ISEGORIA / MATH ENCYCLOPEDIA
Stochastic processes: Brownian motion and noise
Random walks rescaled into Brownian motion, quadratic variation, and densities that obey Fokker–Planck.
Before you begin: Probability, measure theory, and Markov chains
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. From random walk to Brownian motion
Take n coin-flip steps, shrink time by n and space by √n. As n grows, the paths stop looking like staircases and start looking like one universal random curve. Zoom in: a quarter of the time and half of the space looks statistically the same as the whole.
Worked example. At \(t=1\), the endpoint histogram approaches the standard normal density; the shaded bands are \(\pm\sqrt t\) and \(\pm2\sqrt t\).
Watch out. Donsker’s theorem is convergence in distribution of whole paths. Any single simulated path is still a staircase at fine enough scale.
Why scale space by √n rather than n?
The variance of n independent ±1 steps is n, so dividing by \(\sqrt n\) is the only choice that keeps the endpoint variance at 1.
2. Quadratic variation
Cut [0, 1] into 2^m pieces and add up the increments of a Brownian path two ways. The sum of absolute increments grows without bound, so the path has infinite length. The sum of squared increments settles at exactly t. That single fact, dB² = dt, is why stochastic calculus needs an extra term.
Worked example. The left-point (Itô) sum of \(B\,\Delta B\) tends to \(\tfrac12(B_1^2-1)\); the midpoint (Stratonovich) sum tends to \(\tfrac12B_1^2\). The readout computes both.
Watch out. The path is sampled at \(2^{14}\) points, so level \(m=14\) is the finest partition the data can show; beyond that the path is a straight-line interpolation.
For a smooth curve, what happens to the sum of squares?
It tends to zero, because each squared increment is \(O(\Delta t^2)\) and there are only \(1/\Delta t\) of them.
3. Paths and their density
An Ornstein–Uhlenbeck particle is pulled toward zero and kicked by noise. Watch individual paths wander while the density they build obeys a deterministic equation. Drag the time slice: the histogram of the simulated paths matches the exact Gaussian solution.
Worked example. Starting from \(x_0\), \(X_t\sim\mathcal N\!\bigl(x_0e^{-\theta t},\ \tfrac{\sigma^2}{2\theta}(1-e^{-2\theta t})\bigr)\). At \(\theta=0\) this is Brownian motion with variance \(\sigma^2t\).
Watch out. The paths use the exact Gaussian transition over each small step, so there is no discretisation bias, only sampling noise from 600 paths.
What is the long-run variance?
\(\sigma^2/(2\theta)\): the balance between noise pumping variance in and the drift pulling it out.