Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion

We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H$, respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1$, which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called the Riemann–Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.