When TWAP and when VWAP is optimal

(a) With temporary impact cost $\sum_k \frac{\eta}{\tau}n_k^2$ and no risk aversion, prove by Cauchy–Schwarz that equal slices minimise expected cost. (b) Suppose instead temporary impact depends on the participation rate, so the cost of slice $k$ is $\eta' n_k^2/V_k$, where $V_k$ is market volume in interval $k$. Show that the cost-minimising schedule is the VWAP schedule $n_k \propto V_k$, and give the minimum cost.