Brownian Motion

Brownian motion, the Wiener process $W_t$, is the single stochastic object underneath almost all of continuous-time finance. Geometric Brownian motion, the Black-Scholes PDE, Heston, SABR, and every short-rate model are built by feeding $W_t$ through some transformation. Understanding it precisely, especially its strange local behaviour, is what makes the rest of stochastic calculus feel inevitable rather than magical.

Definition

A standard Brownian motion $\{W_t\}_{t\ge 0}$ is a stochastic process satisfying four axioms:

1. Starts at zero: $W_0 = 0$ almost surely.
2. Independent increments: for $0 \le t_0 < t_1 < \dots < t_n$, the increments $W_{t_1}-W_{t_0},\ \dots,\ W_{t_n}-W_{t_{n-1}}$ are mutually independent.
3. Stationary Gaussian increments: $W_t - W_s \sim \mathcal{N}(0,\, t-s)$ for $s < t$, the increment's variance equals the elapsed time.
4. Continuous paths: $t \mapsto W_t$ is continuous almost surely.

Everything else follows from these. The variance scaling in axiom 3 is the crucial one: standard deviation grows like $\sqrt{t}$, not $t$, which is the fingerprint of diffusion and the reason vol scales with the square root of horizon.

Immediate consequences

It is a martingale. Since increments are mean-zero and independent of the past,

$\mathbb{E}[W_t \mid \mathcal{F}_s] = W_s + \mathbb{E}[W_t - W_s \mid \mathcal{F}_s] = W_s, \qquad s < t.$

So $W_t$ is a martingale with respect to its natural filtration, its best forecast is its current value. It is also a Markov process: the future depends on the past only through the present.

Covariance. For $s \le t$, writing $W_t = W_s + (W_t - W_s)$ and using independence,

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

Self-similarity. For any $c > 0$, the rescaled process $\tfrac{1}{\sqrt c}\,W_{ct}$ is again a standard Brownian motion. Brownian motion is a fractal: it looks statistically identical at every zoom level, which is exactly why you cannot draw a tangent to it.

Quadratic variation: the load-bearing property

Ordinary smooth functions have zero quadratic variation. Brownian motion does not, and this single fact is what generates the entire Itô calculus. Partition $[0,t]$ into $n$ equal steps of width $\Delta t = t/n$ and form the sum of squared increments

$Q_n = \sum_{i=1}^{n} (W_{t_i} - W_{t_{i-1}})^2.$

Each squared increment has mean $\mathbb{E}[(\Delta W_i)^2] = \Delta t$ and, because $\Delta W_i \sim \mathcal{N}(0,\Delta t)$ so its square is $\Delta t$ times a $\chi^2_1$,

$\operatorname{Var}\big((\Delta W_i)^2\big) = 2(\Delta t)^2.$

By independence of increments,

$\mathbb{E}[Q_n] = n\,\Delta t = t, \qquad \operatorname{Var}(Q_n) = n \cdot 2(\Delta t)^2 = \frac{2t^2}{n} \xrightarrow[n\to\infty]{} 0.$

So $Q_n \to t$ in $L^2$: the quadratic variation of Brownian motion over $[0,t]$ is the deterministic number $t$. We write this heuristically as

$\boxed{(dW_t)^2 = dt,}$

together with $dt\,dW_t = 0$ and $(dt)^2 = 0$. This box is the entire content of Itô's lemma, the second-order term in a Taylor expansion does not vanish, because $(\Delta W)^2$ is of order $\Delta t$, not $(\Delta t)^2$.

Nowhere differentiability and infinite variation

Because $\Delta W \sim \mathcal{N}(0,\Delta t)$, the difference quotient behaves like

$\frac{\Delta W}{\Delta t} \sim \mathcal{N}\!\Big(0,\ \frac{1}{\Delta t}\Big),$

whose variance explodes as $\Delta t \to 0$. The derivative does not exist at any point: Brownian paths are continuous everywhere but differentiable nowhere. Relatedly, the first-order (total) variation is infinite, a Brownian path has infinite length over any interval, while the second-order (quadratic) variation is finite and equal to $t$. This inversion of the usual calculus is precisely why $dW$ cannot be manipulated with ordinary rules and why the stochastic integral $\int H_s\,dW_s$ must be defined as an $L^2$ limit rather than pathwise (Riemann-Stieltjes), the integrand must be evaluated at the left endpoint of each interval, giving the Itô integral its martingale property.

From Brownian motion to asset prices

Brownian motion itself is a poor model for a stock price, it goes negative and its increments do not scale with price. The fix is geometric Brownian motion, defined by the SDE

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t.$

Applying Itô's lemma to $\ln S_t$ gives the drift correction $-\tfrac12\sigma^2$ and the explicit solution

$S_t = S_0 \exp\!\Big(\big(\mu - \tfrac12\sigma^2\big)t + \sigma W_t\Big),$

so $\ln S_t$ is Gaussian and $S_t$ is lognormal, the foundation of Black-Scholes.

Worked example

A stock follows GBM with $\sigma = 20\%$ annualized. What is the standard deviation of $W_t$ over one trading day, $\Delta t = 1/252$? Since $\operatorname{sd}(W_{\Delta t}) = \sqrt{\Delta t} = \sqrt{1/252} \approx 0.063$, the log-return over a day has standard deviation $\sigma\sqrt{\Delta t} = 0.20 \times 0.063 \approx 1.26\%$. This is the $\sqrt{t}$ rule in action, annual vol divided by $\sqrt{252} \approx 15.9$ gives daily vol, and it is why quoting "16% a year is 1% a day" is a desk rule of thumb.

What breaks in practice

• Gaussian increments understate tails. Real returns are leptokurtic and jump; pure Brownian diffusion assigns essentially zero probability to a $-20\%$ day, which markets deliver. Jump-diffusion and stochastic-volatility models patch this.
• Constant, known $\sigma$. Volatility clusters and mean-reverts, the flat diffusion coefficient is the very assumption the volatility smile exposes as false.
• Continuous paths. Prices gap over weekends, halts, and news. The continuity axiom is convenient for hedging math but fails exactly when hedging matters most.
• Independent increments. Autocorrelation, microstructure, and momentum all violate independence at short horizons.

In interviews

Be able to state the four defining properties from memory and to derive the quadratic variation result, that $\sum (\Delta W)^2 \to t$ because the mean is $t$ and the variance vanishes like $1/n$. The classic follow-ups: "why is $(dW)^2 = dt$ and not zero?" (the squared increment is $O(\Delta t)$, not $O(\Delta t^2)$), "is Brownian motion differentiable?" (no, the difference quotient has variance $1/\Delta t \to \infty$), and "what is $\operatorname{Cov}(W_s, W_t)$?" (the answer $\min(s,t)$ trips up candidates who forget to use independent increments). Everything in Itô, Girsanov, and risk-neutral pricing rests on these facts.