Low frequency estimation of continuous-time moving average L\'evy processes

Abstract

In this paper we study the problem of statistical inference for a continuous-time moving average L\'evy process of the form $$Z_{t} = \int_{\mathbb{R}}\mathcal{K}(t-s)\, dL_{s},\quad t\in\mathbb{R}$$ with a deterministic kernel (\K\) and a L{\'e}vy process (L\). Especially the estimation of the L\'evy 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 technical level, the main challenge is to establish a kind of exponential mixing for continuous-time moving average L\'evy processes.