Fractional Brownian Motion: Local Modulus of Continuity with Refined Almost Sure Upper Bound and First Exit Time from One-sided Barrier

TL;DR

This paper refines the understanding of fractional Brownian motion's local continuity and first exit times, providing sharper bounds and asymptotic estimates that fill gaps in existing probabilistic laws.

Contribution

It introduces a refined almost sure upper bound for fractional Brownian motion's increments and improves the asymptotic estimate of its first exit time from a barrier.

Findings

Refined almost sure upper bound of order |h|^H√loglog(1/|h|) for increments.

New asymptotic estimate for the tail probability of first exit time.

Fills the gap in the law of iterated logarithm for fractional Brownian motion.

Abstract

Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian increments is of order $\mathcal O_{a.s.}\big(|h|^H\sqrt{\log\log |h|^{-1}}\big)$ as $|h|\to0$, and an upper bound of its $p$th moment is provided, for any $p>0$. This result fills the gap of the law of iterated logarithm for fractional Brownian motion, where the moments' information of the random multiplier in the upper bound is missing. With this enhanced upper bound and some new results on the distribution of the maximum of fractional Brownian motion, we obtain a new and refined asymptotic estimate of the upper-tail probability for a fractional Brownian motion to first exit from a positive-valued barrier over time $T$, as $T\to+\infty$.