Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 43 / 5 / With transaction costs / i / Solution

Under a proportional transaction cost, the sample gain $h_x(\theta )=\theta x-\epsilon |\theta |$ is concave when $\epsilon \ge 0$. The utility function is increasing and concave, soTake finite expectations. Thus $\boxed{G\text{ is concave}}$. The monotonicity of $U$ is essential to composing it with the concave transaction-cost payoff.