Consider the short rate process \(r_{t}=\sigma B_{t}\), where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion.

a) Find the probability distribution of the time integral \(\int_{0}^{T} r_{s} d s\).

b) Compute the price

\[\mathrm{e}^{-r T} \mathbb{E}^{*}\left[\left(\int_{0}^{T} r_{u} d u-Kight)^{+}ight]\]

of a caplet on the forward rate \(\int_{0}^{T} r_{s} d s\).