Ito Calculus and Stochastic Integration

Ito calculus extends integration and differentiation to processes driven by Brownian motion, replacing the ordinary chain rule with Ito's formula, which carries an extra term from the quadratic variation.

Definition

The Ito integral is the stochastic integral of a predictable process against Brownian motion, defined so that it is a martingale with variance given by the Ito isometry, and Ito's formula is the resulting change-of-variables rule that adds a second-derivative term reflecting the quadratic variation of the integrator.

Scope

This topic covers the construction of the Ito integral as a limit of left-endpoint Riemann sums against Brownian motion, the Ito isometry, the martingale property of the integral, Ito's formula for functions of diffusions, the multidimensional and product rules, the comparison with the Stratonovich integral, and the quadratic-variation calculus that distinguishes stochastic from ordinary integration.

Core questions

• How is the Ito integral constructed and why must left endpoints be used?
• What is the Ito isometry and how does it control the integral's variance?
• What extra term distinguishes Ito's formula from the ordinary chain rule?
• How does the Ito integral differ from the Stratonovich integral?

Key theories

Ito integral and the Ito isometry

Defining the integral with left-endpoint evaluations makes it a martingale, and the Ito isometry equates the expected squared integral with the expected integral of the squared integrand, giving the integral its L2 structure and stability.

Ito's formula

For a smooth function of a diffusion, Ito's formula expresses the differential as the usual gradient term plus a correction involving the second derivative and the quadratic variation, the rule that makes stochastic calculus computational and yields the Black-Scholes equation.

Clinical relevance

Ito calculus is the working language of mathematical finance, where Ito's formula derives the Black-Scholes partial differential equation and hedging strategies, and of stochastic control, filtering, and physics, wherever systems are modelled as driven by Gaussian white noise.

History

Ito introduced the stochastic integral and his change-of-variables formula in papers of 1944 and 1951 to construct diffusion processes, Stratonovich and Fisk later proposed an alternative integral obeying the ordinary chain rule, and the two formulations were reconciled as the theory matured through the work of McKean, Meyer, and others.

Key figures

• Kiyosi Ito
• Ruslan Stratonovich
• Henry McKean

Seminal works

• oksendal2003

Frequently asked questions

Why does Ito's formula have an extra term?

Because Brownian motion has nonzero quadratic variation, the second-order term in a Taylor expansion does not vanish in the limit, adding a half-times-second-derivative correction absent from ordinary calculus.

What is the difference between the Ito and Stratonovich integrals?

The Ito integral evaluates the integrand at the left endpoint and is a martingale, while the Stratonovich integral uses the midpoint and obeys the ordinary chain rule; they differ by a correction term and suit different applications.