Filtering a Markov modulated random measure

We develop a new exact filter when a hidden Markov
chain influences both the sizes and times of a marked point process.
An example would be an insurance claims process, where we assume
that both the stochastic intensity of the claim arrivals and the
distribution of the claim sizes depend on the states of an economy.
We also develop the robust filter-based and smoother-based EM
algorithms for the on-line recursive estimates of the unknown parameters
in the Markov-modulated random measure. Our development
is in the framework of modern theory of stochastic processes.