Two-Time-Scale Approximation for Wonham Filters

Abstract

This paper is concerned with approximation of Wonham filters. A focal point is that the underlying hidden Markov chain has a large state space. To reduce computational complexity, a two-time-scale approach is developed. Under time scale separation, the state space of the underlying Markov chain is divided into a number of groups such that the chain jumps rapidly within each group and switches occasionally from one group to another. Such structure gives rise to a limit Wonham filter that preserves the main features of the filtering process, but has a much smaller dimension and therefore is easier to compute. Using the limit filter enables us to develop efficient approximations and useful filters for hidden Markov chains. The main advantage of our approach is the reduction of dimensionality.