Exact filters for the estimation of the number of transitions of finite-state continuous-time Markov processes

The problem of estimating the number of transitions of finite-state continuous-time Markov processes observed by a noisy sensor is considered. A finite-dimensional exact filter is derived, and using the EM algorithm (an extension of the Baum-Welch algorithm for the discrete-time case), an application is made to the problem of estimating the unknown transition matrix of a finite-state continuous-time Markov process.>