ABSTRACT

A continuous-time generalized autoregressive conditionally heteroscedastic processes, generalizing the existing COGARCH(1,1) process, is introduced and studied. The resulting COGARCH(p,q) processes 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 COGARCH(1,1) process. The 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. The method of moments is employed to estimate the parameters of the COGARCH(p,q) processes.