On the unnormalized solution of the filtering problem with counting process observations

A general procedure to solve filtering problems for counting process observations is discussed. Linear (nonstochastic, integro-differential) equations describe the evolution of unnormalized conditional distribution of the state process between observation jump times, while at jump times a linear updating is required. Final normalization is the only nonlinear operation to be implemented. Quite general situations may be accommodated in the present setup; the state can be virtually any Markov semimartingale, the observation process may affect the dynamics of the state and vice versa, and there is complete freedom in correlating state and observation martingale terms