Quadratic Variation

Squared Brownian wiggles add up to the elapsed time: [W]_T = T, the theorem behind (dW)^2 = dt.

The idea

The quadratic variation of a path measures its roughness. Partition an interval $[0, T]$ into small pieces, add up the squared increments of the path across the pieces, and refine the partition; the limit of these sums is the quadratic variation of the path on $[0, T]$, written $[W]_{T}$ for a path $W$.

Theorem (Quadratic variation of Brownian motion).

Let $W$ be a standard Brownian motion, and let $Q_{n}$ be the sum of its squared increments over the partition of $[0, T]$ into $n$ equal pieces. Then $Q_{n} \to T$ in probability as $n \to \infty$; that is, $[W]_{T} = T$. By contrast, every continuously differentiable path has quadratic variation $0$.

The two values mark the dividing line between ordinary and stochastic calculus. Discarding second-order terms is what every Taylor expansion cut after the first-order term does, and it is justified exactly when the quadratic variation is $0$. Along a Brownian path the second-order term survives and must be kept.

Ways to work on it

• Walkthrough. Define quadratic variation and see why Brownian paths accumulate it while smooth paths do not.
• Proof. See why the squared-increment sums of Brownian motion concentrate, and why smooth paths score zero.
• Practice. Compute quadratic variations of Brownian motion, Itô processes, and smooth paths.
• Hardest. Evaluate a classic stochastic integral, or find the quadratic variation under time-varying volatility.

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