Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) and assume that \(X\) is a \(d\)-dimensional random variable which is independent of \(\mathscr{F}_{\infty}^{B}\).

a) Show that \(\widetilde{\mathscr{F}}_{t}:=\sigma\left(X, B_{s}: s \leqslant tight)\) defines an admissible filtration for \(\left(B_{t}ight)_{t \geqslant 0}\).

b) The completion \(\overline{\mathscr{F}}_{t}^{B}\) of \(\mathscr{F}_{t}^{B}\) is the smallest \(\sigma\)-algebra which contains \(\mathscr{F}_{t}^{B}\) and all subsets of \(\mathbb{P}\) null sets. Show that \(\left(\overline{\mathscr{F}}_{t}^{B}ight)_{t \geqslant 0}\) is admissible.