Brownian Bridge — Mathematics Knowledge System

Overview

The Brownian bridge is a Brownian motion conditioned to return to zero at time 1. More generally, a Brownian bridge from a to b on [0, T] is Brownian motion conditioned on X(0) = a and X(T) = b. It is a Gaussian process with explicit mean and covariance, widely used in statistics (Kolmogorov-Smirnov test), simulation (conditional path generation), and finance (pricing with fixed endpoints).

Intuition

Imagine a Brownian path that starts at 0 and is forced to return to 0 at time 1. It wanders freely in between but is 'pulled back' toward 0 as time approaches 1. The pulling force is proportional to how far the process is from its expected endpoint: the drift is -(X_t)/(1-t), which increases as t -> 1. This is the Brownian bridge.

Formal Definition

Definition

The standard Brownian bridge B^0 on [0,1] is defined as B^0_t = B_t - t*B_1, where B_t is a standard Brownian motion. Equivalently, it is the Gaussian process with mean 0 and covariance Cov(B^0_s, B^0_t) = s(1-t) for s <= t.

B^0_t = B_t - t B_1, \quad t \in [0,1]

Definition via Brownian motion

\mathbb{E}[B^0_t] = 0, \quad \text{Cov}(B^0_s, B^0_t) = s(1-t) \text{ for } s \le t

Mean and covariance

B^0_0 = 0, \quad B^0_1 = 0 \text{ a.s.}

Boundary conditions

dB^0_t = -\frac{B^0_t}{1-t}\,dt + dB_t, \quad t \in [0,1)

SDE for Brownian bridge

\text{Var}(B^0_t) = t(1-t)

Variance

Notation

Notation	Meaning
$B^0_t$	Standard Brownian bridge on [0,1]: B_t - t*B_1
$s(1-t)$	Covariance of standard Brownian bridge for s <= t

Theorems

Theorem 1: Theorem 1 (Gaussian Process Characterization)

\text{The standard Brownian bridge } B^0 \text{ is the unique (in distribution) continuous Gaussian process on } [0,1] \text{ with mean zero, covariance } \text{Cov}(B^0_s, B^0_t) = \min(s,t) - st = s(1-t) \text{ for } s \le t, \text{ and } B^0_0 = B^0_1 = 0 \text{ a.s.}

Theorem 2: Theorem 2 (Kolmogorov-Smirnov)

\text{Let } F_n \text{ be the empirical CDF of } n \text{ i.i.d. Uniform}[0,1] \text{ samples. Then } \sqrt{n}(F_n(t) - t) \xrightarrow{d} B^0_t \text{ in } C[0,1], \text{ underlying the Kolmogorov-Smirnov statistic.}

Worked Examples

1
The variance of the Brownian bridge at time t is t(1-t).
$\text{Var}(B^0_t) = t(1-t)$
2
At t = 1/2: Var(B^0_{1/2}) = (1/2)(1 - 1/2) = (1/2)(1/2) = 1/4.
$\text{Var}(B^0_{1/2}) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$

✓ Answer

Var(B^0_{1/2}) = 1/4. The variance is maximized at t = 1/2 and equals 0 at both endpoints.

Practice Problems

Mediumfree response

A Brownian bridge goes from B^0_0 = 0 to B^0_1 = 0. What is the distribution of B^0_t for a fixed t in (0,1)?

Mediumfree response

Write down the SDE for the Brownian bridge B^0_t on [0,1) and interpret the drift term.

Common Mistakes

Common Mistake

Thinking the Brownian bridge has constant variance

The variance t(1-t) is NOT constant: it starts and ends at 0 (endpoints are fixed) and peaks at t = 1/2.

Common Mistake

Confusing the Brownian bridge with a Brownian motion conditioned to be at 0

B^0_t = B_t - t B_1 is NOT the same as B_t | B_1 = 0 as a process. However, the conditional distribution of the path given B_1 = 0 is exactly the Brownian bridge.

Quiz

Summary

The standard Brownian bridge B^0_t = B_t - t B_1 satisfies B^0_0 = B^0_1 = 0 a.s.
Mean 0, Covariance Cov(B^0_s, B^0_t) = s(1-t) for s <= t; Var(B^0_t) = t(1-t).
SDE: dB^0_t = -(B^0_t)/(1-t) dt + dB_t -- mean-reverting drift grows to infinity at t=1.
Application: KS statistic; sqrt(n)(F_n - F) converges in distribution to a Brownian bridge.

References

BookRevuz, D. and Yor, M. -- Continuous Martingales and Brownian Motion, Chapter XI
BookBillingsley, P. -- Convergence of Probability Measures
WebsiteWikipedia -- Brownian bridge