Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 5 / c / Solution

The definition of the concave conjugate gives the pointwise Fenchel–Young inequalityApply it to $(X_T,Y_T)$, take expectations, and use part b:If $U^{\prime }(X_T)=Y_T$, the first inequality is equality; if $XY$ is a true martingale, the second is equality.