Before revision:  We solve this problem by conditioning and then using the theory of  reflected backward stochastic differential equations (RBSDEs). As a corollary, we provide the solution to the optimal stopping problem \sup_{\tau\in\mathcal{T_{0,T}}\mathbb{E}B_{(\tau-\eps)^+} recently posed by Shiryaev at the International Conference on Advanced Stochastic Optimization URLanized by the Steklov Institute of Mathematics in September 2012.
After revision:  We solve this problem by conditioning and then using the theory of reflected backward stochastic differential equations _{0,T}}\mathbb{E}B_{(\tau-\eps)^+} recently posed by Shiryaev at the International Conference on Advanced Stochastic Optimization URLanized by the Steklov Institute of Mathematics in September 2012.