Klüppelberg, Claudia; Lindner, Alexander M.; Maller, R. A.
A Continuous Time GARCH Process Driven by a Lévy Process: Stationarity and Second Order Behaviour.
Sonderforschungsbereich 386, Discussion Paper 425
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, tosuggest an extension of the (G)ARCH concept to continuous time processes. Our "COGARCH" (continuous time GARCH) model, based on a single background driving Levy process, is different from, though related to, other continuous time stochastic volatility models that have been proposed. The model generalises the essential features of discrete time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.