Two-step estimation of ergodic L\'evy driven SDE

Abstract

We consider high frequency samples from ergodic L\'evy 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\'evy 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 $\int\varphi(z)\nu_0(dz)$, which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of $(\alpha,\gamma)$, and then, for estimating $\int\varphi(z)\nu_0(dz)$ we makes use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.