Let \(f \in L^{2}([0, T])\), and consider a standard Brownian motion \(\left(B_{t}ight)_{t \in[0, T]}\).

a) Compute the conditional expectation

\[ \mathbb{E}\left[\mathrm{e}^{\int_{0}^{T} f(s) d B_{s}} \mid \mathcal{F}_{t}ight], \quad 0 \leqslant t \leqslant T \]

where \(\left(\mathcal{F}_{t}ight)_{t \in[0, T]}\) denotes the filtration generated by \(\left(B_{t}ight)_{t \in[0, T]}\).

b) Using the result of Question (a), show that the process

\[ t \longmapsto \exp \left(\int_{0}^{t} f(s) d B_{s}-\frac{1}{2} \int_{0}^{t} f^{2}(s) d sight), \quad 0 \leqslant t \leqslant T \]

is an \(\left(\mathcal{F}_{t}ight)_{t \in[0, T]}\)-martingale, where \(\left(\mathcal{F}_{t}ight)_{t \in[0, T]}\) denotes the filtration generated by \(\left(B_{t}ight)_{t \in[0, T]}\).

c) By applying the result of Question

(b) to the function \(f(t):=\sigma \mathbb{1}_{[0, T]}(t)\), show that the geometric Brownian motion process \(\left(\mathrm{e}^{\sigma B_{t}-\sigma t^{2} / 2}ight)_{t \in[0, T]}\) is an \(\left(\mathcal{F}_{t}ight)_{t \in[0, T]}\)-martingale for any \(\sigma \in \mathbb{R}\).