Fair-game gambling induced by an incentive fee ($\mathbb{P}(Y=h)=(w_0-\ell )/(h-\ell )$)

With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts $\ell ,h$ can improve expected utility maximization through a fair lottery paying those two values. The probability of $h$ is $(w_0-\ell )/(h-\ell )$. In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.