We consider a multidimensional diffusion process [math] whose drift and diffusion coefficients depend respectively on a parameter [math] and [math] . This process is observed at [math] equally spaced times [math] , and [math] denotes the length of the `observation window'. We are interested in estimating [math] and/or [math] . Under suitable smoothness and identifiability conditions, we exhibit estimators [math] and [math] , such that the variables [math] and [math] are tight for [math] and [math] . When [math] is known, we can even drop the assumption that [math] . These results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.