03-OCT-2025
We study last passage times of standard Brownian motion and their role in the optimal prediction of the running maximum. Using explicit distributional formulas from Borodin and Salminen’s Handbook of Brownian Motion, we characterise the law of the last passage time $g_a = \sup\{t \leq 1 : B_t = a\}$ and connect it to progressive enlargement of filtrations and the theory of honest times. The Azéma–Yor martingale $M_t = \bar{B}_t – B_t$ is shown to be the key object linking last passage times to optimal stopping. We then solve Shiryaev’s problem of predicting the time at which a Brownian motion achieves its maximum on $[0,1]$, deriving the free boundary $b(t) = z^*\sqrt{1-t}$ (with $z^* \approx 0.84$) explicitly via a parabolic variational inequality, and validating the boundary numerically.