MathIsimple

Brownian Bridge Calculator

Calculate Brownian bridge process properties: covariance, variance, and boundary conditions

Covariance Calculation
Calculate Cov[X(s), X(t)] for Brownian bridge process
Variance Calculation
Calculate Var[X(t)] for Brownian bridge process
About Brownian Bridge Process

Definition

A Brownian bridge process {X(t); 0 ≤ t ≤ 1} is defined as:

X(t) = B(t) - tB(1)

where {B(t)} is standard Brownian motion.

Key Properties

  • • X(0) = 0 (fixed at origin)
  • • X(1) = 0 (fixed at endpoint)
  • • Normal process characteristics
  • • Mean function: μ_X(t) = 0
  • • Variance: Var[X(t)] = t(1-t)

Applications

Physics
  • • Fixed-end elastic string vibration
  • • Particle diffusion with boundaries
  • • Heat conduction in finite rods
Finance
  • • Interest rate modeling
  • • Option pricing with constraints
  • • Risk management scenarios