On first passage time problems of Brownian motion -- The inverse method of images revisited

TL;DR

This paper revisits the method of images for solving first passage time problems of Brownian motion, proposing a duality approach to characterize the measure involved and providing an algorithm for its approximation.

Contribution

It introduces a duality framework to determine the existence of the measure and develops an efficient algorithm for approximating it in first passage time problems.

Findings

Established sufficient conditions for measure existence

Developed an algorithm for measure approximation

Provided theoretical foundations for the inverse method of images

Abstract

Let $W$ be a standard Brownian motion with $W_0 = 0$ and let $b\colon[0,\infty) \to \mathbb{R}$ be a continuous function with $b(0) > 0$. In this article, we look at the classical First Passage Time (FPT) problem, i.e., the question of determining the distribution of $\tau := \inf \{ t\in [0,\infty)\colon W_t \geq b(t) \}.$ More specifically, we revisit the method of images, which we feel has received less attention than it deserves. The main observation of this approach is that the FPT problem is fully solved if a measure $\mu$ exists such that \begin{align*} \int_{(0,\infty)} \exp\left(-\frac{\theta^2}{2t}+\frac{\theta b(t)}{t}\right)\mu(d\theta)=1, \qquad t\in(0,\infty). \end{align*} The goal of this article is to lay the foundation for answering the still open question of the existence and characterisation of such a measure $\mu$ for a given curve $b$. We present a new duality approach that allows us to give sufficient conditions for the existence. Moreover, we introduce a very efficient algorithm for approximating the representing measure $\mu$ and provide a rigorous theoretical foundation.