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. $p_c:=P(\overline{U}(t)=U^{\ast }(t))$ where $\overline{U}(t)$ (respectively $U^{\ast }(t)$) is the maximum of the process $U$ over the time interval $[0,t]$ (resp. $[0,g(t)]$ where $g(t)$ is the last zero of $U$ before $t$).

9 pages ; 1 figure