The adjoint process for a partially observed Markov chain

A finite-state-space Markov chain is considered. Without loss of generality its state-space can be taken to be the set of unit basis vectors of R^N. On the basis of knowing only the total number of jumps, a control problem is discussed in `separated´ form. That is, the Zakai equation for its unnormalized distribution is taken as describing the state of the process. This is a linear vector equation driven by a standard Poisson process in which the control variable also appears in the `diffusion´ coefficient multiplying the noise term. The controls, similar to those employed by J.M. Bismut (1978) and H. J. Kushner (1972), are in the stochastic open-loop form. By adapting the techniques of A. Bensoussan (1982) and calculating a Gateaux derivative, the minimum principle satisfied by an optimal control is obtained. Finally, when the optimal control is Markov, the integrand in the martingale representation can be obtained explicitly, and new forward and backward equations satisfied by the adjoint process can be derived