Thread: Expectation and supremum

If [IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif[/IMG][IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif[/IMG][IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif[/IMG]B_{t} is a Brownian motion starting from x and f is a function on R which is bounded above, is this inequality true?

E[sup_{t}f(B_{t})]≤sup_{t}E[f(B_{t})]

thanks[IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image002.gif[/IMG]

I believe you wanted in-text images. For me, they appear as meaningless text. As for Brownian motion, I haven't studied it, though I know some real analysis, so maybe I could figure it out. Could you attach the pictures?