Jump processes driven by α-stable Lévy processes impose inferential difficulties as their increments are heavy-tailed and the intensity of jumps is infinite. This paper considers the estimation of the functional drift and diffusion coefficients from high-frequency observations of a stochastic differential equation. By transforming the increments suitably prior to a regression, the variance of the emerging quantities may be bounded while allowing for identification of drift and diffusion in a single framework. These findings are applied to obtain a novel nonparametric kernel estimator, for which asymptotic normality and consistency of subsampling approximations are derived, and to a parametric volatility estimator for the Ornstein–Uhlenbeck process. The proposed approach also suggests a semiparametric estimator for the index of stability α. Finite sample properties of the proposed estimators, in terms of mean (integrated) absolute error, are investigated by a simulation study and compared to their non-tempered counterparts. 

Keywords: high-frequency data; infinitesimal generator; jumps; kernel regression; stable process; subsampling