{
  "id": "1609.06854",
  "version": "v1",
  "published": "2016-09-22T08:04:48.000Z",
  "updated": "2016-09-22T08:04:48.000Z",
  "title": "On the joint distribution of first-passage time and first-passage area of drifted Brownian motion",
  "authors": [
    "Mario Abundo",
    "Danilo Del Vescovo"
  ],
  "comment": "13 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "For drifted Brownian motion $X(t)= x - \\mu t + B_t \\ (\\mu >0)$ starting from $x>0,$ 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 equations with boundary conditions for the joint moments $E[\\tau(x)^m A(x)^n],$ and we present an algorithm to find recursively them, for any $m$ and $n.$ Finally, the expected value of the time average of $X$ till the time $\\tau(x)$ is obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-22T08:04:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60H05",
      "60H10"
    ],
    "keywords": [
      "drifted brownian motion",
      "joint distribution",
      "first-passage time",
      "first-passage area",
      "time average"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}