The Itô Isometry

The Itô isometry is the fundamental identity relating the $L^2$ norm of a stochastic integral to the time integral of the squared integrand. It is the cornerstone of stochastic integration theory.

Statement

Theorem5.1Itô isometry

Let $(H_t)$ be a simple adapted process. Then:

$\mathbb{E}\left[\left(\int_0^T H_t \, dB_t\right)^2\right] = \mathbb{E}\left[\int_0^T H_t^2 \, dt\right].$

Proof for simple processes: Write $H_t = \sum_{i=0}^{n-1} H_i \mathbf{1}_{[t_i, t_{i+1})}(t)$ . Then:

$\int_0^T H_t \, dB_t = \sum_{i=0}^{n-1} H_i (B_{t_{i+1}} - B_{t_i}).$

Taking expectations using $\mathbb{E}[H_i (B_{t_{i+1}} - B_{t_i})] = 0$ and $\mathbb{E}[H_i^2 (B_{t_{i+1}} - B_{t_i})^2] = \mathbb{E}[H_i^2] (t_{i+1} - t_i)$ gives the isometry.

Extension

The isometry extends to all $L^2$ integrands by approximation: the space of simple processes is dense in $L^2([0,T] \times \Omega)$ , and the Itô integral is a continuous linear map from $L^2$ integrands to $L^2$ random variables.

RemarkComparison with Riemann integration

For deterministic $f$ , $\int_0^T f(t) \, dB_t$ is a Gaussian random variable with mean 0 and variance $\int_0^T f(t)^2 \, dt$ . The isometry says the variance equals the $L^2$ norm of $f$ .

Summary

The Itô isometry $\mathbb{E}[(\int H \, dB)^2] = \mathbb{E}[\int H^2 \, dt]$ allows us to extend the Itô integral from simple to general $L^2$ processes via completion.