Consider a risky asset whose price \(S_{t}\) is given by

\[d S_{t}=\sigma S_{t} d B_{t}+\sigma^{2} S_{t} d t / 2\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion.

a) Compute the cumulative distribution function and the probability density function of the minimum \(\min _{t \in[0, T]} B_{t}\) over the interval \([0, T]\) ?

b) Compute the price value

\[\mathrm{e}^{-\sigma^{2} T / 2} \mathbb{E}^{*}\left[S_{T}-\min _{t \in[0, T]} S_{t}ight]\]

of a lookback call option on \(S_{T}\) with maturity \(T\).