  We propose a unified framework that extends the inference methods for
classical hidden Markov models to continuous settings, where both the hidden
states and observations occur in continuous time. Two different settings are
analyzed: (1) hidden jump process with a finite state space; (2) hidden
diffusion process with a continuous state space. For each setting, we first
estimate the hidden state given the observations and model parameters, showing
that the posterior distribution of the hidden states can be described by
differential equations in continuous time. Then we consider the estimation of
unknown model parameters, deriving the formulas for the
expectation-maximization algorithm in the continuous-time setting. We also
propose a Monte Carlo method for sampling the posterior distribution of the
hidden states and estimating the unknown parameters.
