Role of measure-valued decompositions in stochastic control

Abstract

Following up the measure-valued decompositions of Kunita [1], and the martingale representation result for L2-processes of Bensoussan [2], we have recently derived in [3], 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 in [3], for the uncorrelated case to be exactly the one's derived in [2], after showing that the adjoint equations derived in [2, 3] are identical. In the present note, independently of the martingale representation result given in [2], we outline the derivation of two stochastic partial differential equations (forward and backward in time), with the forward satisfying the exact adjoint equation derived in [3], and we consider the question if there is a connection between the adjoint equation derived in [4]. We show that even the adjoint equation derived in [4], follows from our adjoint equation as a special case. That is, for the uncorrelated case, even though the adjoint equations derived in [2, 3, 4] 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.