Smoothing, filtering and estimation by Monte Carlo methods for doubly stochastic Poisson processes

Smoothing, filtering and estimation for doubly stochastic Poisson processes are interesting problems which, in general, cannot be solved analytically. Here we discuss some solutions to these problems based on the reversible jump Markov chain Monte Carlo algorithm and on particle filtering. Maximum likelihood estimation will be carried out by some Monte Carlo versions of the EM algorithm. In particular, we will discuss these methods for a class of models with a stochastic intensity given by a jump process with drift. Models in this class can be used to describe ultra-high-frequency stock prices.