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!