 Utility duality with martingale deflators

The concave conjugate $\hat{u}(y)={\sup }_{x>0}\{u(x)-xy\}$ gives the weak-duality bound $\mathbb{E}u(X)\le \mathbb{E}\hat{u}(Y)+X_0{Y}_0$. Equality holds when $u^{\prime }(X)=Y$ and the payoff exhausts the deflated budget.