  We study a stochastic optimal control problem for a partially observed
diffusion. By using the control randomization method in [4], we prove a
corresponding randomized dynamic programming principle (DPP) for the value
function, which is obtained from a flow property of an associated filter
process. This DPP is the key step towards our main result: a characterization
of the value function of the partial observation control problem as the unique
viscosity solution to the corresponding dynamic programming
Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new,
fully non linear partial differential equation on the Wasserstein space of
probability measures. An important feature of our approach is that it does not
require any non-degeneracy condition on the diffusion coefficient, and no
condition is imposed to guarantee existence of a density for the filter process
solution to the controlled Zakai equation, as usually done for the separated
problem. Finally, we give an explicit solution to our HJB equation in the case
of a partially observed non Gaussian linear quadratic model.
