Sufficient conditions of optimality for stochastic systems with controllable diffusions

This paper studies optimal controls for systems governed by Ito´s stochastic differential equations. Both the drift and diffusion terms of the equations are allowed to depend on controls, and the systems are allowed to be degenerate. It is shown that the necessary conditions of optimality, namely, the maximum conditions in terms of the “ℋ-function” (which is a generalization of the usual Hamiltonian and is quadratic with respect to the diffusion coefficients), along with some convexity conditions, constitute sufficient conditions of optimality for such controlled systems