Consider \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)a standard Brownian motion generating the filtration \(\left(\mathcal{F}_{t}ight)_{t \in \mathbb{R}_{+}}\), and let \(\sigma>0\).

a) Compute the mean and variance of the random variable \(S_{t}\) defined as

\[
\begin{equation*}
S_{t}:=1+\sigma \int_{0}^{t} \mathrm{e}^{\sigma B_{s}-\sigma^{2} s / 2} d B_{s}, \quad t \geqslant 0 \tag{5.25}
\end{equation*}
\]

b) Express \(d \log \left(S_{t}ight)\) using (5.25) and the Itô formula.

c) Show that \(S_{t}=e^{\sigma B_{t}-\sigma^{2} t / 2}\) for \(t \geqslant 0\).