Kluppelberg, ClaudiaLindner, AlexanderMaller, Ross2015-12-132015-12-130021-9002http://hdl.handle.net/1885/80909We 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 ARCH and GARCH concepts to continuous-time processes. Our 'COGARCH' (continuous-time GARCH) model, based on a single background driving Lévy 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.Keywords: ARCH model; Conditional heteroscedasticity; GARCH model; Lévy process; Perpetuities; Stability; Stationarity; Stochastic integration200410.1239/jap/10915434132015-12-11