A non-Markov finite dimensional filter

The state space of a general, not necessarily Markov, finite state space process is identified with the set of unit vectors in an Euclidean space. A filtering problem is considered where the observation process records only jumps between certain subsets of the states. A recursive equation for the filtered estimate of the state is obtained. If the transition rates are measurable with respect to the observations this filter is finite dimensional. The corresponding Zakai equation is also derived