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

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 $[0,t]$ (respectively $\big[0,g(t)\big]$ where $g(t)$ is the last zero of $U$ before $t$).