Martingale conditions for the optimal control of continuous time stochastic systems

A martingale condition is shown to be sufficient for optimality in a generally formulated continuous time control problem. Under the additional assumption that the class of admissible control laws has an [epsilon]-lattice property, the same martingale property is shown also to be necessary for optimality. The method makes use of the P-ess inf of a class of measurable functions used by Rishel [4] in a less general formulation. The general result of the paper is applied to more specific Markov and stochastic differential equation models to obtain conditions for optimality for these models.