  We construct fractionally integrated continuous-time GARCH models, which
capture the observed long range dependence of squared volatility in
high-frequency data. Since the usual Molchan-Golosov and Mandelbrot-van-Ness
fractional kernels lead to problems in the definition of the model, we resort
to moderately long memory processes by choosing a fractional parameter
$d\in(-0.5,0)$ and remove the singularities of the kernel to obtain
non-pathological sample paths. The volatility of the new fractional COGARCH
process has positive features like stationarity, and its covariance function
shows an algebraic decay, which make it applicable to econometric
high-frequency data. In an empirical application the model is fitted to
exchange rate data using a simulation-based version of the generalised method
of moments.
