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 the process \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)defined by

\[
S_{t}=S_{0} \exp \left(\int_{0}^{t} \sigma_{s} d B_{s}+\int_{0}^{t} u_{s} d sight), \quad t \geqslant 0
\]

where \(\left(\sigma_{t}ight)_{t \in \mathbb{R}_{+}}\)and \(\left(u_{t}ight)_{t \in \mathbb{R}_{+}}\)are \(\left(\mathcal{F}_{t}ight)_{t \in[0, T]^{-}}\)-adapted processes.

a) Compute \(d S_{t}\) using Itô calculus.

b) Show that \(S_{t}\) satisfies a stochastic differential equation to be determined.