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

TL;DR

This paper develops a two-step estimation method for ergodic Lévy-driven SDEs using high-frequency data, combining Gaussian quasi-likelihood and method of moments to estimate parameters and functionals of the Lévy measure.

Contribution

It introduces a novel two-step estimation procedure for ergodic Lévy-driven SDEs, jointly estimating parameters and Lévy measure functionals with asymptotic normality.

Findings

Establishes joint asymptotic normality of estimators.

Demonstrates effectiveness of combined quasi-likelihood and moment methods.

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.