We consider the optimal problem \sup:=\sup} _{\tau\inT _{\eps,T } \mathbb{E[i=1^n (\tau-\eps^i)^+^i], where } B_{(\tau-\eps)^+} posed by Shiryaev at the International Conference on Advanced Stochastic Optimization URLanized by the Steklov Institute of Mathematics in September 2012. Here } T>0 is a fixed time horizon, ( \phi_t^i )_{0\leq t\leq T} is progressively measurable with respect to the Brownian filtration , \eps ^i \in[0,T] is a constant, i=1,...o,n, and T_{\eps,T} is the set of stopping times that lie between a constant \eps \in[0,T] and T. 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. We also provide its asymptotic order } \eps \to  follows. For large enough }\eps and for small }\eps  as \eps\searrow 0. 
We consider the optimal stopping problem v^{(\eps):=\sup} _{\tau\inT _{0,T } [i=1^n (\tau-\eps^i)^+^i], where } \mathbb{EB_{(\tau-\eps)^+} posed by Shiryaev at the International Conference on Advanced Stochastic Optimization URLanized 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 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 _{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. We also provide its asymptotic order } , 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
R
problem \sup
stopping problem v^{(\eps)
R
_{\eps,T
_{0,T
D
\mathbb{E
A
\mathbb{E
R
\phi_t^i
B_t
R
progressively measurable with respect to the Brownian filtration
the Brownian motion
D
^i
D
i=1,...o,n,
R
that lie between a constant
taking values in
R
\in[0,T] and T. We solve this problem by conditioning and then using the theory of
,T
A
. The solution of this problem is characterized by a path dependent
D
(RBSDEs). As a corollary, we provide the solution to the optimal stopping problem \sup_{\tau\in\mathcal{T
A
, from which the continuity of
A
v^{(\eps)
A
, we obtain an explicit expression for v^{(\eps)
A
we have lower and upper bounds. The main result of the paper is the asymptotics of v^{(\eps)
A
As a byproduct, we also obtain L\'{e