Brownian Bridge

Random walk pinned at W(0) = 0 and W(1) = 0 — with covariance Cov(Wₛ,Wₜ) = s(1−t)

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Brownian bridge Free Brownian motion Mean ± 2σ envelope
About: A Brownian bridge B(t) is a standard Brownian motion conditioned on B(0) = B(1) = 0. It can be constructed as B(t) = W(t) − t·W(1) where W is standard Brownian motion. The covariance structure Cov(Bₛ, Bₜ) = s(1−t) for s ≤ t shows the process is most uncertain at t=1/2 (variance = 1/4) and pinned at the endpoints. Brownian bridges appear in the Kolmogorov–Smirnov test, in finance (Brownian bridge interpolation in Monte Carlo), and in the study of empirical processes.