Consider the asset price process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)given by the stochastic differential equation

\[
d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}
\]

Find the stochastic integral decomposition of the random variable \(S_{T}\), i.e. find the constant \(C\left(S_{0}, r, Tight)\) and the process \(\left(\zeta_{t, T}ight)_{t \in[0, T]}\) such that

\[
\begin{equation*}
S_{T}=C\left(S_{0}, r, Tight)+\int_{0}^{T} \zeta_{t, T} d B_{t} \tag{5.24}
\end{equation*}
\]

Use the fact that the discounted price process \(\left(X_{t}ight)_{t \in[0, T]}:=\left(\mathrm{e}^{-r t} S_{t}ight)_{t \in[0, T]}\) satisfies the relation \(d X_{t}=\sigma X_{t} d B_{t}\).