Non-Gaussian quasi-likelihood estimation of SDE driven by locally stable L\'evy process

TL;DR

This paper develops a non-Gaussian quasi-likelihood estimator for SDEs driven by locally stable Lévy processes, improving estimation accuracy over traditional Gaussian methods, especially for pure-jump processes observed at high frequency.

Contribution

It introduces a novel quasi-likelihood based on small-time stable approximation, extending Gaussian frameworks to handle locally stable Lévy processes, and proves its asymptotic properties.

Findings

Estimator is asymptotically mixed normally distributed.

Outperforms Gaussian quasi-maximum likelihood estimator in simulations.

Effective for pure-jump Lévy processes without ergodicity.

Abstract

We address estimation of parametric coefficients of a pure-jump L\'evy driven univariate stochastic differential equation (SDE) model, which is observed at high frequency over a fixed time period. It is known from the previous study Masuda (2013) that adopting the conventional Gaussian quasi-maximum likelihood estimator then leads to an inconsistent estimator. In this paper, under the assumption that the driving L\'evy process is locally stable, we extend the Gaussian framework into a non-Gaussian counterpart, by introducing a novel quasi-likelihood function formally based on the small-time stable approximation of the unknown transition density. The resulting estimator turns out to be asymptotically mixed normally distributed without ergodicity and finite moments for a wide range of the driving pure-jump L\'evy process, showing much better theoretical performance compared with the Gaussian quasi-maximum likelihood estimator. Extensive simulations are carried out to show good estimation accuracy. The case of large-time asymptotics under ergodicity is briefly mentioned as well, where we can deduce an analogous asymptotic normality result.