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

TL;DR

This paper investigates the asymptotic behavior of the persistence exponent for integrated fractional Brownian motion and Riemann-Liouville processes, confirming conjectures and analyzing limits as the Hurst parameter approaches zero and one.

Contribution

It determines the asymptotic behavior of the persistence exponent for these processes, validating a conjecture for integrated fractional Brownian motion.

Findings

Asymptotic behavior of persistence exponent as H→0 and H→1 for integrated fractional Brownian motion.

Confirmation of Molchan and Khokhlov's conjecture in the asymptotic regime.

Asymptotics of the persistence exponent for Riemann-Liouville process as H→0.

Abstract

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 Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.