On the (mixed) integrated fractional Brownian motion

We consider the integrated fractional Brownian motion with Hurst index 0 <; H <; 1. The main aim consists in studying its links with the quasi-helix with approximately stationary increments class of centered Gaussian processes. More precisely, we compute some well-chosen quantities which depend on the moments of the integrated fractional Brownian motion. Firstly, we check if one can obtain appropriate upper or lower bounds of the above quantities. Secondly, focusing our attention on the small increments of the integrated fractional Brownian motion we derive an equivalent. Some surprising facts are observed. Next, we consider the mixed integrated fractional Brownian motion. Its main properties are studied. We also determine the values of H for which this process is a semi-martingale. Finally we compare the results obtained for the mixed integrated fractional Brownian motion with those obtained for the mixed fractional Brownian motion and for the sub-mixed fractional Brownian motion.