Estimating functions for SDE driven by stable Lévy processes

This paper is concerned with parametric inference for a stochastic differential equation driven by a pure-jump Lévy process, based on high frequency observations on a fixed time period. Assuming that the Lévy measure of the driving process behaves like that of an α-stable process around zero, we propose an estimating functions based method which leads to asymptotically efficient estimators for any value of α ∈ (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 α-stable distribution. This method is an alternative to the non Gaussian quasi-likelihood estimation method proposed by Masuda [20] where the Blumenthal-Getoor index α is restricted to belong to the interval [1, 2)