  Motivated by applications in movement ecology, in this paper I propose a new
class of integrated continuous-time hidden Markov models in which each
observation depends on the underlying state of the process over the whole
interval since the previous observation, not only on its current state. This
class gives a new representation of a range of existing models, including some
widely applied switching diffusion models. I show that under appropriate
conditioning, a model in this class can be regarded as a conventional hidden
Markov model, enabling use of the Forward Algorithm for efficient evaluation of
its likelihood without sampling of its state sequence. This leads to an
algorithm for inference which is more efficient, and scales better with the
amount of data, than existing methods. This is demonstrated and quantified in
some applications to animal movement data and some related simulation
experiments.
