On the relation between partially observed, stochastic optimal control and deterministic infinite dimensional optimal control

The authors explore the relation between partially observed stochastic optimal control and deterministic infinite dimensional control. The former is formulated as the control problem for the Zakai stochastic partial differential equation with adapted controls which is reduced to a family of pathwise deterministic infinite dimensional control problems for the robust Zakai random partial differential equation with possibly anticipating controls. This is done using the robustifying gauge transformation and by introducing the nonanticipativity of the control processes as an equality constraint via a Lagrange multiplier stochastic process. An explicit formula is obtained for this in terms of the adjoint process of the maximum principle for infinite dimensional optimal control applied to the control problem of the robust Zakai equation. This Lagrange multiplier has an interpretation as a price system for small violations of the constraint, in this case small anticipative perturbations of the nonanticipative controls