Evy Driven and Fractionally Integrated Arma Processes with Continuous Time Parameter

The de nition and properties of L evy driven CARMA continuous time ARMA processes are re viewed 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 speci cation 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 Barndor Nielsen and Shephard 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 de ne a class of zero mean fractionally integrated L evy driven CARMA pro cesses 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