Abstract: The definition and properties of Lévy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving Lévy process is Brownian motion. The use of more general Lévy processes permits the specification of CARMA processes with a wide variety of marginal distributions which may be asymmetric and heavier tailed than Gaussian. Non-negative CARMA processes are of special interest, partly because of the introduction by Barndorff-Nielsen and Shephard (2001) of non-negative Lévy-driven Ornstein-Uhlenbeck processes as models for stochastic volatility. Replacing the Ornstein-Uhlenbeck process by a Lévy-driven CARMA process with non-negative kernel permits the modelling of non-negative, heavy-tailed processes with a considerably larger range of autocovariance functions than is possible in the Ornstein-Uhlenbeck framework. We also define a class of zero-mean fractionally integrated Lévy-driven CARMA processes, obtained by convoluting the CARMA kernel with a kernel corresponding to Riemann-Liouville fractional integration, and derive explicit expressions for the kernel and autocovariance functions of these processes. They are long-memory in the sense that their kernel and autocovariance functions decay asymptotically at hyperbolic rates depending on the order of fractional integration. In order to introduce long-memory into non-negative Lévy-driven CARMA processes we replace the fractional integration kernel with a closely related absolutely integrable kernel. This gives a class of stationary non-negative continuous-time Lévy-driven processes whose autocovariance functions at lagalso converge to zero at asymptotically hyperbolic rates.
Key words and phrases: Continuous-time ARMA process, Lévy process, stochastic volatility, long memory, fractional integration.