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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 hasIts explicit solution is $r_s=\bar{r}+(r_t-\bar{r})e^{-\kappa (s-t)}+\eta {\int }_t^s{e}^{-\kappa (s-u)}d{W}_u^Q$. 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 $P(t,T)={\mathbb{E}}_Q[e^{-{\int }_t^T{r}_{s}ds}\mid {\mathcal{F}}_t]$. Write $\tau =T-t$ and $B(\tau )=(1-e^{-\kappa \tau })/\kappa$. Stochastic integration of the explicit solution givesThis Gaussian variable has variance ${\eta }^2{\int }_0^{\tau }B(v{)}^{2}dv$. Evaluating its exponential moment gives exponential-affine bond pricingFor example, ${\int }_0^{\tau }B(v{)}^{2}dv={\kappa }^{-2}[\tau -2B(\tau )+(1-e^{-2\kappa \tau })/(2\kappa )]$, 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 givesThis 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 thatThus 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 ${\eta }^2{e}^{-\kappa |t-s|}/(2\kappa )$. A fixed initial short rate does not give a stationary process.

Put $n=T-t$ and define the finite geometric seriesStarting the autoregressive process of order one at time $t$ gives $r_{t+k}={\beta }^k{r}_t+{\sum }_{j=1}^k{\beta }^{k-j}{\xi }_{t+j}$. ConsequentlyThe future innovations are independent and identically distributed random variables, independent of ${\mathcal{F}}_t$. Factorizing their exponential conditional expectation and using the cumulant-generating function $K$ yieldsThus the exponential-affine bond pricing coefficients areThey vanish at maturity, giving $P(T,T)=1$. Everywhere finiteness of $K$ ensures that each zero-coupon bond price is positive and finite, including when $\beta$ is zero, negative, or one.

A unit-face-value zero-coupon bond pays $1$ at maturity. The martingale deflator pricing identity isUse the affine process Markov property with respect to the market filtration; if that filtration contains extra predictive information, the natural Markov property alone would not suffice. Part (b), with every future coefficient equal to $1$, gives the exponential-affine form. More explicitly, the exponential-affine bond pricing recursion isIndeed, conditioning the first future step in an $(n+1)$-step horizon transforms $e^{(1+\alpha (n))X_{t+1}+\beta (n)}$ into $e^{A(1+\alpha (n))X_t+B(1+\alpha (n))+\beta (n)}$. ThereforeThis uses the true martingale deflator pricing identity; a merely local martingale deflator would not by itself justify replacing prices by conditional terminal expectations.