If $W^0$ is a tied-down Wiener process (Brownian bridge) on the range $(0, 1)$, what is the distribution of
\begin{equation} \sup_{a \leq u \leq b} \frac{|W^0(u)|}{\sqrt{u(1-u)}} \end{equation}
i.e. how would I calculate a range of values for the above, where: $0 < a < b <1$ and I have a defined values for both $a$ and $b$. Here $\sup$ refers to supremum. An approximate solution would be OK.
I've been through the other questions on Brownian bridge on the site and have looked at some tutorials on Brownian motion in general as well as Billingsley's "Convergence of Probability Measures". However this isn't my usual field and I'm having trouble getting started. The equation comes from a paper I was reading on JSTOR.
Would be grateful for any pointers. Thanks!
Not a solution but...
This is also $\sup\limits_{r\leqslant t\leqslant s}|W(t)|/\sqrt{t}$, or the square root of $\sup\limits_{r\leqslant t\leqslant s}W(t)^2/t$, where $W$ is a standard Brownian motion, $r=a/(1-a)$ and $s=b/(1-b)$.
By scaling, the distribution depends only on $s/r=1+v$ hence it suffices to solve the case $\sup\limits_{1\leqslant t\leqslant 1+v}W(t)^2/t$ for every $v\gt0$, or equivalently, $\sup\limits_{0\leqslant t\leqslant v}(Z+W(t))^2/(1+t)$ where $Z$ is standard normal and independent of $W$.