I would like to know if there is a formula for the law of $$ \sup_{l \leq t \leq u} \frac{B_t}{\sqrt{t}} $$ where $B$ is a standard Brownian motion, and $0 < l < u < 1$ are constants?

The law of $\sup_{l \leq t \leq u} B_t$ itself is well-known, but I couldn't find in any textbook at hand the law of the quantity that I am interested in..

Thanks for any help in advance!

  • 1
    $\begingroup$ I think there is a problem at $0$. Brownian motion is not Holder continuous with exponent $1/2$. $\endgroup$ – Michael Aug 1 '18 at 12:32
  • $\begingroup$ So the $\sup$ should be infinite. $\endgroup$ – Michael Aug 1 '18 at 12:32
  • $\begingroup$ @Michael, you are right. I carelessly put 0 and 1, which is not really the case in my work. $\endgroup$ – Dormire Aug 1 '18 at 13:40

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