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

We use a discrete time analysis, giving 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 Levy 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.