Confidence bands for Brownian motion and applications to Monte Carlo simulation

Abstract

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.

International Encyclopedia of Statistical Science

“…, x T is produced according to a distribution density f , all standard statistical tools, including bootstrap, apply to this sample (with the further appeal that more data points can be produced if deemed necessary). As illustrated by Figure 1, of the overall variability of the sequence of approximations are provided in Kendall et al (2007). But the ease with which such methods are analysed and the systematic resort to statistical intuition explain in part why Monte Carlo methods are privileged over numerical methods.…”

Neural Network based Novelty Detection for Incremental Semi-supervised Learning in Multi-class Gesture Recognition

“…Confidence bands are computed in several approaches (e.g. Kendall et al, 2007). In the context of semisupervised learning, the confidence bands of a polynomial classifier are used by Al-Behadili et al (2014) to detect outlier samples.…”

Analytic crossing probabilities for certain barriers by Brownian motion

“…When n = 2, g(t) = a − √ T − tx(t), where x(t) is the largest solution to the equation x exp(−x 2 /2) = (T − t)/b. The function g(t) is, up to a translation and time-reversion, identical to the upper boundary of a symmetric band used in [13] as a confidence region for Brownian motion. Whether our techniques could be used to compute analytically the crossing probability for the two-sided band, which has been approximated numerically in [13], is a question that deserves further investigation.…”