The Ito Integral

The Ito integral makes sense of integrating a random process against Brownian motion, a task ordinary calculus cannot handle because Brownian paths have infinite variation, by exploiting their finite quadratic variation and a clever choice of evaluation points.

Definition

The Ito integral of a predictable process against Brownian motion is the limit, in mean square, of approximating sums that evaluate the integrand at the left endpoint of each subinterval, defined first for simple integrands and extended by the Ito isometry.

Scope

The topic covers the construction of the Ito integral first for simple predictable integrands and then by the Ito isometry for square-integrable ones, the extension to continuous local martingales, the martingale property of the integral and its quadratic variation, the contrast between the Ito and Stratonovich conventions, and the role of predictability and the non-anticipating choice of left endpoints.

Core questions

• Why does integration against Brownian motion require a new definition?
• How does the Ito isometry make the construction work?
• Why must the integrand be evaluated at the left endpoint, and what does predictability ensure?
• How does the Ito integral differ from the Stratonovich integral?

Key concepts

• predictable integrand
• Ito isometry
• quadratic variation
• martingale property
• Ito versus Stratonovich

Key theories

Ito isometry and construction

For square-integrable predictable integrands the mean square of the Ito integral equals the expected time-integral of the squared integrand, an isometry that lets the integral be defined for simple processes and extended by completeness to a large class of integrands.

Martingale property of the integral

The Ito integral of a suitable predictable process against Brownian motion is itself a continuous martingale with quadratic variation given by the time-integral of the squared integrand, which is what makes the left-endpoint, non-anticipating convention the natural one.

Clinical relevance

The Ito integral is the mathematical object representing the gains from a continuously rebalanced trading strategy in mathematical finance, the accumulated effect of noise in models of physical and biological systems, and the innovations term in stochastic filtering; its martingale property is the analytic basis of arbitrage-free pricing.

History

Kiyosi Ito defined the stochastic integral in the 1940s to give meaning to differential equations driven by Brownian motion, and Stratonovich later introduced an alternative convention with ordinary chain-rule behavior; the Ito construction, with its martingale property, became the standard for probability and finance.

Key figures

• Kiyosi Ito
• Ruslan Stratonovich
• Henry McKean

Seminal works

• karatzas1991

Frequently asked questions

Why is the integrand evaluated at the left endpoint?

Using the left endpoint keeps the integrand non-anticipating, so it cannot peek at the future increment of Brownian motion; this is what makes the resulting integral a martingale and reflects the causal nature of strategies and controls.

How does the Ito integral differ from the Stratonovich integral?

The Stratonovich integral evaluates the integrand at the midpoint and obeys the ordinary chain rule but is not a martingale, while the Ito integral uses the left endpoint, is a martingale, and obeys the modified Ito chain rule; the two differ by a correction term involving quadratic variation.