On the one-sided exit problem for fractional Brownian motion

TL;DR

This paper investigates the one-sided exit problem for fractional Brownian motion, focusing on the distribution of its maximum's lower tail, and improves existing bounds while exploring related quantities.

Contribution

It provides improved bounds for the one-sided exit problem of FBM and clarifies the relationship with the quantity I studied by Molchan.

Findings

Enhanced bounds for the lower tail distribution of FBM maximum

Deeper understanding of the relation to Molchan's quantity I

Insights into the behavior of fractional Brownian motion's extremal events

Abstract

We consider the one-sided exit problem for fractional Brownian motion (FBM), which is equivalent to the question of the distribution of the lower tail of the maximum of FBM on the unit interval. We improve the bounds given by Molchan (1999) and shed some light on the relation to the quantity $I$ studied there.