= Solution
{c}
Apply <Itô's formula> to $f(u,s)=e^{-\alpha u}s$. Its second stock derivative is zero, and the two drift terms cancel, giving
$$
d(e^{-\alpha u}S_u)=\sigma e^{-\alpha u}\,dW_u.
$$
Integrating and multiplying by $e^{\alpha T}$ solves the <linear stochastic differential equation>:
$$
\boxed{S_T=e^{\alpha(T-t)}S_t+\sigma\int_t^T e^{\alpha(T-u)}\,dW_u.}
$$
The <Itô integral> has a deterministic integrand, so it is a centered <Gaussian random variable> independent of the current <filtration> $\mathcal F_t$. Its <variance> follows from the <Itô isometry>. With $\tau=T-t$,
$$
\boxed{\mathbb E[S_T\mid\mathcal F_t]=e^{\alpha\tau}S_t,\qquad
\operatorname{var}(S_T\mid\mathcal F_t)=\begin{cases}
\dfrac{\sigma^2}{2\alpha}(e^{2\alpha\tau}-1),&\alpha\ne0,\\
\sigma^2\tau,&\alpha=0.
\end{cases}}
$$
The second line is also the <limit> as $\alpha\to0$. The normal law asserted here is conditional on $\mathcal F_t$; an arbitrary random initial value need not give an unconditional <normal distribution>.
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