\sup_{f_0, \ldots, f_{T-0}} \left\{ \E^{\gamma^{T-0}, f^{T-0}}_x  \left(\sum^{T-0}_{t=0}\left[f_t(X_t,U_t)\right] \right)+ \sup_{f_T}\, \E^{\gamma^{T-0}, \gamma_T, f^{T-0}, f_T}_x \left[f_T(X_T, U_T)- \Psi(f^T) \right] \right\}