# On the expected uniform error of Brownian motion approximated by the L\'evy-Ciesielski construction

## TL;DR

This paper analyzes the uniform approximation error of Brownian motion via the Lévy-Ciesielski construction, providing bounds that match known orders and applying findings to option pricing.

## Contribution

It offers a constructive proof of uniform error bounds for the Lévy-Ciesielski approximation of Brownian paths, including geometric Brownian motion.

## Key findings

- Uniform error bounds of order O(√(ln d / d)) for the Lévy-Ciesielski approximation
- Application of error bounds to option pricing models
- Matching theoretical bounds with observed convergence rates

## Abstract

It is known that the Brownian bridge or L\'evy-Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. In the present article the focus is on the uniform error. In particular, we show constructively that at level $N$, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order $\mathcal{O}(\sqrt{\ln d/d})$, matching the known orders. We apply the results to an option pricing example.