In this paper we develop likelihood based inferential methods for a novel
class of (potentially non-stationary) diffusion driven state space models.
Examples of models in this class are continuous time stochastic volatility
models and counting process models. Although our methods are sampling based,
making use of Markov chain Monte Carlo methods to sample the posterior
distribution of the relevant unknowns, our general strategies and details
are different from previous work on related but simpler models. The proposed
methods are easy to implement and simulation efficient. Importantly, unlike
methods for related models, the performance of our method is not worsened,
in fact it improves, as the degree of latent augmentation is increased to
reduce the bias of the Euler approximation. We also consider the problems of
model choice, model checking and filtering and apply the techniques and
ideas to both simulated and real data.