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COGARCH as a continuous-time limit of GARCH(1,1)

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  • Kallsen, Jan
  • Vesenmayer, Bernhard

Abstract

COGARCH is an extension of the GARCH time series concept to continuous time, which has been suggested by Klüppelberg, Lindner and Maller [C. Klüppelberg, A. Lindner, R. Maller, A continuous-time GARCH process driven by a Lévy process: Stationarity and second order behaviour, Journal of Applied Probability 41 (2004) 601-622]. We show that any COGARCH process can be represented as the limit in law of a sequence of GARCH(1,1) processes. As a by-product we derive the infinitesimal generator of the bivariate Markov process representation of COGARCH. Moreover, we argue heuristically that COGARCH and the classical bivariate diffusion limit of Nelson [D. Nelson, ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38] are probably the only continuous-time limits of GARCH.

Suggested Citation

  • Kallsen, Jan & Vesenmayer, Bernhard, 2009. "COGARCH as a continuous-time limit of GARCH(1,1)," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 74-98, January.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:1:p:74-98
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    References listed on IDEAS

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    1. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    2. Carol Alexandra & Emese Lazar, 2005. "On The Continuous Limit of GARCH," ICMA Centre Discussion Papers in Finance icma-dp2005-13, Henley Business School, University of Reading.
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    Cited by:

    1. Lee, Oesook, 2012. "V-uniform ergodicity of a continuous time asymmetric power GARCH(1,1) model," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 812-817.

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