Uniqueness of first passage time distributions via Fredholm integral equations

Abstract

Let W be a standard Brownian motion with W0=0 and let b:R+→R be a continuous function with b(0)>0. The first passage time (from below) is then defined as τ≔inf{t≥0|Wt≥b(t)}.It is well-known that the distribution F of τ 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.