Itô integral

The Itô integral ${\int }_0^t{H}_{s}d{W}_s$ integrates a predictable square-integrable process against Brownian motion. It is defined first for predictable step processes and then completed using the Itô isometry.

For deterministic square-integrable $h$, approximate by step functions. Their Itô integrals are sums of independent normal Brownian increments. The Itô isometry gives convergence in $L^2$, and the variances converge to $\int h^2$; convergence of characteristic functions proves the normal limit. Each linear combination of finitely many such integrals is another deterministic stochastic integral, so they are jointly Gaussian.

Deterministic Itô integrals are jointly centered Gaussian by approximation from step functions and the Itô isometry. Their covariance is the L2 inner product of their integrands. Orthonormal integrands therefore produce independent standard-normal coordinates, with independence of a countable family understood through all finite subfamilies.

For a bounded continuous adapted process $H$ and a Brownian motion $B$, ${\int }_t^{t+h}{H}_{s}d{B}_s/(B_{t+h}-B_t)$ converges in probability to $H_t$ as $h\downarrow 0$. The Itô isometry makes the error numerator small relative to $\sqrt{h}$; the normal distribution of the denominator controls its chance of being too close to zero.