COGARCH as a continuous-time limit of GARCH(1,1) |
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Authors: | Jan Kallsen Bernhard Vesenmayer |
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Institution: | 1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany;2. HVB-Stiftungsinstitut für Finanzmathematik, Zentrum Mathematik, TU München, Boltzmannstraße 3, 85747 Garching bei München, Germany |
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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. |
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Keywords: | 60F17 60J25 91B70 |
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