Filtering of derived point processes

Abstract

We consider two marked point processes Φ and Ψ on the real half-line such that Ψ is an { F t Φ} -predictable thinning and marking of Φ. Using the method of the probability of reference we derive linear and non-linear filtering equations for the conditional distribution E[g t∣ F t Ψ] , where { g t } is a certain { F t Φ} -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 ⩾ 0 and the corresponding jumpvalue can be expressed explicitly in terms of a solution of a Markov renewal equation.