Near-optimal controls of two-time scale functional diffusion systems

This paper investigates near-optimal controls for a class of fully-coupled stochastic functional differential equations (SFDEs) with two-time scales, in which all coefficients depend on the segment processes of both the fast and slow components. The underlying problem is to minimize a cost functional subject to the SFDEs mentioned above. Our primary tools are probabilistic methods, in particular, weak convergence methods. The main challenge lies in the complete coupling of the fast and slow processes through their segment processes along with the resulting effects on the tightness of the segment process of the slow component. To address these challenges, the boundedness and Hölder continuity for such segment process are established in a continuous function space. In addition, it is also shown that the segment process of a fixed-x SFDE is uniformly bounded, exponentially ergodic, and continuously dependent on the parameter x. By using the relaxed control representation and the martingale problem formulation, it is proved that the slow process and the corresponding value function in the original problem converge to that of a limit problem, where the coefficients of the limit problem are obtained by averaging coefficients of the original problem with respect to the invariant measure of the fixed-x equation. Finally, by solving the optimal control problem for the limit problem, a practically useful control for the original system is constructed. Such constructed controls are shown to be nearly optimal for the original problem.