Joint distributions of partial and global maxima of a Brownian Bridge

We analyze the joint distributions and temporal correlations between the partial maximum $m$ and the global maximum $M$ achieved by a Brownian Bridge on the subinterval $[0,t_1]$ and on the entire interval $[0,t]$, respectively. We determine three probability distribution functions: The joint distribution $P(m,M)$ of both maxima; the distribution $P(m)$ of the partial maximum; and the distribution $\mathrm{\Pi }(G)$ of the gap between the maxima, $G=M-m$. We present exact results for the moments of these distributions and quantify the temporal correlations between $m$ and $M$ by calculating the Pearson correlation coefficient.