Uniqueness of First Passage Time Distributions via Fredholm Integral Equations

Abstract

Let $W$ be a standard Brownian motion with $W_0=0$ and let $b:{\mathbb{R}}_{+}\to \mathbb{R}$ be a continuous function with $b(0)>0$. The first passage time (from below) is then defined as \begin{align*} τ:= \inf \{ t \geq 0 \vert W_t \geq b(t) \}. \end{align*} It is well-known that the distribution $F$ of $\tau$ satisfies a set of Fredholm equations of the first kind, which is used, for example, as a starting point for numerical approaches. For this, it is fundamental that the Fredholm equations have a unique solution. In this article, we prove this in a general setting using analytical methods.