The Itô integral

Brownian-motion paths are nowhere differentiable, so the Riemann-Stieltjes integral ∫ f dW doesn't make sense in the classical way. Kiyosi Itô (1944) constructed an integral that does — but with two crucial properties that differ from ordinary calculus: it's defined via L² convergence, and it satisfies a quadratic-variation identity that breaks the classical chain rule.

What's wrong with ordinary integration?

Riemann-Stieltjes integration ∫ f(t) dg(t) requires g to have bounded variation. Brownian motion has unbounded variation (Σ |W(t_i+1) - W(t_i)| → ∞ as partition refines). The integral, taken naively, depends on the choice of evaluation point in each subinterval — different choices give different limits. Mathematical horror.

Itô's construction

Itô fixed the evaluation point: always at the left endpoint of each subinterval. The integral becomes:

∫_0^T f(t) dW(t) = lim_{||Π||→0} Σ f(t_i) (W(t_{i+1}) - W(t_i))

The 'left-endpoint' rule is not arbitrary — it makes the integrand 'non-anticipating' (f(t_i) is known before W(t_{i+1}) is revealed). The integral is a limit in L² (mean square), not pointwise.

Itô isometry

E[(∫_0^T f(t) dW(t))²] = E[∫_0^T f(t)² dt]

The expected squared integral equals the expected integral of f². This is the L² geometry that makes the integral well-defined: it's the unique extension of the simple-process integral that is continuous in L² of paths and observations.

Properties

• Linearity: ∫(αf + βg) dW = α ∫f dW + β ∫g dW.
• E[∫_0^T f dW] = 0 — the integral is a martingale.
• M(t) = ∫_0^t f dW is a continuous L² martingale.
• Quadratic variation: ⟨M⟩(t) = ∫_0^t f²(s) ds.

Comparison with Stratonovich integral

Use the midpoint instead of the left endpoint: this gives the Stratonovich integral, denoted ∫ f ∘ dW. The Stratonovich integral obeys the ordinary chain rule (no quadratic-variation correction) but is NOT a martingale — its expected value is non-zero.

Finance uses Itô because martingale theory and risk-neutral pricing rely on the martingale property. Engineering and physics sometimes prefer Stratonovich because it makes coordinate changes cleaner.

What can you integrate against dW?

Any process f(t, ω) that is (a) adapted to the filtration (F_t-measurable), and (b) L² in the sense E[∫_0^T f² dt] < ∞. 'Adapted' means f(t) depends only on information available by time t — the non-anticipating condition.

Itô integral of W against W

∫_0^T W(t) dW(t) = (W(T)² - T) / 2

Note the extra -T/2 term. Compare to the classical ∫ x dx = x²/2: the Itô integral has a quadratic-variation correction. This is the canonical example of the new chain rule.

Approximation by simple processes

The full Itô integral is defined first for simple (piecewise-constant) processes, then extended by L² density to general adapted processes. This is the same machinery as the Lebesgue integral construction, just with stochastic dW driving the integral.