Brownian bridge

Abstract

A Brownian bridge is a stochastic process derived from standard Brownian motion by requiring an extra constraint. This gives Brownian bridges unique mathematical properties, fascinating, itself, and useful in statistical and mathematical modeling. Here, we define a Brownian bridge and explain some of its unique mathematical properties. These properties make it useful as a mathematical model for real-world phenomena and as a tool for statistical analysis. We then give some examples of applications. We will see many mathematical ideas that touch Brownian bridges, such as differential equations, linear spaces, topology, metric spaces of functions, measure theory, stochastic integrals, estimation, and hypothesis testing. © 2009 John Wiley & Sons, Inc.