Hybrid filters modulated by Markov chains with two-time scales

We consider a class of hybrid filtering problems in discrete-time. The main feature is that the system is modulated by a Markov chain. Our main effort is to reduce the complexity of the underlying problems. Consider the case that the Markov chain has a large state space. Then the solution of the problem relies on solving a large number of filtering equations. By using the hierarchical structure of the system, we show that a reduced system of filtering equations can be obtained by aggregating the states of each recurrent class into one state. Extensions to inclusion of transient states and nonstationary cases are also treated.