Ito Isometry

The Ito isometry is the fundamental rule for computing the variance of stochastic integrals. If $f(t)$ is a (sufficiently nice) deterministic function, then: $$E\!\left[\left(\int_0^T f(t)\, dW_t\right)^{\!2}\right] = \int_0^T f(t)^2\, dt$$ More generally, for adapted processes $f(t, \omega)$: $$E\!\left[\left(\int_0^T f(t)\, dW_t\right)^{\!2}\right] = E\!\left[\int_0^T f(t)^2\, dt\right]$$ Intuition: Ordinary integrals add up signed areas, so squaring an integral involves cross terms. But Brownian increments $dW_t$ at different times are independent, so all cross terms vanish in expectation. The only surviving terms are the squared ones — just like how $\text{Var}(\sum X_i) = \sum \text{Var}(X_i)$ when the $X_i$ are independent. Concrete example: Compute $\text{Var}\!\left(\int_0^T W_t\, dW_t\right)$. Since $\int_0^T W_t\, dW_t$ is a martingale with zero mean, its variance equals: $$E\!\left[\int_0^T W_t^2\, dt\right] = \int_0^T E[W_t^2]\, dt = \int_0^T t\, dt = \frac{T^2}{2}$$ When to use: Any time you need the variance or second moment of a stochastic integral — pricing, hedging error analysis, or verifying that a process is a true martingale (finite second moment). The isometry also underpins the construction of the Ito integral itself.