The trap of complacency in predicting the maximum

Given a standard Brownian motion B^{\mu}=(B_t^{\mu})_{0\le t\le T} with drift \mu \in \mathbb{R} and letting S_t^{\mu}=\max_{0\le s\le t}B_s^{\mu} for 0\le t\le T, we consider the optimal prediction problem: \[V=\inf_{0\le \tau \le T}\mathsf{E}(B_{\tau}^{\mu}-S_T^{\mu})^2\] where the infimum is taken over all stopping times \tau of B^{\mu}. Reducing the optimal prediction problem to... Show more