Itô's Lemma

The chain rule for Brownian motion: the second-order term survives, so df = f' dW + 12 f'' dt.

The idea

Theorem (Itô's lemma).

Let $W$ be a standard Brownian motion and let $f$ be twice continuously differentiable. Then $df(W) = f'(W)\,dW + \tfrac{1}{2}f''(W)\,dt.$

Itô's lemma is the chain rule for functions of Brownian motion. The first term is the ordinary chain rule. The second is a correction with no counterpart in ordinary calculus, and it is what the roughness of a Brownian path costs: an increment over a short time $\Delta t$ has typical size $\sqrt{\Delta t}$, so its square has size $\Delta t$ — the same order as the elapsed time — and summed across a partition of $[0, t]$ the squared increments converge to $t$ itself. The rule $(dW)^{2} = dt$ records this.

The surviving term is therefore not extra noise but a deterministic drift, contributed by the path's roughness to every function with curvature.

Ways to work on it

• Walkthrough. Expand to second order to get the corrected chain rule for functions of Brownian motion.
• Proof. Why (dW)^2 = dt: the quadratic-variation sum has mean t and vanishing variance, so it concentrates.
• Practice. Extract Itô-correction coefficients for powers, exponentials, and products; derive Brownian moments.
• Hardest. Find the drift that makes an exponential of Brownian motion a fair game.

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