Lévy Driven CARMA Processes: Properties, Applications and Inference

Abstract

This paper reviews Continuous time Auto-Regressive Moving Average (CARMA) models.We define CARMA models as a natural extension of discrete time ARMA models through state space representations. After the continuous time extension to CARMA models, we introduce the causal stationary conditions, derive the explicit forms of covariance and spectral density functions, show the joint distributions and examine the second order properties of regularly sampled CARMA processes. Finally, we review two empirical applications to high frequency data of exchange rates and Brookhaven turbulence data. The contents of the paper are based on the presentation slides of Professor Peter Brockwell for his plenary session in Japanese Joint Statistical Meeting at Shiga University in 2019.