{
  "id": "1803.04184",
  "version": "v1",
  "published": "2018-03-12T11:01:51.000Z",
  "updated": "2018-03-12T11:01:51.000Z",
  "title": "On the first-passage area of a L$\\acute{\\text{e}}$vy process",
  "authors": [
    "Mario Abundo",
    "Sara Furia"
  ],
  "comment": "18 pages, 9 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let be $X(t)= x - \\mu t + \\sigma B_t - N_t$ a L$\\acute{\\text{e}}$vy process starting from $x >0,$ where $ \\mu \\ge 0, \\ \\sigma \\ge 0, \\ B_t$ is a standard BM, and $N_t$ is a homogeneous Poisson process with intensity $ \\theta >0,$ starting from zero. We study the joint distribution of the first-passage time below zero, $\\tau (x),$ and the first-passage area, $A(x),$ swept out by $X$ till the time $\\tau (x).$ In particular, we establish differential-difference equations with outer conditions for the Laplace transforms of $\\tau(x)$ and $A(x),$ and for their joint moments. In a special case $(\\mu = \\sigma =0),$ we show an algorithm to find recursively the moments $E[\\tau(x)^m A(x)^n],$ for any integers $m$ and $n;$ moreover, we obtain the expected value of the time average of $X$ till the time $\\tau(x).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-12T11:01:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60H05",
      "60H10"
    ],
    "keywords": [
      "vy process",
      "first-passage area",
      "special case",
      "standard bm",
      "joint moments"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}