Non-Gaussian quasi-likelihood estimation of locally stable SDE

We address parametric estimation of both trend and scale coefficients of a pure-jump Levy driven univariate stochastic differential equation (SDE) model based on high-frequency data over a fixed time period. It is known from the previous study [35] that the conventional Gaussian quasimaximum likelihood estimator is inconsistent. In this paper, under the assumption that the driving Levy process is locally stable, we propose a novel quasi-likelihood function based on the small-time nonGaussian stable approximation of the unknown transition density. The resulting estimator is shown to be asymptotically mixed-normally distributed and remarkably more efficient than the Gaussian quasimaximum likelihood estimator. We need neither ergodicity nor existence of finite moments. Compared with the existing methods for estimating SDE models, the proposed quasi-likelihood enables us to achieve better performance in a unified manner for a wide range of the driving Levy processes.