We consider the optimal problem \tau\in\mathcal{T_{\eps,T}}E[i=1^n (\tau-\eps^i)^+^i], where 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 \tau\in\mathcal{T_{0,T}}EB_{(\tau-\eps)^+} recently posed by Shiryaev at the International Conference on Advanced Stochastic Optimization URLanized by the Steklov Institute of Mathematics in September 2012. \eps
We consider the optimal problem \tau\in\mathcal{T_{\eps,T}}E[i=1^n (\tau-\eps^i)^+^i], where 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 \tau\in\mathcal{T_{0,T}}EB_{(\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 as\eps\searrow 0.
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We also provide its asymptotic order as
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\searrow 0.