Home Statistical Dictionary Ito's Lemma

Ito's Lemma

The chain rule of stochastic calculus, giving the correct formula for how a function of a diffusion process changes over time, including an extra second-order correction term absent from ordinary calculus.

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In Plain English

In ordinary calculus, the chain rule says if $X$ changes and $f$ depends on $X$ , then $f (X)$ 's change is just $f^{'} (X)$ times $X$ 's change. That simple rule breaks down for Brownian motion, because its path is so jagged that the usual first-order approximation misses a real, non-negligible contribution. Ito's lemma is the corrected chain rule for this setting: it keeps the familiar first-order term but adds a second-order correction term (involving $f^{''}$ ) that would ordinarily vanish for smooth paths but does not vanish here. This one extra term is the single most consequential difference between ordinary and stochastic calculus, and it's the key step in deriving the Black-Scholes option pricing formula.

Definition

Let $X_{t}$ satisfy the stochastic differential equation $d X_{t} = μ (X_{t}, t) d t + σ (X_{t}, t) d W_{t}$ , and let $f (x, t)$ be a twice continuously differentiable function. Ito's lemma states that $Y_{t} = f (X_{t}, t)$ satisfies $d Y_{t} = (f_{t} + μ f_{x} + \frac{1}{2} σ^{2} f_{xx}) d t + σ f_{x} d W_{t}$ , where subscripts denote partial derivatives. The extra term $\frac{1}{2} σ^{2} f_{xx}$ , absent from the ordinary chain rule, arises because Brownian increments satisfy the heuristic multiplication rule $(d W_{t})^{2} = d t$ (rather than the ordinary-calculus convention that products of infinitesimals vanish), a consequence of Brownian motion's quadratic variation being non-zero (equal to $t$ ) despite its paths having infinite ordinary variation. Ito's lemma is the essential tool for deriving the Black-Scholes partial differential equation from geometric Brownian motion, and more generally for solving or transforming any SDE by a well-chosen change of variables.

Formula

df (X_{t}, t) = (\frac{\partial f}{\partial t} + μ \frac{\partial f}{\partial x} + \frac{1}{2} σ^{2} \frac{\partial ^{2} f}{\partial x ^{2}}) d t + σ \frac{\partial f}{\partial x} d W_{t}

Notation

f (x, t)

—A twice-differentiable function applied to the diffusion process $X_t$

\frac{1}{2} σ^{2} f_{xx}

—The Ito correction term, absent from the ordinary chain rule

Properties

The famous derivation of geometric Brownian motion's exact solution, with its $- σ^{2} /2$ term, is a direct application of Ito's lemma to $f (x) = ln x$ : the correction term is exactly what produces that otherwise-mysterious $- σ^{2} /2$ adjustment to the naive exponential guess.
Ito's lemma is often summarized by the informal multiplication table $d W_{t} \cdot d W_{t} = d t$ , $d W_{t} \cdot d t = 0$ , $d t \cdot d t = 0$ , treating these as formal rules makes the lemma's derivation (via a second-order Taylor expansion, keeping the term that these rules say survives) both memorable and mechanically reproducible.
Financial engineering uses Ito's lemma as its core derivation tool: applying it to the value of an option (a function of the underlying asset price, itself geometric Brownian motion) and requiring the result be riskless (no random term) is precisely the derivation that produces the Black-Scholes partial differential equation.

At a Glance

Rolethe chain rule for functions of a diffusion process

Extra term vs. ordinary calculus

\frac{1}{2} σ^{2} f_{xx} d t

, the Ito correction

Source of the correctionthe heuristic rule

(d W_{t})^{2} = d t

, from Brownian motion's quadratic variation

Landmark applicationderiving the Black-Scholes partial differential equation

Last updated September 8, 2026← Back to the Dictionary