Itô's Formula

Itô's formula is the fundamental theorem of stochastic calculus, providing a change-of-variables formula for functions of Itô processes. It is the stochastic analogue of the chain rule.

Statement

Theorem6.1Itô's formula (one-dimensional)

Let $(X_t)$ be an Itô process $dX_t = \mu_t \, dt + \sigma_t \, dB_t$ and $f \in C^2$ . Then:

$df(X_t) = f'(X_t) \, dX_t + \frac{1}{2} f''(X_t) (dX_t)^2,$

or in integral form:

$f(X_t) = f(X_0) + \int_0^t f'(X_s) \, dX_s + \frac{1}{2} \int_0^t f''(X_s) \sigma_s^2 \, ds.$

The second-order term $\frac{1}{2}f''(X_t)\sigma_t^2 \, dt$ is the Itô correction, arising from the non-zero quadratic variation $(dB_t)^2 = dt$ .

ExampleGeometric Brownian motion

Let $X_t = e^{B_t}$ . Apply Itô with $f(x) = e^x$ :

$dX_t = e^{B_t} dB_t + \frac{1}{2} e^{B_t} dt = X_t(dB_t + \frac{1}{2}dt).$

Integrating: $X_t = \exp(B_t + t/2)$ . The drift correction $+t/2$ comes from Itô's formula.

Multiplication table

The key to Itô calculus is the multiplication table:

$dt \cdot dt = 0, \quad dt \cdot dB_t = 0, \quad dB_t \cdot dB_t = dt.$

These rules follow from $(B_t)$ having quadratic variation $t$ and finite variation processes (like $t$ ) having zero quadratic variation.

ExampleItô formula for $B_t^2$

$f(x) = x^2$ , so $f'(x) = 2x$ , $f''(x) = 2$ . Then:

$d(B_t^2) = 2B_t \, dB_t + \frac{1}{2} \cdot 2 \cdot dt = 2B_t \, dB_t + dt.$

Integrating: $B_t^2 = \int_0^t 2B_s \, dB_s + t$ , or $\int_0^t B_s \, dB_s = \frac{1}{2}B_t^2 - \frac{1}{2}t$ .

Multidimensional Itô formula

Theorem6.2Itô's formula (multidimensional)

For $dX_t = \mu_t \, dt + \sigma_t \, dB_t$ in $\mathbb{R}^n$ and $f \in C^2(\mathbb{R}^n)$ :

$df(X_t) = \sum_i \frac{\partial f}{\partial x_i} dX_t^i + \frac{1}{2} \sum_{i,j} \frac{\partial^2 f}{\partial x_i \partial x_j} d\langle X^i, X^j \rangle_t,$

where $d\langle X^i, X^j \rangle_t = \sigma_t^i \cdot \sigma_t^j \, dt$ is the cross-variation.

Summary

Itô's formula is $df(X_t) = f'(X_t) dX_t + \frac{1}{2}f''(X_t) \sigma_t^2 dt$ , with the second-order correction essential for stochastic processes. It is the foundation for solving stochastic differential equations and pricing derivatives.