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How can I evaluate, or at least find an upper bound for, the following integral without the Hölder inequality, is there an alternate way anyone knows of:

$$\mathbb{E}\left[\sup\left|\int_0^t\mu X(u)du\right|^2\right]?$$

Here $dX = \mu X dt + \sigma X dB$ is the Black Scholes model.

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    What did you try?2012-09-21
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    i tried the holder inequality which simplified, not sure if it was a correct application. somebody on this forum must have some idea won't they?2012-09-21
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    You might want to show what you tried, they explicitely recommend to do so, don't they?2012-09-21
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    if someone knew how to solve it they would have solved it, nobody obviously can. I didn't post my solution so i could get someone elses perspective, which i had hoped would have been something different2012-09-21
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    Are you asserting that you have a solution? Then it is definitely recommended to include it in the question. (Note that the implication in your first sentence is wrong.)2012-09-21
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    Why do you not want to use Holder's inequality?2013-03-27

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