Given \(T>0\), let \(\left(X_{t}ight)_{t \in[0, T)}\) denote the solution of the stochastic differential equation

\[\begin{equation*}d X_{t}=\sigma d B_{t}-\frac{X_{t}}{T-t} d t, \quad t \in[0, T) \tag{4.42}\end{equation*}\]

under the initial condition \(X_{0}=0\) and \(\sigma>0\).

a) Show that

\[
X_{t}=(T-t) \int_{0}^{t} \frac{\sigma}{T-s} d B_{s}, \quad 0 \leqslant tHint: Start by computing \(d\left(X_{t} /(T-t)ight)\) using the Itô formula.

b) Show that \(\mathbb{E}\left[X_{t}ight]=0\) for all \(t \in[0, T)\).

c) Show that \(\operatorname{Var}\left[X_{t}ight]=\sigma^{2} t(T-t) / T\) for all \(t \in[0, T)\).

d) Show that \(\lim _{t ightarrow T} X_{t}=0\) in \(L^{2}(\Omega)\). The process \(\left(X_{t}ight)_{t \in[0, T]}\) is called a Brownian bridge.