Fractionally integrated COGARCH processes

We construct fractionally integrated continuous time GARCH models, which capture the observed long range dependence of squared volatilities in high-frequency data. We discuss the Molchan-Golosov kernel and the Mandelbrot-van-Ness kernel with respect to their integrability, and the resulting fractional processes with respect to their increments, as they should be stationary and positively correlated. Since this poses problems we resort to moderately long memory problems by choosing a fractional parameter $d\in(-0.5,0)$ and remove the singularities of the kernel to obtain non-pathological sample path properties. The new fractional COGARCH process has certain positive features like stationarity and algebraically decreasing covariance function.