Noise recovery for L\'evy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums

TL;DR

This paper develops an L^2-consistent estimator for the increments of Lévy processes in CARMA models and analyzes the high-frequency behavior of Riemann sum approximations, providing insights into their autocovariance structures.

Contribution

It introduces a new estimator for Lévy process increments in invertible CARMA models and compares high-frequency Riemann sum approximations with sampled CARMA processes.

Findings

Proposes an L^2-consistent estimator without prior order selection.

Analyzes the autocovariance structure of Riemann sum approximations.

Provides new insights into kernel estimation procedures.

Abstract

We consider high-frequency sampled continuous-time autoregressive moving average (CARMA) models driven by finite-variance zero-mean L\'evy processes. An L^2-consistent estimator for the increments of the driving L\'evy process without order selection in advance is proposed if the CARMA model is invertible. In the second part we analyse the high-frequency behaviour of approximating Riemann sum processes, which represent a natural way to simulate continuous-time moving average processes on a discrete grid. We shall compare their autocovariance structure with the one of sampled CARMA processes, where the rule of integration plays a crucial role. Moreover, new insight into the kernel estimation procedure of Brockwell et al. (2012a) is given.