Lower bounds for distribution of suprema of Brownian increments and Brownian motion normalized by the corresponding modulus functions

TL;DR

This paper derives non-asymptotic bounds for the distribution of Brownian motion increments normalized by modulus functions, providing uniform estimates and extending results to truncated Brownian motion using the Lévy-Ciesielski construction.

Contribution

It introduces new non-asymptotic estimates for Brownian motion increments normalized by modulus functions, including uniform results and bounds for truncated processes.

Findings

Non-asymptotic bounds for maximal deviations of Brownian increments.

Uniform estimates over the parameter delta.

Results applicable to truncated Brownian motion.

Abstract

The L\'evy-Ciesielski Construction of Brownian motion is used to determine non-asymptotic estimates for the maximal deviation of increments of a Brownian motion process $(W_{t})_{t\in \left[ 0,T\right] }$ normalized by the global modulus function, for all positive $\varepsilon $ and $\delta $. Additionally, uniform results over $\delta $ are obtained. Using the same method, non-asymptotic estimates for the distribution function for the standard Brownian motion normalized by its local modulus of continuity are obtained. Similar results for the truncated Brownian motion are provided and play a crucial role in establishing the results for the standard Brownian motion case.