On an optimal stopping problem of an insider

We consider the optimal stopping problem v^(ε):= sup_τ∈T0,T E B_(τ−ε)+ 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≦t≦T is the Brownian motion, ε ∈ [0,T] is a constant, and T_ε,T is the set of stopping times taking values in [ε, T ]. The solution of this problem is characterized by a path dependent reflected backward stochastic differential equation, from which the continuity of ε → v^(ε) follows. For large enough ε, we obtain an explicit expression for v^(ε), and for small ε we have lower and upper bounds. The main result of the paper is the asymptotics of v^(ε) as ε ↘ 0. As a by-product, we also obtain Lévy’s modulus of continuity result in the L¹ sense.