We have a process $Z_t = \frac{W_t}{t}$ on $t \geq 1$. We have to find filtration for which this process is a maritngale and find the distribution of the $\sup_{t \geq 1} Z_t$
I tried to do something with Ito formula. Let's define $f(t,x) = \frac{W_t}{t}$. Then we can say (after some calculation) that:
$Z_t = - \int_0^t \frac{W_u}{u^2}du + \int_0^t \frac{1}{u} dW_u$
But I do not have any idea what I can do with this result.