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We calculate analytically the probability density $P(t_m)$ of the time $t_m$at which a continuous-time Brownian motion (with and without drift) attains itsmaximum before passing through the origin for the first time. We also computethe joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In thedriftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\simt_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. Inpresence of a drift towards the origin, $P(t_m)$ decays exponentially for large$t_m$. The results from numerical simulations are in excellent agreement withour analytical predictions.

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