Asymptotics for the maximum likelihood estimators of diffusion models

In this paper I derive the asymptotics of the exact, Euler, and Milstein MLestimators for diffusion models, including general nonstationary diffusions. Thoughthere have been many estimators for the diffusion model, their asymptotic propertieswere generally unknown. This is especially true for the nonstationary processes, eventhough they are usually far from the standard ones. Using a new asymptotics withrespect to both the time span T and the sampling interval ¢, I find the asymptoticsof the estimators and also derive the conditions for the consistency. With this newasymptotic result, I could show that this result can explain the properties of theestimators more correctly than the existing asymptotics with respect only to thesample size n. I also show that there are many possibilities to get a better estimatorutilizing this asymptotic result with a couple of examples, and in the second part ofthe paper, I derive the higher order asymptotics which can be used in the bootstrapanalysis.