Adjoint processes for Markov chains observed in Gaussian noise

A discrete time partially observed control problem is considered in which the dynamics of the system are described by a finite state Markov chain observed in Gaussian noise. A change of measure is introduced under which the observations are independent random variables. The unnormalized conditional probabilities of the Markov chain can be taken as information states and the problem discussed in separated form. An adjoint process is defined, and a minimum principle is obtained