ARCH model and fractional Brownian motion

We study an extension of the ARCH model that includes the squared fractional Brownian motion. We study the statistical properties of the model as the conditions for the existence of a stationary solution and the moments of the process. We study their asymptotic behavior of the autocorrelation function of the squared of the process and we prove that the long memory property of the model holds. We illustrate our results by numerical simulations.