  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: hidden jump process with a finite state space, and hidden diffusion
process with a continuous state space. For each setting, we first estimate the
hidden states given the observations and model parameters, showing that the
posterior distribution of the hidden states can be described by differential
equations in continuous time. We then consider the estimation of unknown model
parameters, deriving the continuous-time formulas for the
expectation-maximization algorithm. We also propose a Monte Carlo method based
on the continuous formulation, sampling the posterior distribution of the
hidden states and updating the parameter estimation.
