{
  "id": "2202.01904",
  "version": "v1",
  "published": "2022-02-03T23:14:07.000Z",
  "updated": "2022-02-03T23:14:07.000Z",
  "title": "A Note on the Conditional Probabilities of the Telegraph Process",
  "authors": [
    "Fabrizio Cinque"
  ],
  "comment": "15 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the telegraph process with two velocities, $a_1>a_2\\in\\mathbb{R}$, and two rates of reversal, $\\lambda_1,\\lambda_2>0$. We study some of its features with respect to the conditional probability measure where both the initial speed and the number of changes of direction are known. We exhibit a new proof by induction of the (conditional) probability law and a detailed study of the distribution of the motion at time $t>0$ conditioned on its position at a previous time $0<s<t$. In the case of a symmetric process, we present some results on the joint distribution of the position of the motion at time $t>0$, its maximum and its minimum up to that moment.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-03T23:14:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K99",
      "60G50"
    ],
    "keywords": [
      "telegraph process",
      "conditional probability measure",
      "joint distribution",
      "probability law",
      "symmetric process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}