On an Optimal Stopping Problem of an Insider

We consider the optimal stopping problem $v^{(\eps)}:=\sup_{\tau\in\mathcal{T}_{0,T}}\mathbb{E}B_{(\tau-\eps)^+}$ posed by Shiryaev at the International Conference on Advanced Stochastic Optimization Problems organized by the Steklov Institute of Mathematics in September 2012. Here $T>0$ is a fixed time horizon, $(B_t)_{0\leq t\leq T}$ is the Brownian motion, $\eps\in[0,T]$ is a constant, and $\mathcal{T}_{\eps,T}$ is the set of stopping times taking values in $[\eps,T]$. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equations, from which the continuity of $\eps \to v^{(\eps)}$ follows. For large enough $\eps$, we obtain an explicit expression for $v^{(\eps)}$ and for small $\eps$ we have lower and upper bounds. The main result of the paper is the asymptotics of $v^{(\eps)}$ as $\eps\searrow 0$. As a byproduct, we also obtain L\'{e}vy's modulus of continuity result in the $L^1$ sense.