Optimal Linear Mean Square Filter and an Associated Stationary Filter for Hidden Markov Chain

Abstract

In this paper we make further foray on the filtering problem for hidden Markov chain, as described in (2). Previous result in the literature on this problem has been obtained in [1], which has derived an optimal nonlinear filter for this problem. The main contribution of this paper is to devise an optimal linear filter for the Markov chain in conjunction with an associated stationary linear filter which amounts here to obtain the convergence of the error covariance matrix. The optimal linear filter derived here bears the following advantages when compared with the one derived in [1]: (i) The innovation coefficients do not depend upon the estimates, which provides a desirable feature of the Kalman filter; (ii) It allows us to derive a stationary filter which has the advantage that it is easy to implement since this filter gain can be performed offline, leading to a linear time-invariant filter. In addition, relying on Euler-Murayama's stochastic numerical method and the results in [2], we carry out a simulation which shows that the filter performs very well.