Filtering of derived point processes

We consider two marked point processes [Phi] and [Psi] on the real half-line such that [Psi] is an -predictable thinning and marking of [Phi]. Using the method of the probability of reference we derive linear and non-linear filtering equations for the conditional distribution , where {gt} is a certain -adapted process. In particular, we will apply our results to the filtering of a partially observed semi-Markov process. In that case, the conditional distribution of the last jumptime before t [greater-or-equal, slanted] 0 and the corresponding jumpvalue can be expressed explicitly in terms of a solution of a Markov renewal equation.