 Solution

By definition of the concave conjugate, $u(x)\le \hat{u}(y)+xy$ for all $x,y>0$. Put $x=H\cdot P_1$ and $y=Y_1$, take expectations, and use the deflator identity:The maximizing condition in the definition of $\hat{u}$ is $u^{\prime }(x)=y$, so equality holds when $u^{\prime }(H\cdot P_1)=Y_1$ almost surely. This is utility duality with martingale deflators.