Evy - Driven and Fractionally Integrated Armaprocesses with Continuous Time Parameterpeter

The deenition and properties of L evy-driven CARMA (continuous-time ARMA) processes are reviewed. Gaussian CARMA processes are special cases in which the driving L evy process is Brownian motion. The use of more general L evy processes permits the speciication of CARMA processes with a wide variety of marginal distributions which may be asymmetric and heavier tailed than Gaus-sian. Non-negative CARMA processes are of special interest, partly because of the introduction by Barndorr-Nielsen and Shephard (2001) of non-negative L evy-driven Ornstein-Uhlenbeck processes as models for stochastic volatility. Replacing the Ornstein-Uhlenbeck process by a L evy-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 deene a class of zero-mean fractionally integrated L evy-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 L evy-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 L evy-driven processes whose autocovariance functions at lag h also converge to zero at asymptotically hyperbolic rates.