Uniqueness of first passage time distributions via Fredholm integral equations

Sören Christensen, Simon Fischer and Oskar Hallmann

Statistics & Probability Letters, 2023, vol. 203, issue C

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.

Keywords: First passage time problem; Brownian motion; Fredholm integral equations; Uniqueness (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spl.2023.109912

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