Lévy-driven CARMA processes

Abstract

Properties and examples of continuous-time ARMA (CARMA) processes driven by Levy processes are examined. By allowing Levy processes to replace Brownian motion in the definition of a Gaussian CARMA process, we obtain a much richer class of possibly heavy-tailed continuous-time stationary processes with many potential applications in finance, where such heavy tails are frequently observed in practice. If the Levy process has finite second moments, the correlation structure of the CARMA process is the same as that of a corresponding Gaussian CARMA process. In this paper we make use of the properties of general Levy processes to investigate CARMA processes driven by Levy processes {W(t)} without the restriction to finite second moments. We assume only that W (1) has finite r-th absolute moment for some strictly positive r. The processes so obtained include CARMA processes with marginal symmetric stable distributions.