On a Property of the Moment at Which Brownian Motion Attains Its Maximum and Some Optimal Stopping Problems

Abstract

Let $B=(B_t)_{0\le t\le 1}$ be a standard Brownian motion and $\theta$ be the moment at which B attains its maximal value, i.e., $B_\theta=\max_{0\le t\le 1}B_t$. Denote by $({\cal F}^B_t)_{0\le t\le 1}$ the filtration generated by B. We prove that for any $({\cal F}^B_t)$-stopping time $\tau$ $(0\le\tau\le 1)$, the following equality holds: $$ {\bf E}(B_\theta-B_\tau)^2={\bf E}|\theta-\tau|+{\frac{1}{2}}. $$ Together with the results of [S. E. Graversen, G. Peskir, and A. N. Shiryaev, {\em Theory Probab. Appl.}, 45 (2000), pp. 41--50] this implies that the optimal stopping time $\tau_*$ in the problem $$ \inf_\tau{\bf E}|\theta-\tau| $$ has the form $$ \tau_*=\inf\big\{0\le t\le 1: S_t-B_t\ge z_*\sqrt{1-t}\,\big\}, $$ where $S_t=\max_{0\le s\le t}B_s$, $z_*$ is a unique positive root of the equation $4\Phi(z)-2z\phi(z)-3=0$, and $\phi(z)$ and $\Phi(z)$ are the density and the distribution function of a standard Gaussian random variable. Similarly, we solve the optimal stopping problems $$ \inf_{\tau\in{\mathfrak{M}}_\alpha}{\bf E}(\tau-\theta)^+ \quad\mbox{and}\quad \inf_{\tau\in{\mathfrak{N}}_\alpha}{\bf E}(\tau-\theta)^-, $$ where ${\mathfrak{M}}_\alpha=\{\tau :\,{\bf E}(\tau-\theta)^-\le \alpha\}$, and ${\mathfrak{N}}_\alpha=\{\tau :\,{\bf E}(\tau-\theta)^+\le\alpha\}$. The corresponding optimal stopping times are of the same form as above (with other $z_*$'s).