathbb{X}=(\\mathbb{X}_t)_{t\\geq 0}$ be the subdiffusive process defined, for any $t\\geq 0$, by $ \\mathbb{X}_t = X_{\\ell_t}$ where $X=(X_t)_{t\\geq 0}$ is a L\\'evy process and $\\ell_t=\\inf \\{s>0;\\: \\mathcal{K}_s>t \\}$ with $\\mathcal{K}=(\\mathcal{K}_t)_{t\\geq 0}$ a subordinator independent of $X$. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair $(\\mathbb{T}_a^{(\\mathcal{b})}, (\\mathbb{X} - \\mathcal{b})_{\\mathbb{T}_a^{(\\mathcal{b})}})$ where \\begin{equation*} \\mathbb{T}_a^{(\\mathcal{b})} = \\inf \\{t>0;\\: \\mathbb{X}_t > a+ \\mathcal{b}_t \\} \\end{equation*} with $a \\in \\mathbb{R}$ and $\\mathcal{b}=(\\mathcal{b}_t)_{t\\geq 0}$ a (possibly degenerate) subordinator independent of $X$ and $\\mathcal{K}$. We proceed by providing a detailed analysis of the cases where either $\\mathcal{K}$ is a stable subordinator or $X$ is spectrally negative. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable $\\mathbb{T}_a^{(\\mathcal{b})}$ has the same law as the first passage time of a semi-regenerative process of L\\'evy type, a terminology that we introduce to mean that this process satisfies the Markov property of L\\'evy processes for stopping times whose graph is included in the associated regeneration set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-05T17:17:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first passage time",
      "subdiffusive process",
      "stochastic boundaries",
      "subordinator independent",
      "levy process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}