LÉVY-DRIVEN CARMA PROCESSES

P. J. BROCKWELL

Statistics Department, Colorado State University, Fort Collins, Colorado 80523-1877, U.S.A.

(Received April 25, 2000; revised June 23, 2000)

Abstract. Properties and examples of continuous-time ARMA (CARMA) processes driven by Lévy processes are examined. By allowing Lévy 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 Lévy 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 Lévy processes to investigate CARMA processes driven by Lévy 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.

Key words and phrases: Lévy process, CARMA process, stochastic differential equation, stable process.