Ito Isometry

Statement

For a square-integrable adapted process $(Y_t)$,

$$ \mathbb E\left[\left(\int_0^T Y_t,dB_t\right)^2\right]

\int_0^T \mathbb E[Y_t^2],dt. $$

Meaning

The second moment of the Ito integral is exactly the time integral of the second moment of the integrand.

Why it works

For simple processes, expanding the square produces diagonal terms and cross terms.

• Cross terms vanish because different Brownian increments are independent and centered.
• Diagonal terms contribute

$$ \mathbb E[(Y^{(k)})^2]\,(t_{k+1}-t_k). $$

Taking limits yields the full formula.

Source Links

• STA447
• Phase 7 Atomic Reading Order
• Previous atom: Ito Integral Is a Martingale
• Next atom: Linearity of the Ito Integral