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，表示过程首次达到指定阈值所需的时间。在实际建模中，考虑时变阈值通常至关重要，因为它能提供更精确且适应性更强的框架。本文扩展了一种现有的针对恒定阈值的精确模拟方法，使其能够处理时变阈值。所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，可直接模拟首次通过时间。我们展示了该方法有效性的验证结果，包括对时变阈值的扩展、时间复杂度分析、通过数值算例与现有方法的比较，以及在神经元放电时间预测中的应用。