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Martingales & the Optional Stopping Theorem — 2D Chart

Watch a bundle of fair-coin random walks race on a 2D chart toward a stopping barrier and see the martingale property survive: the sample mean at the moment each path stops still hovers at its starting value, exactly as the optional stopping theorem predicts. Pan and zoom the chart freely.

Probability & Statistics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

This simulator races a bundle of independent fair-coin random walks on a 2D step-vs-value chart, each one stopped the instant it first crosses a barrier ±B, and tracks the live sample mean against the theoretical value the optional stopping theorem predicts. It is a direct, hands-on view of a martingale property that underpins fair-game analysis, Wald's identity, and risk-neutral option pricing: no matter how the individual paths wander before they are stopped, their average outcome cannot drift away from the starting value as long as the underlying process stays fair. Drag to pan and scroll to zoom into any stretch of the run.

⚙ Under the hood

Watch a bundle of fair-coin random walks race on a 2D step-vs-value chart toward a stopping barrier and see the martingale property survive: the sample mean at the moment each path stops still hovers at its starting value, exactly as the optional stopping theorem predicts. Drag to pan and scroll to zoom into any stretch of the run.

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install