The Wonham Filter With Random Parameters: Rate of Convergence and Error Bounds

Abstract

Let /spl alpha/(t) be a finite-state continuous-time Markov chain with generator Q=(q/sup ij/)/spl isin/R/sup m/spl times/m/ and state space M={z/sup i/,...,z/sup m/}, where z/sup 1/ for i/sub /spl middot//spl middot//spl middot//spl middot//m are distinct real numbers. When the state-space and the generator are known a priori, the best estimator of /spl alpha/(t) (in terms of mean square error) under noisy observation is the classical Wonham filter. This note addresses the estimation issue when values of the state-space or values of the generator are unknown a priori. In each case, we propose a (suboptimal) filter and prove its convergence to the desired Wonham filter under simple conditions. Moreover, we obtain the rate of convergence using both the mean square and the higher moment error bounds.