The role of measure-valued decompositions in stochastic control

Following up the measure-valued decompositions of Kunita (1981), and the martingale representation result for L²-processes of Bensoussan (1981), we have derived (1993) necessary conditions of optimizing nonlinear partially observed controlled diffusions with integral cost, when the signal and the observation processes are correlated. These necessary conditions were shown for the uncorrelated case to be exactly the ones derived by Bensoussan, after showing that the adjoint equations derived in the two papers are identical. In the present note, independently of the martingale representation result, we outline the derivation of two stochastic partial differential equations (forward and backward in time), with the forward satisfying the exact adjoint equation, and we consider the question if there is a connection with the adjoint equation derived by Bensoussan (1992). We show that even that adjoint equation follows from our adjoint equation as a special case. That is, for the uncorrelated case, even though the adjoint equations appear to be different, they are in fact identical as expected. Finally, we comment on the use of measure-valued decompositions in deriving necessary conditions for optimizing an exponential-of-integral cost.