{
  "id": "2412.13060",
  "version": "v1",
  "published": "2024-12-17T16:25:42.000Z",
  "updated": "2024-12-17T16:25:42.000Z",
  "title": "Exact simulation of the first-passage time of SDEs to time-dependent thresholds",
  "authors": [
    "Devika Khurana",
    "Sascha Desmettre",
    "Evelyn Buckwar"
  ],
  "categories": [
    "math.PR",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "The first-passage time (FPT) is a fundamental concept in stochastic processes, representing the time it takes for a process to reach a specified threshold for the first time. Often, considering a time-dependent threshold is essential for accurately modeling stochastic processes, as it provides a more accurate and adaptable framework. In this paper, we extend an existing Exact simulation method developed for constant thresholds to handle time-dependent thresholds. Our proposed approach utilizes the FPT of Brownian motion and accepts it for the FPT of a given process with some probability, which is determined using Girsanov's transformation. This method eliminates the need to simulate entire paths over specific time intervals, avoids time-discretization errors, and directly simulates the first-passage time. We present results demonstrating the method's effectiveness, including the extension to time-dependent thresholds, an analysis of its time complexity, comparisons with existing methods through numerical examples, and its application to predicting spike times in a neuron.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-17T16:25:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37M05",
      "65C30",
      "60G05",
      "60H35",
      "68Q87",
      "92C20"
    ],
    "keywords": [
      "first-passage time",
      "existing exact simulation method",
      "handle time-dependent thresholds",
      "avoids time-discretization errors",
      "simulate entire paths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}