Compute Var(∫_0^T W_s ds) for standard Brownian motion, and give Cov(W_T, ∫_0^T W_s ds).

Compute Var(∫_0^T W_s ds) for standard Brownian motion, and give Cov(W_T, ∫_0^T W_s ds).

Approach: Rewrite the time integral as a stochastic integral with a deterministic kernel using integration by parts, then apply the isometry. Check the result against a direct double integral of the covariance.

T^3/3. Integration by parts on d(sW_s) gives ∫_0^T W_s ds = T W_T - ∫_0^T s dW_s = ∫_0^T (T - s) dW_s, which is the same rearrangement stochastic Fubini produces by writing W_s = ∫_0^s dW_u and swapping the order of integration. The Ito isometry then gives the variance as ∫_0^T (T - s)^2 ds = T^3/3. The direct check is ∫_0^T ∫_0^T min(u,v) du dv = 2 ∫_0^T ∫_0^v u du dv = ∫_0^T v^2 dv = T^3/3. The pair (W_T, ∫_0^T W_s ds) is jointly Gaussian, since both are limits of linear combinations of the same increments, and their covariance is E[W_T ∫_0^T W_s ds] = ∫_0^T min(s,T) ds = T^2/2, giving a correlation of sqrt(3)/2 = 0.866.

Follow-up: What is the conditional distribution of the time average given W_T, and what does it give for pricing an arithmetic Asian option by conditioning?

Key concepts: stochastic Fubini, Ito isometry, integration by parts, jointly Gaussian.