First Passage Time of Skew Brownian Motion
Thilanka Appuhamillage, Daniel Sheldon
TL;DR
This paper derives the first passage time distribution for skew Brownian motion, a problem that has remained unsolved for nearly fifty years, by generalizing previous results on excursion heights.
Contribution
It provides the first explicit formulas for the distribution of first passage times of skew Brownian motion, extending prior work on excursion heights.
Findings
Derived formulas for ranked excursion heights of skew Brownian motion.
Obtained the first passage time distribution as a corollary.
Extended previous results to a broader class of stochastic processes.
Abstract
Nearly fifty years after the introduction of skew Brownian motion by It\^o and McKean (1963), the first passage time distribution remains unknown. In this paper, we generalize results of Pitman and Yor (2001) and Cs\'aki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion. We then derive the first passage time distribution as a simple corollary.
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Taxonomy
TopicsFinancial Risk and Volatility Modeling · Complex Systems and Time Series Analysis · Stochastic processes and financial applications