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MathidiaMathematics Academy
Stochastic processes
Lesson 40 of 75
Review

Brownian motion

Objective. Apply Gaussian increments and diffusion scaling.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Brownian motion is a continuous-time stochastic process Bₜ with independent, stationary increments that are normally distributed, serving as the continuum limit of a scaled random walk. It satisfies B₀ = 0, Bₜ has continuous paths, and for any s < t the increment Bₜ − Bₛ is Normal with mean 0 and variance t − s, independent of the path up to time s. The diffusion scaling property states that Bₜ and (1/√c)B_{ct} have the same distribution for any c > 0, reflecting how variance grows linearly with time while the spatial spread grows like √t. Example: the probability that B₄ lies between −2 and 2 is found using B₄ ~ Normal(0,4), a standard deviation of 2.

Worked example

For standard Brownian motion Bₜ, find the standard deviation of B₉ to illustrate the diffusion scaling of Brownian motion.

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  1. 1By definition, Bₜ ~ Normal(0, t) for standard Brownian motion.
  2. 2At t = 9, the variance of B₉ is Var(B₉) = 9.
  3. 3The standard deviation is the square root of the variance: √9.
  4. 4This gives standard deviation 3, confirming spread scales as √t rather than t.

Answer: Standard deviation of B₉ is 3

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Common mistake. A common mistake is treating the standard deviation of Bₜ as equal to t; because the variance is t, the standard deviation is √t, so spatial spread grows like the square root of elapsed time, not linearly.

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