Itô integrals and nonanticipating accumulation

For a deterministic partition, an elementary predictable integrand holds a random coefficient Hᵢ constant on each interval (tᵢ,tᵢ₊₁]. The coefficient is measurable using information at tᵢ, before the increment it multiplies is revealed. The integral is the finite sum of those products. Brownian future increments have conditional mean zero, so each product has zero expectation when its integrability is adequate. This temporal ordering is the nonanticipating feature that makes the resulting accumulation a martingale.

A coefficient can still be random and depend strongly on the past. For example, selecting Hᵢ from the current Brownian position is allowed. Selecting it from the sign of the next Brownian increment is not allowed in this elementary Itô construction. Adaptedness alone can require measurability qualifications in full generality; a standard sufficient setting uses predictable or progressively measurable square-integrable processes. The definition is not a blanket rule that every arbitrary time-indexed random coefficient can be integrated against Brownian motion.

When the finite sum is squared, the expected cross terms vanish. For a later increment, condition on its starting information; everything multiplying that increment is already known and the increment has conditional mean zero. A diagonal term contributes the expected squared coefficient times the interval length. Summing yields the Itô isometry. The right side is an ordinary integral of a nonnegative expectation, giving a computable measure of stochastic energy without assigning a derivative to a Brownian sample path.

Approximate a predictable square-integrable H by elementary processes in the norm given by expected integrated square. The isometry guarantees that their stochastic integrals form a Cauchy sequence in L2, so completeness supplies a limit independent of the chosen approximation. This is a definition through controlled approximation, not an instruction to integrate a wildly oscillating drawn curve. Under these assumptions the integral has mean zero, and its second moment equals its variance. Locally square-integrable settings can extend the theory by localization, but need additional care.

For the integrand Hₜ=Bₜ, the left-endpoint sum has a simple identity obtained by expanding each squared increment. The sum of B(tᵢ) times its next increment equals half the terminal squared displacement minus half the sum of increment squares. Quadratic variation makes the latter term approach T, so the integral is one-half of Bₜ squared minus T. Its expectation is zero, as the isometry-based construction requires. Omitting T would produce a nonzero mean and expose the error.

Right-endpoint sums differ from left-endpoint sums by the sum of squared increments, which tends to T rather than zero. A midpoint convention leads to Stratonovich integration, with its own transformation rule and hypotheses. These constructions are related but cannot be interchanged silently. Nor can the formal coefficient B_T be treated as a known constant throughout [0,T] in an Itô integral: before T it contains future information. Different anticipation theories exist, but they are outside the nonanticipating model developed here.