Exact simulation of the first-passage time of SDEs to time-dependent thresholds

TL;DR

This paper introduces an exact simulation method for the first-passage time of stochastic processes to time-dependent thresholds, improving accuracy and efficiency by avoiding path discretization errors.

Contribution

It extends an existing exact simulation approach to handle time-dependent thresholds using Girsanov's transformation, eliminating the need for path discretization.

Findings

Method accurately simulates first-passage times to time-dependent thresholds.

It reduces computational complexity compared to traditional methods.

Effective in applications like neuron spike time prediction.

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.