Brownian Bridge

A random walk conditioned to return to its starting point

Paths 8

Steps 200

Sigma (volatility) 1.2

End Y (drag canvas) 0

Bridge paths

Mean ± std envelope

Theoretical std

A Brownian bridge B(t) is a standard Brownian motion W(t) conditioned on W(T) = 0. It can be constructed as B(t) = W(t) − (t/T)W(T), or via the SDE dB = −B/(T−t)dt + dW. The bridge has mean 0 and variance σ²t(T−t)/T — highest in the middle, zero at endpoints. Drag on the canvas to set the endpoint height. Click "Animate" to watch paths grow step by step.