Before revision:  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).
After revision:  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 ,T reflected backward stochastic differential equations (RBSDEs).