Solution
= Solution
Apply the <Itô product rule> with deterministic <integrating factor> $e^{\alpha u}$:
$$
d(e^{\alpha u}x_u)=\alpha\widetilde\mu e^{\alpha u}du+\sigma e^{\alpha u}dZ_u.
$$
Integrate from $t$ to $T$ and multiply by $e^{-\alpha T}$ to obtain
$$
\boxed{x_T=x_t e^{-\alpha(T-t)}+\widetilde\mu(1-e^{-\alpha(T-t)})+\sigma e^{-\alpha T}\int_t^T e^{\alpha u}\,dZ_u}.
$$
This is the explicit solution behind the <Ornstein-Uhlenbeck transition distribution>.