  GARCH is one of the most prominent nonlinear time series models, both widely
applied and thoroughly studied. Recently, it has been shown that the COGARCH
model (which was introduced a few years ago by Kl\"{u}ppelberg, Lindner and
Maller) and Nelson's diffusion limit are the only functional continuous-time
limits of GARCH in distribution. In contrast to Nelson's diffusion limit,
COGARCH reproduces most of the stylized facts of financial time series. Since
it has been proven that Nelson's diffusion is not asymptotically equivalent to
GARCH in deficiency, in the present paper, we investigate the relation between
GARCH and COGARCH in Le Cam's framework of statistical equivalence. We show
that GARCH converges generically to COGARCH, even in deficiency, provided that
the volatility processes are observed. Hence, from a theoretical point of view,
COGARCH can indeed be considered as a continuous-time equivalent to GARCH.
Otherwise, when the observations are incomplete, GARCH still has a limiting
experiment, which we call MCOGARCH, which is not equivalent, but nevertheless
quite similar, to COGARCH. In the COGARCH model, the jump times can be more
random than for the MCOGARCH, a fact practitioners may see as an advantage of
COGARCH.
