A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour

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 sin...

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Bibliographic Details
Published in:Journal of applied probability 2004-09, Vol.41 (3), p.601-622
Main Authors: Klüppelberg, Claudia, Lindner, Alexander, Maller, Ross
Format: Article
Language:English
Subjects:
60G10
60J25
91B28
91B70
91B84
Applied sciences
ARCH model
Autocorrelation
conditional hetero-scedasticity
Differential equations
Econometrics
Economic models
Exact sciences and technology
GARCH model
Integration by parts
Lévy process
Markov processes
Martingales
Mathematical moments
Mathematics
Operational research and scientific management
Operational research. Management science
perpetuities
Poisson process
Probability and statistics
Probability theory and stochastic processes
Random variables
Research Papers
Sciences and techniques of general use
stability
stationarity
stochastic integration
Stochastic models
Stochastic processes
Studies
Sufficient conditions
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Summary: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.
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1091543413