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

@article{Klppelberg2004ACG,
  title={A continuous-time GARCH process driven by a L{\'e}vy process: stationarity and second-order behaviour},
  author={Claudia Kl{\"u}ppelberg and Alexander M. Lindner and Ross A. Maller},
  journal={Journal of Applied Probability},
  year={2004},
  volume={41},
  pages={601 - 622}
}
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… 

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References

SHOWING 1-10 OF 34 REFERENCES

Stationarity and Persistence in the GARCH(1,1) Model

This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift

Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process

The asymptotic theory for the sample autocorrelations and extremes of a GARCH(I, 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close

Dynamic models of long-memory processes driven by Lévy noise

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential

Stationarity of Garch processes and of some nonnegative time series

The Variance Gamma (V.G.) Model for Share Market Returns

A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance

Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics

Non‐Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures.

Modelling by Lévy Processess for Financial Econometrics

This paper reviews some recent work in which Levy processes are used to model and analyse time series from financial econometrics. A main feature of the paper is the use of posi- tive