Ito's Formula

Ito's formula is the chain rule of stochastic calculus: when a smooth function is applied to an Ito process, the differential picks up not only the usual first-order terms but an extra second-order term driven by the quadratic variation.

Definition

Ito's formula expresses the stochastic differential of a smooth function of an Ito process as the sum of the ordinary chain-rule terms and an additional term involving the second derivative and the quadratic variation of the process.

Scope

The topic covers the statement of Ito's formula for functions of Brownian motion and of general Ito processes, the multidimensional version with cross-variation terms, the formula for continuous semimartingales, and its principal consequences including integration by parts, the derivation of the Black-Scholes equation, the Feynman-Kac representation, and Girsanov's change-of-measure theorem.

Core questions

• Why does the stochastic chain rule include a second-order term absent from ordinary calculus?
• How does Ito's formula extend to several processes and to general semimartingales?
• How does it lead to the partial differential equations governing diffusions?
• How do change-of-measure results such as Girsanov's theorem follow from it?

Key concepts

• stochastic chain rule
• quadratic-variation correction
• integration by parts
• Feynman-Kac formula
• Girsanov theorem

Key theories

Ito's formula

For a twice-differentiable function of an Ito process the differential equals the first derivative times the process differential plus one half the second derivative times the quadratic variation, the correction term arising because squared Brownian increments accumulate at a definite rate.

Feynman-Kac and Girsanov consequences

Applying Ito's formula yields the Feynman-Kac representation of solutions to parabolic partial differential equations as expectations over diffusions and Girsanov's theorem describing how Brownian motion transforms under an equivalent change of probability measure.

Clinical relevance

Ito's formula is the computational workhorse of stochastic modeling: it produces the Black-Scholes partial differential equation and option-pricing formulas in finance, derives the equations of stochastic filtering and control, and connects diffusion processes to the partial differential equations of physics through the Feynman-Kac representation.

History

Ito proved his formula in the 1940s as the cornerstone of the new stochastic calculus; Kac's earlier path-integral ideas combined with it to give the Feynman-Kac formula, and Girsanov's 1960 change-of-measure theorem, derived through the same calculus, became essential to filtering and finance.

Key figures

• Kiyosi Ito
• Mark Kac
• Igor Girsanov

Seminal works

• karatzas1991

Frequently asked questions

Why does Ito's formula have an extra term compared with the ordinary chain rule?

Because the squared increments of Brownian motion do not vanish in the limit but accumulate proportionally to time, a second-order Taylor term survives and contributes the characteristic one-half second-derivative term.

What is Ito's formula used for in finance?

Applying it to the discounted price of a derivative as a function of an underlying Ito process produces the Black-Scholes partial differential equation, from which option prices and hedging strategies are obtained.