  The statistical problem of parameter estimation in partially observed
hypoelliptic diffusion processes is naturally occurring in many applications.
However, due to the noise structure, where the noise components of the
different coordinates of the multi-dimensional process operate on different
time scales, standard inference tools are ill conditioned. In this paper, we
propose to use a higher order scheme to approximate the likelihood, such that
the different time scales are appropriately accounted for. We show consistency
and asymptotic normality with non-typical convergence rates. When only partial
observations are available, we embed the approximation into a filtering
algorithm for the unobserved coordinates, and use this as a building block in a
Stochastic Approximation Expectation Maximization algorithm. We illustrate on
simulated data from three models; the Harmonic Oscillator, the FitzHugh-Nagumo
model used to model the membrane potential evolution in neuroscience, and the
Synaptic Inhibition and Excitation model used for determination of neuronal
synaptic input.
