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

TL;DR

This paper develops a method to estimate the Lévy measure of a continuous-time moving average Lévy process from low-frequency data, addressing technical challenges related to mixing properties and demonstrating the estimator's effectiveness.

Contribution

It introduces a consistent estimator for the Lévy measure from low-frequency observations and establishes its convergence rates, advancing statistical inference for such processes.

Findings

The estimator is consistent and converges at a quantifiable rate.

Numerical examples demonstrate the estimator's practical performance.

The paper addresses the technical challenge of establishing exponential mixing.

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.