The trap of complacency in predicting the maximum

Abstract

Given a standard Brownian motion $B^μ=(B_t^μ)_{0\le t\le T}$ with drift $μ\in \mathbb{R}$ and letting $S_t^μ=\max_{0\le s\le t}B_s^μ$ for $0\le t\le T$, we consider the optimal prediction problem: \[V=\inf_{0\le τ\le T}\mathsf{E}(B_τ^μ-S_T^μ)^2\] where the infimum is taken over all stopping times $τ$ of $B^μ$. Reducing the optimal prediction problem to a parabolic free-boundary problem we show that the following stopping time is optimal: \[τ_*=\inf \{t_*\le t\le T\mid b_1(t)\le S_t^μ-B_t^μ\le b_2(t)\}\] 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 $b_2(T)=1/2μ$. If $μ>0$, 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 $μ\le 0$, then $t_*=0$ and $b_2\equiv +\infty$ so that $τ_*$ 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 $μ=0$, and the system of two Volterra equations reduces to one Volterra equation. If $μ=0$, 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 $μ\ne 0$ and the present paper settles the remaining cases using a different approach.