Low-frequency estimation of continuous-time moving average Lévy processes

Abstract

In this paper, we study the problem of statistical inference for a continuous-time moving average Lévy process of the form \[Z_{t}=\int_{\mathbb{R}}\mathcal{K}(t-s)\,dL_{s},\qquad t\in\mathbb{R},\] with a deterministic kernel $\mathcal{K}$ and a Lévy process $L$. Especially the estimation of the Lévy measure $\nu$ of $L$ from low-frequency observations of the process $Z$ is considered. We construct a consistent estimator, derive its convergence rates and illustrate its performance by a numerical example. On the mathematical level, we establish some new results on exponential mixing for continuous-time moving average Lévy processes.