I'm really struggling with the dual Problem of the AV@R
$AV@R (Y)=\underset{Z}{\sup} \{ \, \mathbb{E}(YZ) \, , \, 0\le Z \le 1/(1-\alpha) \, , \, \mathbb{E}=(Z) = 1 \} $
By formulating the linear dual Problem one should get:
$AV@R (Y)=\underset{x \in \mathbb{R}}{\min} \,\{ x \, + \, 1/(1-\alpha) \, \mathbb{E} \, (\max \{0,Y-x\})\} $
According to different papers it seems tob trivial but unfortunately I couldn't calculate it. Could someone pleas explain it to me?
Thank you very much