Brownian Motion

Brownian motion — also called the Wiener process and written (W_t)_{t \ge 0} — is the canonical continuous-time stochastic process, the continuous cousin of the random walk. It is the building block of nearly all of mathematical finance: the noise that drives stock prices in the Black–Scholes world is built from W_t.

It is pinned down completely by four defining properties:

Starts at zero: W_0 = 0.
Independent increments: over disjoint time intervals the increments W_{t_2} - W_{t_1},\, W_{t_4} - W_{t_3}, \dots are mutually independent — what the process does next does not depend on its past.
Gaussian stationary increments: for s < t, W_t - W_s \sim N(0,\, t - s), a normal with mean 0 and variance equal to the elapsed time — the law depends only on the gap t - s, not on where it sits.
Continuous paths: almost every sample path t \mapsto W_t(\omega) is continuous (no jumps).

Taking s = 0 gives the headline fact W_t \sim N(0, t), so \mathbb{E}[W_t] = 0 and \operatorname{Var}(W_t) = t. The spread grows like \sqrt{t}: uncertainty fans out with time.

Mean and variance, derived from the axioms

Let us not take those numbers on faith — every one of them falls straight out of the four properties, with no skipped steps. Start with a single time t > 0 and write W_t as an increment off the start:

W_t = W_t - W_0,

which is allowed because W_0 = 0 by the first property. The increment W_t - W_0 runs over the interval [0, t] of length t - 0 = t, so by the Gaussian-increments property

W_t - W_0 \sim N(0,\, t - 0) = N(0, t).

Reading the mean and variance straight off this normal law:

\mathbb{E}[W_t] = 0,

\operatorname{Var}(W_t) = \operatorname{Var}(W_t - W_0) = t.

So the variance at time t is exactly t — nothing more was needed than "starts at zero" plus "increments are N(0, \text{gap})".

The covariance: \operatorname{Cov}(W_s, W_t) = \min(s, t)

Values at two different times are linked through the history they share. Fix two times and, without loss of generality, label them so that s < t. The idea is to split the later value into "what had happened by s" plus "what happened afterwards", because those two parts are independent. Write

W_t = W_s + (W_t - W_s),

a plain algebraic identity. Now expand the covariance using its bilinearity (covariance is linear in each argument):

\operatorname{Cov}(W_s, W_t) = \operatorname{Cov}\big(W_s,\, W_s + (W_t - W_s)\big),

= \operatorname{Cov}(W_s, W_s) + \operatorname{Cov}(W_s,\, W_t - W_s).

Take the two terms in turn. The first is just a variance,

\operatorname{Cov}(W_s, W_s) = \operatorname{Var}(W_s) = s,

by the result we just derived. For the second term, notice that W_s = W_s - W_0 is the increment over [0, s] and W_t - W_s is the increment over the disjoint interval [s, t]. By the independent-increments property these two increments are independent, and the covariance of independent random variables is zero:

\operatorname{Cov}(W_s,\, W_t - W_s) = 0.

Adding the two pieces back together,

\operatorname{Cov}(W_s, W_t) = s + 0 = s.

We assumed s < t, so s = \min(s, t); had we labelled them the other way the same argument would give t = \min(s, t). Either way the answer is the smaller of the two times:

\operatorname{Cov}(W_s, W_t) = \min(s, t).

For a standard Brownian motion (W_t)_{t \ge 0} and any times s, t \ge 0, \mathbb{E}[W_t] = 0, \qquad \operatorname{Var}(W_t) = t, \qquad \operatorname{Cov}(W_s, W_t) = \min(s, t). In particular W_t \sim N(0, t). These three facts follow from the defining properties alone — W_0 = 0, independent increments, and W_t - W_s \sim N(0, t - s) — and they characterise the covariance structure of the process completely. (For a Gaussian process the mean and covariance fix the law, so they fix Brownian motion itself.)

The four properties are easy to state, but it is not obvious that any process satisfies all of them at once — continuous paths and independent Gaussian increments pull in different directions. That a Brownian motion actually exists is a theorem, and there are two famous ways to see it.

Direct construction (Lévy / Wiener). One can build W explicitly on [0, 1] by the Lévy–Ciesielski method: fill in the value at the midpoint with the correct conditional Gaussian (a "Brownian bridge" interpolation), then the quarter-points, then the eighths, and so on. The series converges uniformly with probability one, so the limit has continuous paths by construction, and one checks it has the right Gaussian increments. This is essentially Wiener's original existence proof, which is why the process bears his name.

Scaling limit of a random walk (Donsker). Take a simple symmetric random walk S_n = \xi_1 + \cdots + \xi_n with \pm 1 steps, speed it up n-fold in time and shrink it by \sqrt{n} in space:

W^{(n)}_t = \frac{1}{\sqrt{n}}\, S_{\lfloor n t \rfloor}.

Donsker's invariance principle says these rescaled walks converge in distribution (as processes on path space) to Brownian motion. The 1/\sqrt{n} is exactly the central-limit scaling, which is why the limiting increments are Gaussian — Brownian motion is the continuous-time central limit theorem made into a process, the precise sense in which it is "the continuous cousin of the random walk".

One path of the noise

Below is a single sample path of W_t over t \in [0,1], built as a running sum of many tiny independent normal increments \Delta W \sim N(0, \Delta t). Notice the texture: it is continuous, yet violently jagged at every scale, and it shows no preferred direction. Refresh to draw a fresh path — a different \omega.

A note on its history

The name honours the botanist Robert Brown, who in 1827 watched pollen grains jitter erratically in water. The mathematics that turned that jitter into a rigorous object was supplied by Norbert Wiener in the 1920s — which is why the process bears his name too. Remarkably, it had already been used to model the stock market in 1900 by Louis Bachelier, five years before Einstein's celebrated physical account — the first time chance was given a continuous, time-evolving shape in finance.

Independent increments make variances add: split [0, t] into n equal pieces and W_t is the sum of n independent increments, each of variance t/n, so \operatorname{Var}(W_t) = n \cdot (t/n) = t. The standard deviation is therefore \sqrt{t} — typical displacement scales with the square root of time, the signature of diffusion. Doubling the elapsed time only multiplies the typical wander by \sqrt{2}, not by 2.