  A family of continuous-time generalized autoregressive conditionally
heteroscedastic processes, generalizing the $\operatorname {COGARCH}(1,1)$
process of Kl\"{u}ppelberg, Lindner and Maller [J. Appl. Probab. 41 (2004)
601--622], is introduced and studied. The resulting $\operatorname
{COGARCH}(p,q)$ processes, $q\ge p\ge 1$, exhibit many of the characteristic
features of observed financial time series, while their corresponding
volatility and squared increment processes display a broader range of
autocorrelation structures than those of the $\operatorname {COGARCH}(1,1)$
process. We establish sufficient conditions for the existence of a strictly
stationary nonnegative solution of the equations for the volatility process
and, under conditions which ensure the finiteness of the required moments,
determine the autocorrelation functions of both the volatility and the squared
increment processes. The volatility process is found to have the
autocorrelation function of a continuous-time autoregressive moving average
process.
