Itô's lemma

Itô's lemma gives the differential of a smooth function of an Itô process, adding a second-derivative term that accounts for quadratic variation.

Itô process: A stochastic process described by a drift term and a diffusion term driven by Brownian motion. The lemma applies to functions of this process and transforms its drift and diffusion.
Stochastic calculus: The mathematical study of integration and differential equations involving random processes. Itô's lemma is a foundational change-of-variables rule within this broader subject.
Geometric Brownian motion: A positive stochastic process whose logarithm follows Brownian motion with drift. Applying the lemma to its logarithm yields a simple linear stochastic differential.
Kiyoshi Itô: Japanese mathematician who founded much of modern stochastic calculus. He introduced the lemma and the integration theory underlying it.
Quadratic variation: The limiting accumulated sum of squared increments of a stochastic process. Brownian motion has nonzero quadratic variation, producing the lemma’s second-derivative correction.
Stochastic differential equation: A differential equation whose evolution includes random forcing, commonly modeled with Brownian motion. The lemma transforms functions of solutions to these equations.
Black–Scholes equation: A partial differential equation governing prices of European options under the Black–Scholes assumptions. Itô's lemma helps derive the equation by tracking an option price as its underlying asset moves.
Itô calculus: A calculus for stochastic processes built around the Itô integral and quadratic variation. The lemma is its characteristic chain rule.
Brownian motion: A continuous stochastic process with independent, normally distributed increments and variance proportional to elapsed time. Its increments scale so their squared contributions persist in the differential limit.
Drift (stochastic process): The predictable rate of change in a stochastic process, represented by its finite-variation component. The lemma combines the original drift with a correction derived from diffusion.

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