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GARCH modelling in continuous time for irregularly spaced time series data

: Maller, R.A.; Müller, G.; Szimayer, A.


Bernoulli 14 (2008), Nr.2, S.519-542
ISSN: 1350-7265
Fraunhofer ITWM ()

The discrete-time GARCH methodology which hits had such a profound influence on the modelling of heteroscedasticity in time series is intuitively Well motivated in capturing many 'stylized facts' concerning financial series, and is now almost routinely used in a wide range of situations, often including some where the data are not observed at equally spaced intervals of time. However, such data is more appropriately analyzed with a continuous-time model which preserves the essential features of the successful GARCH paradigm. One possible such extension is the diffusion limit of Nelson, but this is problematic in that the discrete-time GARCH model and its continuous-time diffusion limit tire not statistically equivalent. As till alternative, Kluppelberg et al. recently introduced a continuous.-time version of the GARCH (the 'COGARCH' process) which is constructed directly from a background driving Levy process, The present paper how to fit this model to irregularly spaced time series data using discrete-time GARCH methodology, by approximating the COGARCH with an embedded sequence of discrete-time, GARCH series which converges to the continuous-time model in a strong sense (in probability, in the Skorokhod metric), as the discrete/approximating grid grows finer. This property is also especially useful in certain other applications, such as options pricing. The way is then open to using, for the COGARCH, similar statistical techniques to those already worked out for GARCH models and to illustrate this, an empirical investigation using stock index data is carried out.