On the maximum of the generalized Brownian bridge

We present some extensions of the distributions of the maximum of the Brownian bridge in $[0,t]$ when the conditioning event is placed at a future time $u>t$ or at an intermediate time $u<t$. The standard distributions of Brownian motion and Brownian bridge are obtained as limiting cases. These results permit us to derive also the distribution of the first-passage time of the Brownian bridge. Similar generalizations are carded out for the Brownian bridge with drift $\mu$; in this case, it is shown that the maximal distribution is independent of $\mu$ (when $u\geqt$). Finally, the case of the two-sided maximal distribution of Brownian motion in $[0,t]$, conditioned on $B(u)=\eta$ (for both $u>t$ and $u<t$), is considered.