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Hi, I'm looking for conditions on $G(t,x)$ such that $$ \sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)] $$ where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [0,1]}E[G(t,X)]\leq E[\sup\limits_{t\in [0,1]}G(t,X)]$). Any suggestion or reference is greatly appreciated! |
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Hi, I'm looking for conditions on $G(t,x)$ such that $$ \sup\limits_{t\in I}E[G(t,X)]=E[\sup\limits_{t\in I}G(t,X)] $[0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)] $where $ I=[0,1]$ and $X$ is a random variable (it's easy to see that $\sup\limits_{t\in I}E[G(t,X)]\leq [0,1]}E[G(t,X)]\leq E[\sup\limits_{t\in I}G(t,X)]$).[0,1]}G(t,X)]$). Any suggestion or reference is greatly appreciated! |
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Hi, I'm looking for conditions on $G(t,x)$ such that $$ \sup\limits_{t\in I}E[G(t,X)]=E[\sup\limits_{t\in I}G(t,X)] $$ where $I\subset \mathbb{R}$ I=[0,1]$ and $X$ is a random variable (it's easy to see that $\sup\limits_{t\in I}E[G(t,X)]\leq E[\sup\limits_{t\in I}G(t,X)]$). Any suggestion or reference is greatly appreciated! |
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