Some conditional crossing results of Brownian motion over a piecewise-linear boundary

Explicit formulae are found for the probability that the Brownian motion, Bt, up-crosses, in [0,T], a piecewise-linear function S(t), with the condition that the value of Bt is assigned at a future time u>T or at an intermediate time u<T. Also, the analogous conditional probability concerning Bt is obtained. These results permit to obtain an explicit expression for the first-passage-time distribution function of the generalized Brownian bridge, over S(t).