Estimating functions for SDE driven by stable Levy processes

This paper is concerned with parametric inference for a stochastic differential equation driven by a pure jump Levy process, based on high frequency observations on a fixed time period. Assuming that the Levy measure of the driving process behaves like that of an alpha-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of alpha is an element of (0, 2) and does not require any integrability assumptions on the process. The main limit theorems are derived thanks to a control in total variation distance between the law of the normalized process, in small time, and the alpha-stable distribution. This method is an alternative to the non Gaussian quasi-likelihood estimation method proposed by Masuda (Stochastic Process. Appl. (2018) To appear) where the Blumenthal-Getoor index alpha is restricted to belong to the interval [1, 2).