2.2
Increment explorer
What do we know about how Brownian motion changes over an arbitrary window?
Goal: verify experimentally that W(t) − W(s) ~ N(0, t − s), and that only the window's length matters, not its position.
Drag the shaded window to slide it, its edges to resize. The coloured paths are emphasised inside the window; the thick bar at t is the increment W(t) − W(s) itself.
t − s = 0.40 · sd √(t − s) = 0.632
Each path's segment inside the window is shifted to start at 0. The fan's width depends only on the elapsed time u (±√u); at its end, the histogram of increments sits beside the N(0, t − s) density.