Compute d(t W_t), use it to express ∫_0^T s dW_s in terms of T W_T and ∫_0^T W_s ds, and give the exact distribution of ∫_0^T s dW_s.

Compute d(t W_t), use it to express ∫_0^T s dW_s in terms of T W_T and ∫_0^T W_s ds, and give the exact distribution of ∫_0^T s dW_s.

Approach: Apply the Ito product rule to the product of the deterministic function t and W_t, noting that the cross variation of a smooth function with W is zero, then use the isometry for the variance.

d(t W_t) = W_t dt + t dW_t, so ∫_0^T s dW_s = T W_T - ∫_0^T W_s ds, and that integral is N(0, T^3/3). The product rule carries no second order term because t has finite variation, so the cross variation d[t, W]_t is zero. Integrating the differential from 0 to T and rearranging gives the integration by parts identity. For the law, the integrand s is a deterministic integrand, so the Ito integral is an L^2 limit of sums of independent Gaussian increments with fixed weights, which makes it a Gaussian integral with mean zero. The Ito isometry gives its variance as ∫_0^T s^2 ds = T^3/3.

Follow-up: From the same identity, what are Cov(W_T, ∫_0^T s dW_s) and the correlation between the two?

Key concepts: integration by parts, Ito isometry, Gaussian integral, deterministic integrand.