Geodesic
Stopping Brownian motion without anticipation as close as possible to its ultimate maximum
Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 1, pp. 125-136

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Let $B=(B_t)_{0 \le t \le 1}$ be the standard Brownian motion started at 0, and let $S_t=\max_{ 0 \le r \le t} B_r$ for $0 \le t \le 1$. Consider the optimal stopping problem $$ V_*=\inf_\tau{\mathsf E}(B_\tau-S_1)^2, $$ where the infimum is taken over all stopping times of $B$ satisfying $0 \le \tau \le 1$. We show that the infimum is attained at the stopping time $\tau_*=\inf\{0\le t\le 1\mid S_t-B_t\ge z_*\sqrt{1-t}\}$, where $z_*=1.12 \ldots$ is a unique root of the equation $4\Phi(z_*)-2z_*\varphi(z_*)-3=0$ with $\varphi(x)=(1/\sqrt{2 \pi })\,e^{-x^2/2}$ and $ \Phi (x)=\int_{-\infty}^x \varphi(y) dy$. The value $V_*$ equals $2 \Phi (z_*)-1$. The method of proof relies upon a stochastic integral representation of $S_1$, time-change arguments, and the solution of a free-boundary (Stefan) problem.
Keywords: Brownian motion, optimal stopping, ultimate maximum, free-boundary (Stefan) problem, Ito–Clark representation theorem, Markov process
Mots-clés : anticipation, diffusion. 
@article{TVP_2000_45_1_a5,
     author = {S. E. Graversen and G. Peskir and A. N. Shiryaev},
     title = {Stopping {Brownian} motion without anticipation as close as possible to its ultimate maximum},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {125--136},
     publisher = {mathdoc},
     volume = {45},
     number = {1},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2000_45_1_a5/}
}
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S. E. Graversen; G. Peskir; A. N. Shiryaev. Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 1, pp. 125-136. http://geodesic.mathdoc.fr/item/TVP_2000_45_1_a5/