Uniqueness of First Passage Time Distributions via Fredholm Integral Equations

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*} \tau := \inf \{ t \geq 0 \vert W_t \geq b(t) \}. \end{align*} It is well-known that...

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Bibliographic Details
Published in:arXiv.org 2023-03
Main Authors: Christensen, Sören, Fischer, Simon, Hallmann, Oskar
Format: Article
Language:English
Subjects:
Brownian motion
Continuity (mathematics)
Fredholm equations
Integral equations
Mathematical analysis
Online Access:Get full text
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Summary: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*} \tau := \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.
ISSN:2331-8422