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 $$Z_t={\int }_{\mathbb{R}}\mathcal{K}(t-s)d{L}_s,t\in \mathbb{R}$$ with a deterministic kernel (\K\) and a L{é}vy process (L\). Especially the estimation of the Lévy measure (ν\) 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 Lévy processes.

32 pages, 3 figures