Journal of Applied Probability · 2004 · 243 citations · 18 references
EngineeringFinanceMathematicsFinancial EconometricsStochastic ProcessesTime Series AnalysisContinuous-time Stochastic ProcessesDiscrete-time Stochastic ProcessesLévy ProcessesMarkov ProcessesStochastic VolatilityVolatility ModelingAlmost Sure ConvergenceArch ModelsContinuous-time ModelsContinuous-time Time Series ModelsDiscrete-time ModelsLévy-driven Stochastic Differential EquationsStationarity AssumptionStochastic Volatility Models
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 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.
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The Variance Gamma (V.G.) Model for Share Market Returns
Dilip B. Madan, E. Seneta · The Journal of Business · 1990 · 1.6K citations
Stationarity and Persistence in the GARCH(1,1) Model
Daniel B. Nelson · Econometric Theory · 1990 · 1.1K citations
ARCH models as diffusion approximations
Daniel B. Nelson · Journal of Econometrics · 1990 · 964 citations