Article2004

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

Claudia Klüppelberg · Alexander Lindner · Ross Maller

20 citations0/0 files storedJournal of Applied Probability

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Stochastic processes and financial applications Financial Risk and Volatility Modeling Complex Systems and Time Series Analysis

Abstract

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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References

2001Non-Gaussian Ornstein–Uhlenbeck-based Models and Some of Their Uses in Financial Economics 1990The Variance Gamma (V.G.) Model for Share Market Returns 1990Stationarity and Persistence in the GARCH(1,1) Model 1990ARCH models as diffusion approximations 1992Stationarity of Garch processes and of some nonnegative time series 2000Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process 1997Augmented GARCH (p,q) process and its diffusion limit 1996Closing the GARCH gap: Continuous time GARCH modeling

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