On a property of the moment at which Brownian motion attains its maximum

M. A. Urusov

Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 1, pp. 184-190

Résumé

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 $(\mathscr{F}^B_t)_{0\le t\le 1}$ the filtration generated by $B$. We prove that for any $(\mathscr{F}^B_t)$-stopping time $\tau$ $(0\le\tau\le 1)$, the following equality holds: $$ E(B_\theta-B_\tau)^2=E|\theta-\tau|+\frac{1}{2}. $$ Together with the results of [S. E. Graversen, G. Peskir, and A. N. Shiryaev, Theory Probab. Appl., 45 (2000), pp. 41–50] this implies that the optimal stopping time $\tau_*$ in the problem $$ \inf_\tauE|\theta-\tau| $$ 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 $4\Phi(z)-2z\phi(z)-3=0$, $\phi(z)$ and $\Phi(z)$ are the density and the distribution function of a standard Gaussian random variable. Similarly, we solve the optimal stopping problems $$ \inf_{\tau\in\mathfrak{M}_\alpha}E(\tau-\theta)^+ \quadand\quad \inf_{\tau\in\mathfrak{N}_\alpha}E(\tau-\theta)^-, $$ where $\mathfrak{M}_\alpha=\{\tau\colon\,E(\tau-\theta)^-\le \alpha\}$, and $\mathfrak{N}_\alpha=\{\tau\colon\,E(\tau-\theta)^+\le\alpha\}$. The corresponding optimal stopping times are of the same form as above (with other $z_*$'s).

Détail

Keywords: moment of attaining the maximum, Brownian motion, optimal stopping.