Financial Mathematics · Lecture 6 of 12

From Random Walks to Brownian Motion

Shrink the coin-flip walk in time and scale it by √n, and a continuous but nowhere-smooth random path appears. Its strange properties are why ordinary calculus breaks.

November 15, 2026By Diganta Bhattacharya5 min read

Scaling a random walk

Take the ±1 walk from Lecture 1 with $n$ steps in unit time. Each step is shrunk to $\pm1/\sqrt n$ (not $1/n$, which would wash out to zero by the law of large numbers):

$$W^{(n)}(t)=\frac{1}{\sqrt n}\sum_{i=1}^{\lfloor nt\rfloor}\xi_i,\qquad \operatorname{Var}W^{(n)}(t)\approx t.$$

As $n\to\infty$ the CLT makes $W^{(n)}(t)$ normal with variance $t$, and the whole path converges (Donsker's theorem) to Brownian motion.

Definition

Standard Brownian motionA process $W_t$ with
  1. $W_0=0$;
  2. independent increments: $W_t-W_s$ is independent of the past before $s$;
  3. stationary Gaussian increments: $W_t-W_s\sim\mathcal N(0,t-s)$;
  4. continuous paths.

Consequences: $\mathbb E W_t=0$, $\operatorname{Var}W_t=t$, $\operatorname{Cov}(W_s,W_t)=\min(s,t)$, and the scaling property $W_{ct}\overset{d}{=}\sqrt c\,W_t$.

Strange but crucial properties

1. Nowhere differentiable

Over a step $h$, $\Delta W\sim\sqrt h$, so $\frac{\Delta W}{h}\sim\frac{1}{\sqrt h}\to\infty$. The path is continuous but has no slope anywhere. "$dW/dt$" doesn't exist, which is why we write SDEs with $dW$ rather than ODEs with $dW/dt$.

2. Infinite total variation

Summing $|\Delta W|$ over a partition of $[0,t]$ gives roughly $n\cdot\sqrt{t/n}=\sqrt{nt}\to\infty$. Classical (Riemann–Stieltjes) integration against $W$ is therefore impossible.

3. Finite quadratic variation

But the sum of squared increments settles down:

$$\sum_{i=1}^n\big(W_{t_i}-W_{t_{i-1}}\big)^2\;\longrightarrow\;t\qquad\text{as the mesh}\to0.$$

Each term has mean $t/n$ and variance $2t^2/n^2$, so the sum has mean $t$ and variance $2t^2/n\to0$. A random sum becomes deterministic. This is written informally as

$$(dW_t)^2=dt,\qquad dW_t\,dt=0,\qquad (dt)^2=0.$$

In the random-walk figure, the printed sum of squared steps is exactly 1 for any path. Lecture 8 shows the same convergence for true Brownian increments. These multiplication rules are all you need to use Itô's lemma.

Brownian motion with drift and scale

$X_t=\mu t+\sigma W_t$ is normal with mean $\mu t$ and variance $\sigma^2t$. Over short horizons the noise ($\sigma\sqrt t$) dominates the drift ($\mu t$). This is why daily returns look like pure noise even for a stock with a healthy expected return, and why μ is so hard to estimate.

Estimation factWith σ = 20% and μ = 8%, distinguishing μ from 0 at two standard errors needs about $(2\sigma/\mu)^2=25$ years of data. Volatility, on the other hand, can be estimated from a few weeks of intraday data. Black–Scholes needs σ but not μ, which is a gift.

Key takeaways

Exercises

  1. Compute Cov(W₂, W₅) and Var(W₅ − 2W₂).
  2. Show that −W_t and W_{ct}/√c are also Brownian motions.
  3. If daily σ = 1.25%, what is the annualised σ (252 days)? Why is it not 252 × 1.25%?