Itô calculus

Itô calculus is the stochastic calculus developed by Kiyoshi Itô for manipulating integrals and derivatives of stochastic processes, particularly Wiener processes. Unlike ordinary calculus, where the chain rule is straightforward, Itô calculus introduces an additional second-order term that accounts for the fact that a Wiener process has non-zero quadratic variation.

The central formula is Itô's lemma: if X_t follows a stochastic differential equation, then a function f(X_t) evolves with an additional (1/2)f(X_t)(dX_t)^2 term. This term is the signature of stochastic calculus: it encodes the fact that random walks accumulate variance in a way that deterministic flows do not.

Itô calculus is not a minor technical adjustment. It is the reason that stochastic differential equations are not merely ODEs with noise but a distinct class of mathematical object. Without Itô's correction, the naive chain rule would produce wrong answers, and the entire edifice of mathematical finance — from the Black-Scholes equation to risk-neutral pricing — would collapse.