Brownian Bridge Recipe | The Gaussian Cookbook

Simulating the Brownian Bridge

Continuous-time Recipe for Brownian Bridge

Intuition

The Brownian bridge is a Brownian motion conditioned to return to zero at time 1. Its covariance kernel is $$ K(s, t) = \min(s, t) - s t $$ The Karhunen–Loève expansion gives $$ B_t = \sqrt{2} \sum_{k=1}^\infty Z_k \frac{\sin(k \pi t)}{k \pi} $$ where $ Z_k \sim \mathcal{N}(0, 1) $ are independent.

General Recipe

Define the process and covariance kernel $ K(s, t) = \min(s, t) - s t $.
Set up and solve the integral eigenvalue problem: $$ \int_0^1 K(s, t) e_k(s) ds = \lambda_k e_k(t) $$
Obtain eigenfunctions and eigenvalues: $$ e_k(t) = \sqrt{2} \sin(k \pi t), \quad \lambda_k = \frac{1}{(k \pi)^2} $$
Construct the Karhunen–Loève expansion for simulation.
Discretize the expansion for practical computation.

Python Implementation

import numpy as np

def brownian_bridge_ctr(steps=100, n_terms=500, paths=1):
    t = np.linspace(0, 1, steps)
    Bt = np.zeros((paths, steps))
    for p in range(paths):
        Z = np.random.normal(0, 1, n_terms)
        k = np.arange(1, n_terms + 1)
        for i, ti in enumerate(t):
            sin_terms = np.sin(k * np.pi * ti) / (k * np.pi)
            Bt[p, i] = np.sqrt(2) * np.dot(Z, sin_terms)
    return Bt

Summary: This approach yields sample paths of the Brownian bridge with the correct covariance structure, as guaranteed by the Karhunen–Loève theorem.

Discrete-time Recipe for Brownian Bridge

Intuition

In discrete time, we want to construct a vector $ X $ such that $$ \mathbb{E}[X] = 0, \quad \text{Cov}(X) = \Gamma $$ where $ \Gamma_{ij} = \min(t_i, t_j) - t_i t_j $ is the covariance matrix for the Brownian bridge.

General Recipe

Generate the desired covariance matrix $ \Gamma $.
Decompose $ \Gamma = Q \Lambda Q^T $ (eigendecomposition).
Generate a Gaussian vector $ Z \sim N(0, I_n) $.
Set $ X = Q \Lambda^{1/2} Z $.

Python Implementation

import numpy as np

def brownian_bridge_dtr(steps=100, paths=1, T=1):
    t = np.linspace(0, T, steps)
    Gamma = np.zeros((steps, steps))
    for i in range(steps):
        for j in range(steps):
            Gamma[i, j] = min(t[i], t[j]) - t[i]*t[j]
    eigenvals, eigenvecs = np.linalg.eigh(Gamma)
    Z = np.random.normal(0, 1, (steps, paths))
    Bt = eigenvecs @ np.diag(np.sqrt(eigenvals)) @ Z
    return Bt.T

Summary: This approach produces Brownian bridge sample paths with the correct covariance structure by diagonalizing the covariance matrix and scaling standard normal draws.

Simulation

Method:

Sample Paths

Covariance

Covariance Error

Theoretical Covariance

Discretized Equation

$$ B_t = \\sqrt{2} \\sum_{k=1}^N Z_k \\frac{\\sin(k \\pi t)}{k \\pi} $$

Relevant Articles

Roman Paolucci Columbia University

June 30, 2025

Recipes for simulating stochastic processes

This note discusses the necessary steps to simulate a stochastic process with a desired covariance structure. In this context, I outline the Karhunen–Loéve theorem and provide intuition and general recipes to decompose a stochastic process by establishing the integral eigenvalue problem (continuous-time) or, equivalently, the covariance matrix diagonalization problem (discrete-time). I then apply these general recipes to a Brownian motion and Brownian bridge to numerically simulate paths.

Keywords: Brownian motion, Brownian bridge, Stochastic processes, Simulation

View on SSRN

Suggested Citation:
Paolucci, Roman, Recipes for simulating stochastic processes (Brownian bridge) (June 30, 2025). Available at SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5332011