  We study properties and parameter estimation of finite-state homogeneous
continuous-time bivariate Markov chains. Only one of the two processes of the
bivariate Markov chain is observable. The general form of the bivariate Markov
chain studied here makes no assumptions on the structure of the generator of
the chain, and hence, neither the underlying process nor the observable process
is necessarily Markov. The bivariate Markov chain allows for simultaneous jumps
of the underlying and observable processes. Furthermore, the inter-arrival time
of observed events is phase-type. The bivariate Markov chain generalizes the
batch Markovian arrival process as well as the Markov modulated Markov process.
We develop an expectation-maximization (EM) procedure for estimating the
generator of a bivariate Markov chain, and we demonstrate its performance. The
procedure does not rely on any numerical integration or sampling scheme of the
continuous-time bivariate Markov chain. The proposed EM algorithm is equally
applicable to multivariate Markov chains.
