Ito Integration¶

This section builds the Ito integral from scratch — starting from the failure of classical calculus for Brownian motion and arriving at a rigorous framework for integrating against stochastic processes.

The entire construction is driven by a single fact:

\[ (\Delta B)^2 \sim \Delta t \]

Brownian increments have non-vanishing quadratic variation. This forces a fundamentally different notion of integration and produces the correction terms that distinguish stochastic calculus from ordinary calculus.

Section Roadmap¶

The pages in this section follow a deliberate progression from motivation through construction to consequences. Each concept has exactly one canonical home — other pages reference rather than re-derive.

Quadratic Variation — why classical calculus fails¶

Canonical home for \([W]_t = t\), the conceptual break that forces a new theory of integration.

High School Integral — the baseline¶

The ordinary time integral reviewed as a pathwise object. Establishes the baseline: everything works because the integrator is smooth.

Ito Integral (Intuitive) — what we are trying to define¶

Left-endpoint sums, non-anticipation, and the trading interpretation. No proofs — just the target object and its financial meaning.

Construction of the Ito Integral — the rigorous definition¶

The proof hub: simple processes, Ito isometry proof, \(L^2\) extension, first martingale proof. Canonical home for the formal definition.

Ito Isometry — why the construction is stable¶

Hilbert space and orthogonality interpretation of the isometry. Conceptual meaning, not a second proof.

Properties of Ito Integrals — what we get for free¶

Clean statement of consequences: martingale property, continuity, quadratic variation, linearity. References proofs to the construction page.

Ito Processes — putting drift and diffusion together¶

Drift plus diffusion decomposition. The general framework for modeling asset prices, preparing the ground for Ito's formula.

Stratonovich Integral — the alternative definition¶

Midpoint evaluation, classical chain rule, conversion formulas. The alternative convention with different trade-offs.

Conceptual Flow¶

The logical dependencies between pages form a clear pipeline:

Transitions

After High School Integral: Everything works because \(ds\) is smooth. For Brownian motion, this fails due to quadratic variation.
After Quadratic Variation: The non-vanishing of \([W]_t\) forces a new definition of integration — one that respects the roughness of Brownian paths.
After Intuitive Ito: We know what the integral should be. The construction page makes this rigorous.
After Construction: The Ito isometry explains why the \(L^2\) extension is well-defined and stable.
After Properties: The integral is now a complete tool. Combining it with a drift term gives Ito processes, the general building block for SDEs.
After Ito Processes: The Stratonovich integral provides an alternative convention with different trade-offs.

Role in the Chapter¶

The Ito integral is the primitive object of stochastic calculus. Everything later in the book treats it as given and extends it in specific directions:

Section 3.3 (Ito's Formula) — transforms Ito integrals via the chain rule with quadratic variation corrections
Section 3.4 (SDEs) — uses Ito integrals as the solution space for stochastic differential equations
Section 3.5 (Infinitesimal Generator) — extracts the local drift and diffusion of Ito processes
Chapter 4 (Girsanov) — uses stochastic exponentials of Ito integrals to construct new probability measures

One concept, one identity, everything else references it: the isometry is the single fact that makes the whole theory stable, and \((\Delta B)^2 \sim \Delta t\) is the single fact that forces the theory to exist.