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 uT or at an intermediate time uT. Also, the analogous conditional probability concerning Bt...

We consider the double-barrier inverse first-passage time (IFPT) problem for Wiener process X(t), starting from a random position η. Let ab such that P(aηb)=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the first-exit time of X(t)...