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 a parabolic free-boundary problem we show that the following stopping time is optimal: $$\tau_*=\inf \{t_*\le t\le T\mid b_1(t)\le S_t^{\mu}-B_t^{\mu}\le b_2(t)\}$$ where $t_*\in [0,T)$ and the functions $t\mapsto b_1(t)$ and $t\mapsto b_2(t)$ are continuous on $[t_*,T]$ with $b_1(T)=0$ and $b_2(T)=1/2\mu$. If $\mu>0$, then $b_1$ is decreasing and $b_2$ is increasing on $[t_*,T]$ with $b_1(t_*)=b_2(t_*)$ when $t_*\ne 0$. Using local time-space calculus we derive a coupled system of nonlinear Volterra integral equations of the second kind and show that the pair of optimal boundaries $b_1$ and $b_2$ can be characterized as the unique solution to this system. This also leads to an explicit formula for $V$ in terms of $b_1$ and $b_2$. If $\mu \le 0$, then $t_*=0$ and $b_2\equiv +\infty$ so that $\tau_*$ is expressed in terms of $b_1$ only. In this case $b_1$ is decreasing on $[z_*,T]$ and increasing on $[0,z_*)$ for some $z_*\in [0,T)$ with $z_*=0$ if $\mu=0$, and the system of two Volterra equations reduces to one Volterra equation. If $\mu=0$, then there is a closed form expression for $b_1$. This problem was solved in [Theory Probab. Appl. 45 (2001) 125--136] using the method of time change (i.e., change of variables). The method of time change cannot be extended to the case when $\mu \ne 0$ and the present paper settles the remaining cases using a different approach.