Two-step estimation of ergodic Lévy driven SDE

We consider high frequency samples from ergodic Lévy driven stochastic differential equation with drift coefficient $$a(x,\alpha )$$ a ( x , α ) and scale coefficient $$c(x,\gamma )$$ c ( x , γ ) involving unknown parameters $$\alpha $$ α and $$\gamma $$ γ . We suppose that the Lévy measure $$\nu _{0}$$ ν 0 , has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of $$\alpha $$ α , $$\gamma $$ γ and a class of functional parameter $$\int \varphi (z)\nu _0(dz)$$ ∫ φ ( z ) ν 0 ( d z ) , which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of $$(\alpha ,\gamma )$$ ( α , γ ) ; and then, for estimating $$\int \varphi (z)\nu _0(dz)$$ ∫ φ ( z ) ν 0 ( d z ) we make use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.