Itô's formula is the chain rule of stochastic calculus: for a C² function f and a process X(t) = X(0) + ∫μ ds + ∫σ dW, we have df(X) = f'(X)dX + (1/2)f''(X)σ²dt. The extra second-order term (1/2)f''σ²dt arises because (dW)² = dt — the non-zero quadratic variation of Brownian motion prevents the second-order Taylor term from vanishing as it does in classical calculus. This formula is the single most used tool in stochastic calculus.
Itô's formula is to stochastic calculus what the chain rule is to ordinary calculus — it tells you how to compute df(X) when X is a stochastic process. But unlike the classical chain rule, the stochastic version has an extra term. If X(t) satisfies dX = μ(t)dt + σ(t)dW and f is C², then df(X) = f'(X)μ dt + f'(X)σ dW + (1/2)f''(X)σ² dt. The third term, (1/2)f''σ²dt, is the Itô correction — it is absent from classical calculus and arises entirely from the non-zero quadratic variation of Brownian motion.
The derivation follows from a second-order Taylor expansion: f(X + dX) ≈ f(X) + f'(X)dX + (1/2)f''(X)(dX)². In classical calculus, (dX)² is second-order infinitesimal and vanishes. But for stochastic processes, (dX)² = (μ dt + σ dW)² = σ²(dW)² + 2μσ(dt)(dW) + μ²(dt)². Using the multiplication rules dt·dt = 0, dt·dW = 0, and dW·dW = dt (the quadratic variation), only the σ²dt term survives. This is a rigorous consequence of the quadratic variation computation you studied in the previous topic: Σ(ΔWᵢ)² → T in L², so squared Brownian increments behave like dt at the infinitesimal level.
The formula's power lies in its ability to transform one stochastic differential equation into another. To analyze a complicated process Y(t) = f(X(t)), apply Itô's formula to find the SDE for Y directly. For example, if S follows geometric Brownian motion dS = μS dt + σS dW, then applying Itô's formula to f(S) = ln(S) gives d(ln S) = (μ - σ²/2)dt + σ dW. The logarithm converts the multiplicative SDE into an additive one with constant coefficients — immediately showing that ln S(T) is normally distributed and S(T) is lognormally distributed. This single computation underlies the Black-Scholes option pricing model.
The multidimensional version handles functions of several Itô processes simultaneously. If X₁, ..., Xₙ are Itô processes driven by possibly correlated Brownian motions, then for f(X₁, ..., Xₙ) the formula includes all partial derivatives ∂f/∂xᵢ (first order), all cross terms (1/2)∂²f/∂xᵢ∂xⱼ times the quadratic covariation d[Xᵢ, Xⱼ] (second order), and the time derivative ∂f/∂t if f depends explicitly on t. The structure is always the same: classical chain rule plus second-order corrections from quadratic (co)variation. Mastering this formula is the single most important skill in stochastic calculus.