 Solution

For positive times the transformed process is centered and jointly Gaussian, andIts paths are continuous away from zero. To verify continuity at zero, we need $B_u/u\to 0$ almost surely as $u\to \infty$. The strong law of large numbers applied to the independent increments $B_{n+1}-B_n$ gives $B_n/n\to 0$. On the unit interval after $n$, the Doob L2 maximal inequality givesThe Borel-Cantelli first lemma, first for each positive rational $\epsilon$, shows that these between-integer fluctuations divided by $n$ tend to zero. Hence $B_u/u\to 0$ for all real $u\to \infty$, and $t{B}_{1/t}\to 0$ as $t\downarrow 0$.

The Gaussian-process characterization of Brownian motion now proves $X$ is standard Brownian motion, with respect to its natural filtration. This is time inversion of Brownian motion; the transformed process need not be adapted to the original forward filtration.