Itô Isometry Theorem

Theorem5.1Itô isometry

For any adapted process $H$ with $\mathbb{E}[\int_0^T H_t^2 \, dt] < \infty$ :

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

Moreover, $\int_0^T H_t \, dB_t$ is a martingale with mean 0 and variance $\mathbb{E}[\int_0^T H_t^2 \, dt]$ .

The isometry allows extension of the Itô integral from simple to general $L^2$ processes by completion, making stochastic integration a well-defined $L^2$ isometry.