Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and set \(M_{t}:=\sup _{s \leqslant t} B_{s}\) and \(I_{t}=\int_{0}^{t} B_{s} d s\).

a) Show that the two-dimensional process \(\left(B_{t}, M_{t}ight)_{t \geqslant 0}\) is a Markov process for \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\).

b) Show that the two-dimensional process \(\left(B_{t}, I_{t}ight)_{t \geqslant 0}\) is a Markov process for \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\).

c) Are \(\left(M_{t}ight)_{t \geqslant 0}\) and \(\left(I_{t}ight)_{t \geqslant 0}\) Markov processes for \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\) ?