A revisit to stochastic near-optimal controls: The critical case

This paper revisits the stochastic near-optimal control problem considered in Zhou (1998), where the stochastic system is given by a controlled stochastic differential equation with the control variable taking values in a general control space and entering both the drift and diffusion coefficients. A necessary condition of near-optimality is derived using Ekeland's variational principle, spike variation techniques, and some delicate estimates for the state and the adjoint processes. We improve the error bound of order from "almost" ε1³ in Zhou (1998) to "exactly" ε1³.