Stopping Brownian motion without anticipation as close as possible to its ultimate maximum (Q2752966)

Let \(\{B(t)\), \(0<t<1\}\) be a standard Brownian motion started at 0 and let \(S_t=\max_{0\leq r\leq t} B_r\), \(0<t<1\). The authors focus on a problem of computation of \(V_{*}=\inf_{\tau} E(B_{\tau}-S_1)^2\), where the infimum is taken over all stopping times of \(B\) satisfying \(0\leq {\tau} \leq 1\). They prove that infimum is attained at the stopping time \(\tau_{*}=\inf\{0\leq t\leq 1\mid S_t-B_t \geq z_{*}\sqrt{(1-t)}\}\), where \(z_{*}\) is a unique root of the equation \(4\Phi(z_{*})-2z_{*}\varphi(z_{*})-3=0\) with \(\varphi(.)\) and \(\Phi(.)\) being the standard normal density and standard normal distribution function, respectively. The corresponding value \(V_{*}= 2\Phi(z_{*})-1\). The method of proof is based on a stochastic integral representation of the maximum process, time-change arguments, and the solution of a free-boundary (Stefan) problem. Such approach can be extended to a large class of processes with stationary independent increments.