= Solution
Multiply the linear <stochastic differential equation> by $e^{-at}$. The <Itô product rule> gives
$$
Z_t=e^{at}z+b\int_0^t e^{a(t-s)}\,dW_s.
$$
The integrand is deterministic, so the <Itô integral> has a centered <normal distribution>. Its <variance>, by the <Itô isometry>, is
$$
b^2\int_0^t e^{2a(t-s)}ds
=\begin{cases}\displaystyle\frac{b^2}{2a}(e^{2at}-1),&a\ne0,\\b^2t,&a=0.\end{cases}
$$
Hence
$$
\boxed{Z_t\sim N\left(e^{at}z,\frac{b^2}{2a}(e^{2at}-1)\right)\quad(a\ne0).}
$$
At $a=0$ the correct continuous-limit formula is $N(z,b^2t)$. A zero <variance>, for example when $b=0$, denotes the deterministic distribution. For $a<0$ the <variance> is still positive because both numerator and denominator in its quotient are negative. This is the <explicit Ornstein-Uhlenbeck solution>, allowing either sign of the linear drift coefficient.
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