Two-step estimation of ergodic Lévy driven SDE

We consider high frequency samples from ergodic Lévy driven stochastic differential equation (SDE) with drift coefficient $a(x,α)$ and scale coefficient $c(x,γ)$ involving unknown parameters $α$ and $γ$. We suppose that the Lévy measure $ν_{0}$, has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of $α$, $γ$ and a class of functional parameter $\intφ(z)ν_0(dz)$, which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of $(α,γ)$, and then, for estimating $\intφ(z)ν_0(dz)$ we makes use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.

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