= Solution
A <Gaussian> <one-factor short-rate model> is most naturally specified under a chosen <risk-neutral measure> $Q$, not by using a physical drift unaltered in a pricing formula. The <Vasicek model> has
$$
dr_t=\kappa(\bar r-r_t)dt+\eta dW_t^Q,\qquad\kappa>0.
$$
Its explicit solution is $r_s=\bar r+(r_t-\bar r)e^{-\kappa(s-t)}+\eta\int_t^s e^{-\kappa(s-u)}dW_u^Q$. It is a <Markov process>, and its conditional mean and variance are $\bar r+(r_t-\bar r)e^{-\kappa(s-t)}$ and $\eta^2(1-e^{-2\kappa(s-t)})/(2\kappa)$. Hence all finite-dimensional distributions are Gaussian. The model permits negative rates; Gaussianity does not enforce their positivity.
The unit <zero-coupon bond> price is $P(t,T)=\mathbb E_Q[e^{-\int_t^T r_sds}\mid\mathcal F_t]$. Write $\tau=T-t$ and $B(\tau)=(1-e^{-\kappa\tau})/\kappa$. Stochastic integration of the explicit solution gives
$$
\int_t^T r_sds=\bar r\tau+(r_t-\bar r)B(\tau)+\eta\int_t^T B(T-u)dW_u^Q.
$$
This Gaussian variable has variance $\eta^2\int_0^\tau B(v)^2dv$. Evaluating its exponential moment gives <exponential-affine bond pricing>
$$
\boxed{P(t,T)=A(\tau)e^{-B(\tau)r_t},\quad\log A(\tau)=\left(\bar r-\frac{\eta^2}{2\kappa^2}\right)(B(\tau)-\tau)-\frac{\eta^2B(\tau)^2}{4\kappa}.}
$$
For example, $\int_0^\tau B(v)^2dv=\kappa^{-2}[\tau-2B(\tau)+(1-e^{-2\kappa\tau})/(2\kappa)]$, which gives the displayed expression by direct simplification. The <term structure of interest rates> is therefore analytically tractable, and bond-price changes are driven by the current short rate alone.
The <Hull-White model> replaces the drift by $\alpha(t)-\kappa r_t$ with deterministic $\alpha$. The same calculation gives
$$
P(t,T)=\exp\!\left[-B(T-t)r_t-\int_t^T B(T-u)\alpha(u)du+\frac{\eta^2}{2}\int_t^T B(T-u)^2du\right].
$$
This retains the one-factor Gaussian and Markov structure while allowing an exact initial-curve fit through $\alpha$. If $f_0(t)$ is the initial <instantaneous forward rate> and $r_0=f_0(0)$, differentiating the bond formula shows that
$$
\alpha(t)=f_0'(t)+\kappa f_0(t)+\frac{\eta^2}{2\kappa}(1-e^{-2\kappa t}).
$$
Thus calibration flexibility is separated from stochastic mean reversion. Time-dependent calibration generally destroys time stationarity even though the state remains Markov. In the time-homogeneous Vasicek case, a stationary initialization is $r_0\sim N(\bar r,\eta^2/(2\kappa))$ independent of future increments; its covariance is $\eta^2e^{-\kappa|t-s|}/(2\kappa)$. A fixed initial short rate does not give a stationary process.
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