Stochastic Near-Optimal Controls: Necessary and Sufficient Conditions for Near-Optimality

Abstract

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal controls, for systems governed by the Ito stochastic differential equations (SDEs), where both the drift and diffusion terms are allowed to depend on controls and the systems are allowed to be degenerate. Necessary and sufficient conditions for a control to be near-optimal are studied. It is shown that any near-optimal control nearly maximizes the "$\HH$-function" (which is a generalization of the usual Hamiltonian and is quadratic with respect to the diffusion coefficients) in some integral sense, and vice versa if certain additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximum of the $\HH$-function are obtained, based on some delicate estimates of the adjoint processes. Examples are presented to demonstrate the results.