Abstract
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 ϵ-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.
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