  Parametric estimation for diffusion processes is considered for high
frequency observations over a fixed time interval. The processes solve
stochastic differential equations with an unknown parameter in the diffusion
coefficient. We find easily verified conditions on approximate martingale
estimating functions under which estimators are consistent, rate optimal, and
efficient under high frequency (in-fill) asymptotics. The asymptotic
distributions of the estimators are shown to be normal variance-mixtures, where
the mixing distribution generally depends on the full sample path of the
diffusion process over the observation time interval. Utilising the concept of
stable convergence, we also obtain the more easily applicable result that for a
suitable data dependent normalisation, the estimators converge in distribution
to a standard normal distribution. The theory is illustrated by a simulation
study comparing an efficient and a non-efficient estimating function for an
ergodic and a non-ergodic model.
