Probability that the maximum of the reflected Brownian motion over a finite interval $[0,t]$ is achieved by its last zero before $t$

TL;DR

This paper derives the probability that the maximum of a reflected Brownian motion over a finite interval occurs during its last zero before the endpoint, providing insights into the process's extremal behavior.

Contribution

It provides an explicit calculation of the probability that the maximum occurs during the last zero interval, a novel result in the study of reflected Brownian motions.

Findings

Explicit formula for the probability $p_c$.

Insight into the timing of maxima in reflected Brownian motion.

Enhanced understanding of extremal properties of stochastic processes.

Abstract

We calculate the probability $p_c$ that the maximum of a reflected Brownian motion $U$ is achieved on a complete excursion, i.e. $p_c:=P\big(\overline{U}(t)=U^*(t)\big)$ where $\overline{U}(t)$ (respectively $U^*(t)$) is the maximum of the process $U$ over the time interval $[0,t]$ (resp. $\big[0,g(t)\big]$ where $g(t)$ is the last zero of $U$ before $t$).