Online minimum search for a Brownian bridge

TL;DR

This paper investigates an efficient numerical method for accurately finding the minimum and arg-min of a Brownian bridge, leveraging classical results to ensure small errors and low failure probability.

Contribution

It demonstrates that the bisection method effectively computes the minimum of a Brownian bridge with high accuracy and reliability, based on established theoretical results.

Findings

Bisection method achieves small error in minimum estimation

Low probability of failure for the proposed method

Utilizes classical results to validate the approach

Abstract

In this short note we consider the computational problem of numerically finding the minimum and arg-min of a Brownian bridge. Using well-known results by Pitman, Tanaka, Vervaat and Williams we are able to show that the bisection method has both a small error and a small probability of failure.