Brownian Motion and Its Defining Properties
Independent normal increments, continuous paths, and no derivative anywhere. It is the scaling limit of a random walk, and it is the process most continuous-time models are written against.
What a learner can do afterwards
- State the defining properties and compute a covariance from them
- Describe the scaling that turns a random walk into this process
- Say why the paths are continuous and yet nowhere differentiable
1 · Read
Standard Brownian motion starts pinned at the origin, so B of 0 is 0. Its increments over disjoint intervals are independent and normal, with variance equal to elapsed time. Over 4 units of time the variance is 4, since unit variances add: position at a fixed time is a familiar bell curve.
Covariances come from overlap: the covariance of B of s and B of t is the smaller of s and t. So the covariance of B of 5 and B of 7 is 5, and the covariance of B of 5 minus B of 2 with B of 7 is 5 minus 2, which is 3. Rescaling works the same way: B of 9 behaves like 3 times a standard normal.
The process is the scaling limit of a random walk: many small independent pieces turn normal, and whole walks settle into Brownian paths. That upgrade of the central limit idea is called Donsker in textbooks. Sample means settling down is the one dimensional preview of walks settling into motion.
The paths are continuous everywhere yet differentiable nowhere, which is what scaling implies about roughness. Zoom in and the jaggedness never smooths out. When a model assumes smooth velocity for this motion, that assumption is the first thing to distrust.
Brownian motion starts at zero, spreads with variance equal to time, arises from scaled walks, and stays rough at every zoom.
2 · Watch
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.