A Continuous Time GARCH Process of Higher Order

A continuous time GARCH model of order (p,q) is introduced, which
is driven by a single Lévy process. It extends many of the features
of discrete time GARCH(p,q) processes to a continuous time setting.
When p=q=1, the process thus defined reduces to the COGARCH(1,1)
process of Klüppelberg, Lindner and Maller (2004). We give
sufficient conditions for the existence of stationary solutions and
show that the volatility process has the same autocorrelation
structure as a continuous time ARMA process. The autocorrelation of
the squared increments of the process is also investigated, and
conditions ensuring a positive volatility are discussed.