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.

翻译：首次通过时间（FPT）是随机过程中的一个基本概念，表示一个过程首次达到指定阈值所需的时间。在实际建模中，考虑时间依赖的阈值对于准确描述随机过程至关重要，因为它提供了一个更精确且适应性更强的框架。本文针对时间依赖阈值，扩展了一种现有的、为恒定阈值开发的精确模拟方法。我们提出的方法利用布朗运动的FPT，并以一定概率接受其作为给定过程的FPT，该概率通过Girsanov变换确定。该方法无需在特定时间区间内模拟完整路径，避免了时间离散化误差，并可直接模拟首次通过时间。我们展示了该方法有效性的结果，包括向时间依赖阈值的扩展、其时间复杂度的分析、通过数值算例与现有方法的比较，以及其在预测神经元放电时间中的应用。