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I have a question in my homework about Brownian motion. Does someone have a idea about the following question?

Let $X=B^+$ or $|B|$ where $B$ is a standard BM, $p>1$ be a real number and $q$ its conjugate number($1/p+1/q=1$). \

Prove that the r.v. $J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$ is a.s. strictly positive and finite and has the same law as $\sup_{t\geq 0}(X_t/(1+t^{\frac{p}{2}}))^q$

Thanks a lot!

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I deleted a solution since this is homework, but i think the idea is after using brownian scaling $\sqrt(c)X_{\frac t c}$ make a clever choice of c. Where are you getting these problems ? – mike Oct 2 '12 at 16:27
Thanks for your help. This problem is marked in the book Continuous Martingales and Brownian Motion Chapitre 1, 1.21. I have tried brownian scaling, but it works only when $t=1$. – Higgs88 Oct 3 '12 at 7:13

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