Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

We calculate analytically the probability density P(t(m)) of the time tm at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density P(M, tm) of the maximum M and t(m). In the driftless case, we find that P(t(m)) has power-law tails: P(tm) similar to t-(-3/2)(m) m for large tm and P(t(m)) similar to t(m)(-1/2) m for small tm. In the presence of a drift towards the origin, P(t(m)) decays exponentially for large tm. The results from numerical simulations are in excellent agreement with our analytical predictions.