Asymptotics of the persistence exponent of integrated FBM and Riemann-Liouville process

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