Markov Chains in Brownian Motion

In this chapter, a continuous-time, finite-state Markov chain is observed through a Brownian motion with drift. The filtered estimate of the state is the Wonham filter (1965). The smoothed estimate of the state is given in Clements and Anderson (1975). A finite-dimensional filter for the number of jumps $\mathcal{J}^{ij}_{t}$ was obtained by Dembo and Zeitouni (1986) and Zeitouni and Dembo (1988), and used to estimate the parameters of the Markov chain and the observation process. However, this estimation also involves $\mathcal{O}^{i}_{t}$ and $\mathcal{T}^{i}_{t}$ for which finite-dimensional filters are not given in Zeitouni and Dembo (1988). Our filters allow, therefore, the application of the EM algorithm, an extension of the discrete-time Baum-Welch algorithm (Dembo and Zeitouni 1986; Zeitouni and Dembo 1988). Unlike the Baum-Welch method our equations are recursive and can be implemented by the usual methods of discretization; no backward estimates are required.