Ito's formula
/ EE-toh /
Ito's formula is the change-of-variables formula of stochastic calculus — the analogue of the fundamental theorem of calculus and the chain rule, but corrected for the fact that the integrator is rough. It is the single most-used result in the whole subject: it is how you compute with stochastic integrals, derive SDEs, prove Levy's characterization, build the stochastic exponential, and connect SDEs to PDEs via Feynman-Kac. If you know one theorem in stochastic analysis, it is this one.
For a smooth function f and Brownian motion B, the formula reads f(B_t) = f(B_0) + integral_0^t f'(B_s) dB_s + (1/2) integral_0^t f''(B_s) ds. In differential shorthand, df(B) = f'(B) dB + (1/2) f''(B) dt; the second term, absent from ordinary calculus, is the Ito correction, and it comes directly from (dB)^2 = dt in the second-order Taylor expansion. The genuinely useful version is for a continuous semimartingale X_t = X_0 + integral b ds + integral sigma dB (or, more abstractly, X = M + A with M a continuous local martingale and A of bounded variation): then for f twice continuously differentiable, f(X_t) = f(X_0) + integral_0^t f'(X_s) dX_s + (1/2) integral_0^t f''(X_s) d[X]_s, where [X] is the quadratic variation of X (which equals integral sigma^2 ds in the SDE form). The dX integral splits into a dM (martingale) part and a dA (drift) part; the (1/2) f'' d[X] term is the new piece. Two features are worth stressing: f need only be C^2 (no more), and the integral integral f'(X) dX is itself a stochastic integral defined by the earlier construction, so the formula expresses a transformed semimartingale again as a semimartingale.
Its reach is enormous. It turns the verification of martingale properties into differentiation (a process is a local martingale iff its drift term vanishes), it produces the SDE satisfied by any smooth function of a diffusion, and it is the bridge from probability to PDE: applying Ito to f(X_t) and setting the drift to give the generator yields Dynkin's formula and the Feynman-Kac representation. The honest caveats: (1) the C^2 hypothesis matters — for non-smooth f (e.g. |x| or (x-a)^+) the naive formula fails and one needs the Tanaka formula with local time as the correction. (2) The formula is for the Ito integral with its left-endpoint convention; the Stratonovich version obeys the ordinary chain rule with no correction term, precisely because its symmetric convention absorbs the (1/2) f'' d[X]. (3) The d[X] in the correction is the quadratic variation of the integrator, not of time — for a general continuous martingale M it is d[M], not dt.
Take f(x) = e^x applied to X_t = sigma B_t - (sigma^2/2) t. Here [X]_t = sigma^2 t. Ito gives d(e^X) = e^X dX + (1/2) e^X d[X] = e^X (sigma dB - (sigma^2/2) dt) + (1/2) e^X sigma^2 dt = sigma e^X dB. The drift cancels exactly, so e^{sigma B_t - (sigma^2/2) t} is a (local) martingale — this is the geometric Brownian / stochastic exponential, and the cancellation is engineered by the -(sigma^2/2) t term precisely to kill the Ito correction.
Ito's formula computes d(e^X) with its second-order term; the deliberate -(sigma^2/2)t drift cancels it, producing a martingale.
Ito's formula needs f in C^2 and applies to the Ito (left-endpoint) integral; for non-smooth f use the Tanaka formula, and remember the correction term carries d[X] (quadratic variation of the integrator), which is dt only for standard Brownian motion.