Confidence bands for Brownian motion and applications to Monte Carlo simulation

Minimal area regions are constructed for Brownian paths and perturbed Brownian paths. While the theoretical optimal region cannot be obtained in closed form, we provide practical confidence regions based on numerical approximations and local time arguments. These regions are used to provide informal convergence assessments for both Monte Carlo and Markov Chain Monte Carlo experiments, via the Brownian asymptotic approximation of cumulative sums.