Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(\sigma, \tau\) be two stopping times such that \(\mathbb{E} \tau, \mathbb{E} \sigma<\infty\).

a) Show that \(\sigma \wedge \tau\) is a stopping time.

b) Show that \(\{\sigma \leqslant \tau\} \in \mathscr{F}_{\sigma \wedge \tau}\).

c) Show that \(\mathbb{E}\left(B_{\tau} B_{\sigma}ight)=\mathbb{E}(\tau \wedge \sigma)\)

Consider \(\mathbb{E}\left(B_{\sigma} B_{\tau} \mathbb{1}_{\{\sigma \leqslant \tau\}}ight)=\mathbb{E}\left(B_{\sigma \wedge \tau} B_{\tau} \mathbb{1}_{\{\sigma \leqslant \tau\}}ight)\) and use optional stopping.

d) Deduce from \(\mathrm{c})\) that \(\mathbb{E}\left(\left|B_{\tau}-B_{\sigma}ight|^{2}ight)=\mathbb{E}|\tau-\sigma|\).