Necessary conditions for partially observed diffusions

We present a new approach to deriving necessary conditions for stochastic partially observed control problems when the control enters the drift coefficient, and correlation between signal and observation noise is allowed. The problem is formulated as one of complete information but, instead of considering the unnormalized conditional density of nonlinear filtering, using Kunita´s decomposition this equation is decomposed into two measure-valued processes. The minimum principle and the stochastic partial differential equation satisfied by the adjoint process, are then derived, and are shown to be the exact necessary conditions when the correlation is zero