{
  "id": "1212.5287",
  "version": "v4",
  "published": "2012-12-20T22:21:21.000Z",
  "updated": "2015-03-18T12:36:50.000Z",
  "title": "First passage times of two-dimensional correlated diffusion processes: analytical and numerical methods",
  "authors": [
    "Laura Sacerdote",
    "Massimiliano Tamborrino",
    "Cristina Zucca"
  ],
  "comment": "18 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries. This quantity depends on the joint density of the first passage time of the first crossing component and of the position of the second crossing component before its crossing time. First we show that the these densities are solutions of a system of Volterra-Fredholm first kind integral equations. Then we propose a numerical algorithm to solve it and we describe how to use the algorithm to approximate the joint density of the first passage times. The convergence of the method is theoretically proved for any bivariate process. We also derive explicit expressions for these and other quantities of interest in the case of a bivariate Wiener process, correcting some previous misprints appearing in the literature. Finally we illustrate the application of the method through a set of examples.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-07-13T14:47:17.000Z",
      "abstract": "Given a two-dimensional correlated diffusion process, we determine the joint density of the first passage times of the process to some constant boundaries, which we assume to be either crossing or absorbing. This joint density depends on the transition densities of the process constrained to be below these levels. For general bivariate stochastic processes, closed expressions are not available. We show that the transition densities are solutions of a system of Volterra-Fredholm first kind integral equations. We propose a numerical algorithm to solve the system and thus to approximate the joint density of the first passage times, and we prove the convergence of the method. Given two correlated Wiener processes with drift, we provide explicit expressions of several distributions of interest, including the transition densities and the joint density of the passage times. A comparison between theoretical and numerical results for two correlated Wiener and a numerical illustration for two correlated Ornstein-Uhlenbeck process are carried out.",
      "comment": "17 pages, 3 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-03-18T12:36:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G40",
      "60J60",
      "65R20",
      "60J65",
      "60J70"
    ],
    "keywords": [
      "first passage times",
      "two-dimensional correlated diffusion processes",
      "joint density",
      "numerical methods",
      "transition densities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.5287S"
    }
  }
}