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Brownian motion
You will learn: Refine a consistent Gaussian time grid and distinguish finite increments from path-limit claims.
Start with: Random walk · Normal distribution
A random walk has discrete steps. Brownian motion is a continuous-time process whose increments over an interval of length h are normal with variance h. Review normal distributions and joint models: normal marginals alone do not specify how values at different times fit together.
Standard Brownian motion W starts at zero, has continuous paths with probability one, and has independent increments over disjoint time intervals. Its increment law is W(t+h)−W(t)∼Normal(0,h). With drift μ and scale σ>0, set X(t)=μt+σW(t):
Solid blue: 64 intervals. Dashed red: the same path at 32 intervals. Dotted black: population mean μt. Refinement adds conditional Gaussian midpoints and preserves every existing grid value. Straight connecting segments are drawing aids, not Brownian motion between the sampled times. The vertical range includes every displayed value.
| Quantity | Value |
|---|---|
| Time step Δt | 0.01562500 |
| One increment: mean / variance | 0.000000 / 0.01562500 |
| Population endpoint mean / variance | 0.000000 / 1.000000 |
| This path's endpoint | 0.7910457 |
| Fine-grid sum of squared increments | 1.191970 |
| Coarse-grid sum of squared increments | 1.164118 |
| Fine-grid sum after removing drift | 1.191970 |
| Expected drift-removed sum; limiting variation | 1.000000 |
| Expected raw fine-grid sum | 1.000000 |
The finite squared-increment sums fluctuate with the path and need not improve monotonically as the grid is refined. Population endpoint variance is not estimated from this single endpoint. Changing T rescales one unit-time realization; it does not append independent future observations.
Inspect the first eight grid increments
| Time | Position | Increment | Squared increment |
|---|---|---|---|
| 0.01562500 | -0.1163168 | -0.1163168 | 0.01352960 |
| 0.03125000 | -0.09141025 | 0.02490657 | 0.0006203370 |
| 0.04687500 | -0.08522776 | 0.006182489 | 0.00003822317 |
| 0.06250000 | -0.4078334 | -0.3226056 | 0.1040744 |
| 0.07812500 | -0.5718098 | -0.1639764 | 0.02688827 |
| 0.09375000 | -0.7637862 | -0.1919763 | 0.03685492 |
| 0.1093750 | -0.7845109 | -0.02072471 | 0.0004295137 |
| 0.1250000 | -0.7824849 | 0.002025987 | 0.000004104621 |
Only the first eight rows are listed. Every one of the 64 increments contributes to the totals.
Normal at a time, dependent across times
For s≤t, write W(t)=W(s)+[W(t)−W(s)]. The bracketed increment is independent of W(s), so Cov(W(s),W(t))=Var(W(s))=s. Thus:
Sampling independent Normal(0,t) values at each time would get each marginal right but this covariance wrong. A continuous path also carries more structure than any finite list of marginal distributions.
Take μ=0.5, σ=2, T=1. The endpoint has mean 0.5 and variance 4. On a quarter-unit step, an increment has mean 0.125 and variance 1, hence standard deviation one. Four independent such increments add to the stated endpoint law. Making the step four times shorter halves its random standard deviation; it does not divide that standard deviation by four.
Make a prediction
Does zero drift mean that the displayed path stays at zero?
Explore the answer
No. The population mean is zero at each time, while a single path fluctuates. An average over many independent paths estimates that mean; one path’s endpoint is not its population variance.
Refinement should preserve what you already sampled
The experiment first samples W(1). It then inserts a midpoint between each pair of known endpoints. For an interval [a,b], conditional on W(a)=u and W(b)=v, the midpoint is normal with mean (u+v)/2 and variance (b−a)/4. Independent conditional midpoint draws refine disjoint intervals.
This follows by conditioning two equal-variance independent Gaussian increments on their sum. If the total increment is v−u, the first half has conditional mean (v−u)/2 and conditional variance (b−a)/4. Adding u gives the midpoint formula.
At every refinement level the old endpoints remain exactly the same. The newly sampled grid has the correct Brownian joint law, up to finite-precision simulation. This is conditioning on both endpoints to fill a grid, not forecasting a future endpoint from only the past. Changing T instead rescales the whole unit-time realization by √T and the time coordinates by T.
Roughness survives smaller steps
Add the squares of the grid increments. For n equal intervals, independence and the normal increment variance give:
Subtracting the known drift μΔt from each increment removes the second term in expectation. With μ=0, σ=2, T=1, the expected sum is four at every resolution. A realized sum differs and need not move toward four at each individual refinement. For zero drift its variance is 2σ⁴T²/n, so its fluctuations shrink as the grid becomes finer.
Along the deterministic dyadic refinements used here, the squared-increment sum converges almost surely to σ²T. This is quadratic variation. It differs from the squared total displacement and from the sum of absolute increments.
The straight lines drawn between sampled points are not a full Brownian path. Subdividing those fixed straight lines would make their squared-increment sum tend to zero. Correct refinement adds fresh conditional randomness. Similarly, a path may cross a barrier between sampled times even when the endpoints do not reveal the crossing; this display does not compute continuous-time hitting events.
Make a prediction
Can you recover a Brownian maximum by taking the maximum of a coarse grid?
Explore the answer
The grid maximum is a lower bound on the continuous path’s maximum for that realization. Refinement can reveal a larger value between old endpoints. A finite grid alone does not prove that no still-higher point or hidden barrier crossing exists.
With zero drift, Brownian motion is a martingale relative to its natural history. Stopping it safely requires more than noticing its mean is constant.
Sources
Random Services: standard Brownian motion gives the process and increment structure; its Brownian-bridge lesson covers conditional Gaussian paths. Lalley’s Brownian-motion lecture develops scaling and dyadic quadratic variation. The increment, midpoint, covariance, and finite-grid expectation calculations are shown explicitly above.