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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? –  Did Sep 21 '12 at 5:11
    
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? –  Becky D Sep 21 '12 at 7:38
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You might want to show what you tried, they explicitely recommend to do so, don't they? –  Did Sep 21 '12 at 8:55
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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.) –  Did Sep 21 '12 at 9:22
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Why do you not want to use Holder's inequality? –  Nate Eldredge Mar 27 '13 at 2:24
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