Ito's lemma
/ EE-toh /
In ordinary calculus the chain rule tells you how a function changes when its input changes: if y = f(x), then a small change in x produces a change df = f'(x) dx. Ito's lemma is the chain rule for functions of Brownian motion — and it carries a surprise extra term. If you apply a smooth function f to a Brownian path to get f(B(t)), how does f(B(t)) change moment to moment? The answer is NOT just f'(B) dB. There is an additional second-derivative term that has no counterpart in ordinary calculus, and it appears for one reason only: the quadratic variation of Brownian motion does not vanish.
Here is where it comes from. Taylor-expand f(B + dB) to second order: df = f'(B) dB + (1/2) f''(B) (dB)^2 + smaller terms. In ordinary calculus you throw away the (dB)^2 term because it is negligible compared to dB. But for Brownian motion (dB)^2 is NOT negligible — by quadratic variation it equals dt, a genuine first-order quantity. So the second-order term survives, and you get Ito's lemma: df(B) = f'(B) dB + (1/2) f''(B) dt. The first piece is the familiar chain-rule term (and is a martingale, the random integral against dB); the second piece, (1/2) f''(B) dt, is a smooth, predictable drift that ordinary calculus never sees. For a general process X driven by dX = a dt + b dB, the rule reads df(X) = f'(X) dX + (1/2) f''(X) b^2 dt.
Ito's lemma is the single most-used tool in all of stochastic calculus — it is the workhorse you reach for constantly, the way you reach for the chain rule in ordinary calculus. It is what lets you solve stochastic differential equations, derive the Black-Scholes equation, and turn one process into another. The crucial intuition to carry: convexity creates drift. Because of the +(1/2) f'' term, a convex function of a fair (zero-mean) random process tends to drift UPWARD even though the underlying process has no trend — this is the exact mechanism behind 'volatility pumping' and why E[f(B)] differs from f(E[B]) (Jensen's inequality, made dynamic). The honest reminder: this is the deep end, stated here for the constant-coefficient case; the general theorem requires the full Ito calculus to state and prove carefully.
Apply f(x) = x^2 to Brownian motion. Here f' = 2x and f'' = 2, so Ito's lemma gives d(B^2) = 2B dB + (1/2)(2) dt = 2B dB + dt. Integrating recovers the earlier surprise: B(T)^2 = 2 'integral of B dB' + T, i.e. 'integral of B dB' = B(T)^2/2 - T/2. The extra +dt is pure quadratic variation.
The extra (1/2) f'' dt term comes entirely from (dB)^2 = dt; convexity creates an upward drift.
The extra second-order term exists only because (dB)^2 = dt does not vanish. Drop it and you have the wrong answer — Ito's lemma is NOT the ordinary chain rule.