Parametric inference for discretely observed non-ergodic diffusions

We consider a multidimensional diffusion process X whose drift and diffusion coefficients depend respectively on a parameter lambda and theta. This process is observed at n + 1 equally spaced times 0, Delta(n), 2 Delta(n),..., n Delta(n), and T-n = n Delta(n) denotes the length of the 'observation window'. We are interested in estimating lambda and/or theta. Under suitable smoothness and identifiability conditions, we exhibit estimators lambda(n) and theta(n) such that the variables root n(theta(n) - theta) and root T-n(lambda(n) -lambda) are tight for Delta(n) -> 0 and T-n -> infinity. When lambda is known, we can even drop the assumption that T-n -> infinity. These results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.