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I have some diffculties with the following question; Prove that $ES_{\alpha}(L) = \frac{1}{1- \alpha}inf_{c \in {R}}\{E[(L-c)^{+}] + (1-\alpha)c\}$

Hint use

$ES_{\alpha}(L) = \frac{1}{1-\alpha}E\big[I_{\{F^{-1}(\alpha) \leq L\}} L\big]$

and assume that

$f(c) = E[(L-c)^{+}] + (1 - \alpha)c$

is a differentiable function

Floris
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  • What have you tried? See how to ask a good question and welcome to MSE! Include your work into the post to avoid downvotes and prevent closing and explain what exactly you have difficulties with so the community members can help you. – PinkyWay May 03 '20 at 12:05
  • You can get the proper font and spacing for $\inf$ using \inf. For operators that don't have a command of their own, you can use \operatorname{name}. – joriki May 03 '20 at 12:23

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