Brownian Motion

The random walk run in continuous time: normal increments whose variance is the elapsed time.

The idea

Standard Brownian motion is the continuous-time counterpart of the random walk: the model for a quantity that moves randomly at every instant, such as a pollen grain struck by water molecules or a continuously quoted price.

We construct it as a limit. Run the simple symmetric walk at $n$ steps per unit of time, shrink each step to size $\frac{1}{\sqrt{n}}$, and let $n$ grow. The shrink rate is forced: larger steps give a motion whose spread grows without bound, smaller steps give a motion that collapses to zero, and $\frac{1}{\sqrt{n}}$ is the one rate that keeps the spread finite and positive. The limit is standard Brownian motion, written $W(t)$.

Three properties characterize $W$, each inherited from the walk. It starts at $W(0) = 0$. Its increments over disjoint intervals of time are independent, because the steps taken in one interval have nothing to do with those taken in another. And each increment is normally distributed with mean $0$ and variance equal to the elapsed time,

$W(t) - W(s) \sim \mathcal{N}(0,\ t - s),$

because a sum of many small independent steps is normal. Every further property of $W$ follows from these three.

Ways to work on it

• Walkthrough. Brownian motion as the limit of random walks, and the properties that define it.
• Practice. Compute probabilities, variances, and covariances for a Brownian path.
• Hardest. Analyze a rescaled Brownian motion across two points in time.

Not sure where to start? Take the ten-question placement test.