Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\).

a) Show that \(X_{t}:=\left|B_{t}ight|, t \geqslant 0\), is also a Markov process for the filtration \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\), i.e. for all \(s, t \geqslant 0\) and \(u \in \mathcal{B}_{b}(\mathbb{R})\)

\[\mathbb{E}\left(u\left(X_{t+s}ight) \mid \mathscr{F}_{s}ight)=\mathbb{E}\left(u\left(X_{t+s}ight) \mid X_{s}ight) .\]

b) Find in a) the function \(g_{u, s, t+s}(x)\) with \(\mathbb{E}\left(u\left(X_{t+s}ight) \mid X_{s}ight)=g_{u, s, t+s}\left(X_{s}ight)\).

c) Solve a) and b) for the process \(Y_{t}:=\sup _{s \leqslant t} B_{s}-B_{t}, t \geqslant 0\).

d) Compare the transition functions from b) and c) for \(X\) and \(Y\), respectively.