The Maximum Principle for Optimal Control of Diffusions with Partial Information

We derive necessary conditions for the optimal control of a system that satisfies an Ito equation with control entering the drift term. The control is a function of a noise corrupted observation of the state, and the cost is the expectation of an integral and a final term. The robust form of the Zakai equation from nonlinear filtering is used to compute the variation of the cost due to a strong or “needle” variation of a control. This gives rise to an explicit formula for the adjoint process, much as in the case with complete information.