Nonparametric Gaussian inference for stable processes

Jump processes driven by $$\alpha $$ 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 $$\alpha $$ 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.