# Law of supremum of time-scaled Brownian motion

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!

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