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

Part of: Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  14 July 2016

Claudia Klüppelberg ,
Alexander Lindner  and
Ross Maller
Claudia Klüppelberg*
Affiliation:
Munich University of Technology
Alexander Lindner*
Affiliation:
Munich University of Technology
Ross Maller*
Affiliation:
Australian National University
*
∗ Postal address: Center for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany
∗ Postal address: Center for Mathematical Sciences, Munich University of Technology, D-85747 Garching, Germany
∗∗∗∗ Postal address: Centre for Mathematical Analysis and School of Finance and Applied Statistics, Australian National University, Canberra, ACT 0200, Australia. Email address: ross.maller@anu.edu.au
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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.

Keywords

ARCH modelGARCH modelstabilitystationarityconditional heteroscedasticityperpetuitiesstochastic integrationLévy process

MSC classification

Primary: 60G10: Stationary processes 60J25: Continuous-time Markov processes on general state spaces 91B70: Stochastic models
Secondary: 91B84: Economic time series analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 601 - 622
Copyright
Copyright © Applied Probability Trust 2004 

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