A stochastic integral defined for adapted, square-integrable processes with respect to a semimartingale (commonly Brownian motion), constructed as the mean-square limit of nonanticipative Riemann sums using left-endpoint sampling; it satisfies an isometry relating second moments to the predictable quadratic variation of the integrator.

Principle

Nonanticipative (adapted) discretization plus quadratic-variation correction: limits of left-point Riemann sums produce an L2-isometry and an extra second-order term in change-of-variable formulas.

Demonstration

For a predictable process σ(t) with ∫_0^T E[σ(t)^2] dt<∞ and Brownian motion W_t, the Itô integral ∫_0^T σ(t)dW_t exists as an L2 limit and satisfies E[(∫_0^T σ dW)^2]=E[∫_0^T σ^2 dt].

Misapplication

Applying the ordinary chain rule to f(X_t) where X_t solves dX_t=σ(t)dW_t and omitting the Itô correction term (½ f''(X_t)σ(t)^2 dt) — i.e., treating the Itô integral like a classical Riemann integral.

Consequence

Itô integrals produce martingales when integrands are square-integrable and zero-mean; they enter the Itô formula which adds a second-order term proportional to quadratic variation, altering stochastic differential equations' solutions.

Reversal

Stratonovich integrals arise from symmetric (midpoint) Riemann sums and obey the classical chain rule, exchanging the Itô correction for a drift adjustment when converted.

Boundary

Defined for predictable processes integrable in mean-square against a semimartingale; not a pathwise Lebesgue–Stieltjes integral for arbitrary rough paths and requires probabilistic structure (filtration and integrator's quadratic variation).

Semantic Tension

Often contrasted with the Stratonovich and pathwise (Young/rough path) integrals — same symbol ∫ hides differing sampling conventions and transformation rules.

Synthesis

A probabilistic integral built from left-point (nonanticipative) Riemann sums whose L2-isometry and quadratic-variation behavior produce the distinctive Itô correction in stochastic calculus.