Dual effect, certainty equivalence, and separation in stochastic control

In this paper the various policies in fixed end-time stochastic control are discussed first. The emphasis is on the difference between the feedback and closed-loop policies. It is shown how the closed-loop policy has the important property that it can be actively adaptive, while the feedback policy can only be passively adaptive. The feature of being actively adaptive is possible when the control has a dual effect, i.e., in addition to its effect on the state it affects the state uncertainty. The intimate connection between the neutrality (lack of dual effect) and certainty equivalence properties for a class of problems is proved. This new result is then used to widen the class of problems for which it was previously known that the certainty equivalence property holds.