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.

翻译：众所周知，布朗桥或Lévy-Ciesielski构造的布朗路径几乎必然一致收敛于真实布朗路径。本文重点研究均匀误差问题。具体而言，我们构造性地证明：在层级$N$（对应布朗路径上$d=2^N$个评估点）下，均匀误差及其平方项、以及几何布朗运动的均匀误差的上界均为$\mathcal{O}(\sqrt{\ln d/d})$的量级，这与已知阶数一致。我们还将所得结果应用于期权定价示例。