The Itô Integral

Core Idea

The Itô integral ∫₀ᵀ H(t) dW(t) extends integration to allow Brownian motion as the integrator. Because Brownian paths have infinite variation, Riemann-Stieltjes integration fails, and the integral must be constructed as an L² limit of integrals of simple (step) processes. The key choice is evaluating the integrand at the left endpoint of each subinterval, which produces an integral that is a martingale with zero expectation and the Itô isometry E[(∫H dW)²] = E[∫H² dt].

Explainer

Classical integration theory — whether Riemann or Lebesgue — integrates functions against smooth or bounded-variation integrators. Brownian motion has infinite variation on every interval, so these tools fail. The Itô integral resolves this by constructing ∫₀ᵀ H(t) dW(t) as an L² limit rather than a pathwise limit. The construction proceeds in three steps: define the integral for simple (step function) processes as a finite sum Σ Hᵢ(W(tᵢ₊₁) - W(tᵢ)), prove the Itô isometry for these simple integrals, then extend to general adapted square-integrable processes by approximation in L².

The critical design choice is left-endpoint evaluation: the integrand H is evaluated at the left endpoint tᵢ of each subinterval [tᵢ, tᵢ₊₁], not at the midpoint or right endpoint. This is not arbitrary — it ensures that H(tᵢ) is known (adapted to ℱ_{tᵢ}) before the increment W(tᵢ₊₁) - W(tᵢ) is realized. The consequence is that the Itô integral is a martingale: E[∫₀ᵗ H dW | ℱₛ] = ∫₀ˢ H dW for s ≤ t. The integrand is "decided" before the randomness arrives, so no information about the future leaks in. The Stratonovich convention (midpoint evaluation) produces a different integral that satisfies the ordinary chain rule but is not a martingale — the Itô convention sacrifices the classical chain rule to gain the martingale property, a trade that turns out to be enormously profitable.

The Itô isometry E[(∫₀ᵀ H dW)²] = E[∫₀ᵀ H² dt] is the engine of the construction. It says the L² norm of the stochastic integral equals the L² norm of the integrand computed against ordinary Lebesgue measure. The proof is elegant: expand the square of the Riemann sum and observe that cross-terms vanish by independence of increments, leaving only diagonal terms. This isometry makes the map H ↦ ∫H dW a bounded linear operator from L²(Ω × [0,T]) to L²(Ω), and bounded linear operators on Hilbert spaces extend uniquely and continuously to the closure — completing the construction.

The price of left-endpoint evaluation appears immediately in the simplest example. Computing ∫₀ᵀ W(t) dW(t) from the Riemann sum Σ W(tᵢ)(W(tᵢ₊₁) - W(tᵢ)) and using the identity ab = (1/2)((a+b)² - a² - b²) yields (1/2)W(T)² - (1/2)Σ(ΔWᵢ)². The sum of squared increments converges to T (the quadratic variation), giving ∫₀ᵀ W dW = (1/2)W(T)² - (1/2)T. The "-T/2" correction is the signature of Itô calculus — it is absent in Stratonovich calculus and absent in ordinary calculus. This correction generalizes to Itô's formula, the chain rule of stochastic calculus, which is the next topic.