On the excursions of drifted Brownian motion and the successive passage times of Brownian motion

TL;DR

This paper derives the distribution of the n-th passage time for drifted Brownian motion crossing a linear boundary, extending to space-time transformations in the zero-drift case, using excursion theory.

Contribution

It introduces new distributional results for passage times of drifted Brownian motion and extends these to space-time transformations when drift is zero.

Findings

Distribution of the n-th passage time for drifted Brownian motion is obtained.

Results are extended to space-time transformations in the zero-drift case.

Provides analytical tools for understanding Brownian excursions and crossing times.

Abstract

By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)= a + bt.$ In the special case when $b = 0,$ we extend the result to a space-time transformation of Brownian motion.