Two-step estimation of ergodic Lévy driven SDE

Abstract

We consider high frequency samples from ergodic Lévy driven stochastic differential equation (SDE) with drift coefficient $a(x,\alpha )$ and scale coefficient $c(x,\gamma )$ involving unknown parameters $\alpha$ and $\gamma$. We suppose that the Lévy measure ${\nu }_0$, has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of $\alpha$, $\gamma$ and a class of functional parameter $\int \phi (z){\nu }_0(dz)$, which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of $(\alpha ,\gamma )$, and then, for estimating $\int \phi (z){\nu }_0(dz)$ we makes use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.