Estimation of L\'evy-driven CARMA models under renewal sampling

TL;DR

This paper develops a Whittle estimation method for Le9vy-driven CARMA models observed at renewal times, proving consistency and asymptotic normality under mild conditions, thus enabling flexible modeling of irregular high-frequency data.

Contribution

It introduces a novel estimation approach for Le9vy-driven CARMA models under renewal sampling, with theoretical guarantees of consistency and asymptotic normality.

Findings

Whittle estimator is consistent for the model parameters.

The estimator is asymptotically normal under mild conditions.

The paper establishes the asymptotic normality of the integrated periodogram.

Abstract

Continuous-time autoregressive and moving average (CARMA) models are extensively used to model high-frequency and irregularly sampled data. We study Whittle estimation for the model parameters when the process is observed at renewal times. The driving noise is assumed to be a L\'evy process allowing for more flexibility including heavy-tailed marginal distributions and jumps in the sample paths. We show that the Whittle estimator based on the integrated periodogram is consistent and asymptotically normal under very mild conditions. To obtain these results, we establish the asymptotic normality of the integrated periodogram.