  This work aims at making a comprehensive contribution in the general area of
parametric inference for discretely observed diffusion processes. Established
approaches for likelihood-based estimation invoke a time-discretisation scheme
for the approximation of the intractable transition dynamics of the Stochastic
Differential Equation (SDE) model over finite time periods. The scheme is
applied for a step-size that is either user-selected or determined by the data.
Recent research has highlighted the critical ef-fect of the choice of numerical
scheme on the behaviour of derived parameter estimates in the setting of
hypo-elliptic SDEs. In brief, in our work, first, we develop two weak second
order sampling schemes (to cover both hypo-elliptic and elliptic SDEs) and
produce a small time expansion for the density of the schemes to form a proxy
for the true intractable SDE transition density. Then, we establish a
collection of analytic results for likelihood-based parameter estimates
obtained via the formed proxies, thus providing a theoretical framework that
showcases advantages from the use of the developed methodology for SDE
calibration. We present numerical results from carrying out classical or
Bayesian inference, for both elliptic and hypo-elliptic SDEs.
