Compute Var(∫_0^T W_s^2 dW_s) for standard Brownian motion, and state the exact distribution of ∫_0^T f(s) dW_s when f is a deterministic square integrable function.

Compute Var(∫_0^T W_s^2 dW_s) for standard Brownian motion, and state the exact distribution of ∫_0^T f(s) dW_s when f is a deterministic square integrable function.

Approach: The integrand is adapted and square integrable, so apply the Ito isometry and then the fourth moment of a Gaussian. For the second part argue from limits of sums of independent Gaussians.

T^3. The stochastic integral has mean zero, so its variance equals its second moment, and the Ito isometry gives Var = E[∫_0^T W_s^4 ds] = ∫_0^T E[W_s^4] ds. The fourth moment of a centred Gaussian is three times the squared variance, so E[W_s^4] = 3 s^2 and the integral is ∫_0^T 3 s^2 ds = T^3. For a deterministic integrand the Ito integral is a Gaussian integral: it is the L^2 limit of sums sum_i f(t_i)(W_{t_{i+1}} - W_{t_i}) of independent Gaussian terms, and a limit in L^2 of Gaussians is Gaussian, so ∫_0^T f(s) dW_s is N(0, ∫_0^T f(s)^2 ds). That fails once the integrand is random and correlated with the driving increments, which is why ∫ W dW is not Gaussian.

Follow-up: What is the full distribution of ∫_0^T W_s dW_s, and why is it asymmetric about zero?

Key concepts: Ito isometry, fourth moment, Gaussian integral, deterministic integrand.