Stationarity and second order behaviour of discrete and continuous time GARCH(1,1) processes

We use a discrete time analysis, giver necessary and sufficient conditions for the almost sure convergence of ARCH(1) and GARCH(1,1) discrete time models, to suggest an extension of the (G)ARCH concept to continuous time processes. The models, based on a single background driving Lévy process, are different from, though related to, other continuous time stochastic volatility models that have been proposed, Our models generalise the essential features of discrete time GARCH processes, and are amenable to further analysis, possessing useful Markovian and stationarity properties.