Two-step estimation of ergodic Lévy driven SDE

We consider high frequency samples from ergodic L\'{e}vy 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 L\'{e}vy measure $\nu_{0}$, has all order moments but is not fully specified. The goal of this paper is to construct estimators of $\alpha$, $\gamma$ and a functional parameter $\int\varphi(z)\nu_0(dz)$ for some function $\varphi$ and derive their asymptotic behavior. It turns out that the functional estimator is asymptotically biased when the scale coefficient $c(x,\gamma)$ is not a constant, and we show how to remove the bias in an explicit way. In particular, the resulting stochastic expansion can be used to approximate moments of the driving-noise distribution, without assuming a closed form of $\nu_0$.