Necessary conditions of optimization for partially observed controlled diffusions

Necessary conditions are derived 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 directly the equation satisfied by the unnormalized conditional density of nonlinear filtering, measure-valued decompositions are used to decompose it into two processes. The minimum principle and the stochastic partial differential equation satisfied by the adjoint process are then derived, and the optimality conditions are shown to be the exact necessary conditions derived by Bensoussan [Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions, Stochastics, 9 (1983), pp. 169-222; Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, UK, 1992] when the correlation is zero.