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For $\kappa >0$, this error correlation decreases exponentially with elapsed time and permits irregular observation times. It is the stationary correlation of an Ornstein-Uhlenbeck process. Within a Gaussian linear mixed model, apply it to the errors conditional on random effects; the marginal correlation also includes those effects. nlme::corCAR1 parametrizes the same correlation by $\rho =e^{-\kappa }\in (0,1)$ and permits separate grouped time series.

For $\lambda >0$, set ${\mathcal{G}}_t={\mathcal{F}}_{e^{2\lambda t}}$ and $W_t=(2\lambda {)}^{-1/2}{\int }_1^{e^{2\lambda t}}{u}^{-1/2}d{B}_u$. This is a Brownian motion in the new filtration; its independent Gaussian increments have variance equal to elapsed time. The Itô product rule gives $dY=\sqrt{2\lambda }dW-\lambda Ydt$. Its initial value is $B_1$, independent of future W increments, and its covariance function is $e^{-\lambda |t-s|}$. Consequently it is a stationary Ornstein-Uhlenbeck process. The quadratic variation of the unscaled time change is $e^{2\lambda t}-1$, not $e^{2\lambda t}$.

The stationary exponential covariance function isThe corresponding Gaussian process is Markov and is a stationary Ornstein-Uhlenbeck process.

After a sudden change from quadratic kernel $K_I(q)$ to $K_F(q)$, each Fourier mode is an Ornstein-Uhlenbeck process with decay rate $r(q)=\Gamma K_F(q)$. Its equal-time covariance interpolates exponentially between $k_{B}T/K_I(q)$ and $k_{B}T/K_F(q)$.

For Ornstein-Uhlenbeck process volatility, matching powers of $\sigma$ reduces the transform PDE to a scalar Riccati equation for $R$, a linear equation for $Q$ and an integral for $P$. All coefficients start at zero. The representation exists up to the Riccati moment-explosion horizon; it need not remain finite on every horizon for arbitrary exponential powers.

The Gaussian Hull-White model has pricing dynamics $d{r}_t=(\beta (t)-a{r}_t)dt+\eta d{W}_t^Q$. Its deterministic drift function can fit the initial yield curve exactly, while its Ornstein-Uhlenbeck process part controls stochastic variation. Like the Vasicek model, it permits negative rates.

Near the stable Adler phase equation point $\alpha =\arcsin (\omega /ϵ)$, linearized noise obeys an Ornstein-Uhlenbeck process with relaxation rate $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. Its harmonic-model raw correlation is ${\alpha }^2+(T_{\mathrm{eff}}/\kappa )e^{-\kappa |\tau |}$. Nonlinear drift also shifts the local mean at order $T_{meff}$. This describes times after relaxation but before appreciable phase slips; it is not a stationary infinite-time correlation of the unwrapped phase at fixed nonzero noise.

A linear fluctuating interface smooths height gradients by diffusion and receives additive Gaussian white noise. With periodic or Neumann boundary conditions, its nonzero Fourier modes are independent Ornstein-Uhlenbeck processes if the noise is diagonal in that basis. The unconstrained uniform height is a Brownian zero mode of a fluctuating interface.

Mean reversion is a tendency for a stochastic process to move back towards a reference level. For an Ornstein-Uhlenbeck process, the conditional expected value of a future deviation is the present deviation multiplied by $e^{-\theta h}$ after time $h$. Individual sample paths can move away from that level: mean reversion describes a conditional tendency, not monotone motion or a guaranteed return by a deadline. It also does not by itself imply a stationary process; the initial probability distribution matters.

For a unit-jump population chain with one-step probabilities $\alpha k/n$ and $\beta k^2/n^2$, its equilibrium density is $u_0=\alpha /\beta$. Speed time by $n$ and center and scale the population as $(k-n{u}_0)/\sqrt{n}$. The accelerated conditional drift is $-\alpha x-\beta x^2/\sqrt{n}$, and the second moment is $2{\alpha }^2/\beta +3\alpha x/\sqrt{n}+\beta x^2/n$. Their compact-set limits give the displayed Ornstein-Uhlenbeck process. If $2{\alpha }^2<\beta$, the invalid-probability threshold stays a fixed positive density distance above equilibrium. Before that threshold, the generator of $x^2$ is at most one because its drift contribution is nonpositive. The stopped Lyapunov function bound proves compact containment and makes the probability of hitting that threshold on a fixed accelerated horizon tend to zero. This supplies the global hypotheses of the diffusion approximation theorem.

The probability-density equation for the Ornstein-Uhlenbeck process with drift $-v$ and diffusion coefficient $\sqrt{2}$. The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.

The one-dimensional semigroup associated with the standard Ornstein-Uhlenbeck process has invariant standard Gaussian measure and generator $Lf=f^{\prime \prime }-x{f}^{\prime }$. Its Mehler formula for the Ornstein-Uhlenbeck semigroup makes positivity and invariance explicit. The normalized Probabilists' Hermite polynomials diagonalize it, with $P_t{h}_n=e^{-nt}{h}_n$. It has spectral gap one and satisfies the sharp Gaussian Poincaré inequality.

Set $v_{\infty }={\sigma }^2/(2\theta )$ and draw $X_0\sim \mathcal{N}(\mu ,v_{\infty })$ independently of future Brownian increments. The Ornstein-Uhlenbeck process is then a strictly stationary process, withEvery time has the same normal distribution, and the covariance function depends only on the time lag. For a Gaussian process these properties also make all its finite-dimensional distributions invariant under a common time shift. The stationary distribution balances the continual random forcing against the contracting drift.^[ref]

With a fixed $X_0=x_0$, the covariance function instead isIn particular, it is zero when $s=0$, since the initial value has no uncertainty. The second term records the fixed start; it is the general scaling of the zero-start Ornstein-Uhlenbeck covariance. The pure exponential covariance function displayed above assumes stationary initialization. For a fixed start, that function emerges as the late-time limit at fixed lag, while the one-time probability distributions approach $\mathcal{N}(\mu ,v_{\infty })$ by convergence in distribution.

For the stipulated $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$, stationary initialization has variance one from the outset. Its autocorrelation is $e^{-|h|}$, so the correlation time is one: a lag of one reduces the autocorrelation to $e^{-1}\approx 0.3679$. Its raw second moment $\mathbb{E}[X_s{X}_t]=4+e^{-|t-s|}$ includes the squared expected value; it must not be confused with the covariance.

The upper panel shows three simulated sequences with independent stationary initial values and exact transitions at spacing $0.05$. Joining their points only guides the eye; it does not simulate a continuous sample path between those points. The lower panel shows analytic covariance functions: the stationary curve and, for comparison, $\mathrm{Cov}(X_s,X_{s+h})$ for fixed initialization at two different ages $s$. These are simulated samples and model curves, not measured trajectories or empirical covariance estimates.

For a fixed initial value $X_0=x_0$, the Ornstein-Uhlenbeck process has the normal distributionThe two parameters of $\mathcal{N}(m,v)$ are the expected value and variance, not the standard deviation. At $t=0$ the variance is zero and the value is fixed. More generally, its transition kernel over any interval of length $h>0$ has conditional expected value $\mu +(x-\mu )e^{-\theta h}$ and conditional variance ${\sigma }^2(1-e^{-2\theta h})/(2\theta )$ when the current value is $x$. The coefficients depend on elapsed time; the initial probability distribution determines whether the resulting stochastic process is stationary.^[ref]

In the stipulated example $\theta =1$, $\mu =2$, $\sigma =\sqrt{2}$, $x_0=6$, the expected value at time two is $2+4e^{-2}\approx 2.5413$ and the variance is $1-e^{-4}\approx 0.9817$. The initial distance from two has fallen by a factor $e^{-2}$, but the standard deviation has grown to about $0.9908$. Approaching the reference level in expected value is compatible with growing uncertainty from a perfectly known initial value.

The solid probability density functions use times $0.25$, $1$ and $2$; the dashed curve is the limiting normal distribution with expected value two and variance one. The curves are exact model predictions. There is no ordinary density for the fixed value at time zero.

A short derivation uses the integrating factor $e^{\theta t}$. The Itô product rule gives $d[e^{\theta t}(X_t-\mu )]=\sigma e^{\theta t}d{W}_t$, henceThe stochastic integral has zero expected value. The Itô isometry gives its variance as ${\sigma }^2{\int }_0^t{e}^{-2\theta (t-u)}du$. Its deterministic integrand makes it a Gaussian random variable, yielding the displayed normal distribution. Shifting the integration interval gives the same transition kernel from any later time.

Differentiate the material derivative of the passive scalar component by component. With ${\sigma }_{ij}={\partial }_j{u}_i$,The chain rule along a Lagrangian trajectory $\dot{X}=u(X,t)$ therefore gives the covector evolution of a material scalar gradientThe transpose matters: a gradient is transported as a covector, rather than as a material separation vector. When $u_3=0$, the third row of $\sigma$ vanishes, andLet $k_h=(k_1,k_2{)}^T$ and setFor $c>0$, the matrix exponential gives the exact horizontal solutionThe two projections are onto the eigenspaces with eigenvalues $-c$ and $+c$. If $p_{-}\ne 0$, write $p_{-}=k_0(\cos \varphi ,\sin \varphi {)}^T$ with $k_0=|p_{-}|>0$. ThenThe stated growing asymptotic requires this nonzero projection. An initial gradient entirely in the other eigenspace decays, and when $c=0$ the horizontal gradient is constant.

For constant vertical shear, write $h=({\sigma }_{13},{\sigma }_{23}{)}^T$. Integrating the third component using the exact horizontal solution yieldsConsequently, if $D=h\cdot (\cos \varphi ,\sin \varphi )$ is nonzero, $k_3\sim -k_{0}D{e}^{ct}/c$, andIf $D=0$, exponential growth of $k_3$ is absent, but the ratio still tends to zero on the growing horizontal branch. Thus exponential vertical growth is generic, not unconditional. The scalar-gradient inclination in a steady planar strain is the tangent of the angle between the scalar's normal and the horizontal plane. Equivalently, it is the cotangent of the inclination of an isoconcentration surface to that plane. Large $\alpha$ corresponds to nearly horizontal scalar surfaces; small $\alpha$ corresponds to nearly vertical ones.

For the probability calculation, take $\Gamma ,\Lambda >0$. The independent random variables $a,b$ with normal distributions of variance ${\Gamma }^2$ make $c$ distributed according to the Rayleigh distribution, withThe advertised Gaussian-over-Rayleigh inclination distribution also needs $D$ to be independent of $(a,b)$; its Gaussian marginal alone does not imply this. Under that intended assumption, for $A>0$,The cumulative distribution function is zero for $A\le 0$. Differentiating givesand the probability density function is zero for negative $A$. Its integral is one because the cumulative distribution function tends to one at infinity. Independence is a real hypothesis: $D=(\Lambda /\Gamma )a$ has exactly the stated Gaussian marginal but makes $\alpha \le \Lambda /\Gamma$ almost surely, contradicting the unbounded support of this density. A sufficient physical model is isotropic Gaussian vertical shear, independent of $a,b$, whose projection in any horizontal direction has variance ${\Lambda }^2$.

If the strain varies slowly, the horizontal gradient must align with the current growing direction, and the vertical-to-horizontal ratio must adjust before that strain changes. Both adjustment times are of order $c^{-1}$. Thus the local condition is $c{\tau }_{\sigma }\gg 1$, and the typical condition for the specified statistics isThis is an adiabatic approximation, not a uniform statement about arbitrarily weak instantaneous strains: samples with $c{\tau }_{\sigma }\lesssim 1$ retain memory of earlier strain. The instantaneous eigenvector must likewise turn slowly compared with $c$.

For rapidly fluctuating vertical shear, insert the large-time horizontal solution into its stochastic differential equation:The linear combination $W=\cos \varphi W^{(1)}+\sin \varphi W^{(2)}$ is a Brownian motion: it has independent Gaussian increments and increment variance $({\cos }^2\varphi +{\sin }^2\varphi )dt=dt$. For the signed inclination $\beta =k_3{e}^{-ct}/k_0$, the Itô product rule introduces no extra cross term because the exponential factor is deterministic. It follows thatThis is an Ornstein-Uhlenbeck process, explaining why white-noise vertical shear gives an Ornstein-Uhlenbeck inclination. Its Fokker-Planck equation isIn a stationary state the Fokker-Planck probability current $j=-cBp-(g^2/2)p^{\prime }$ is constant. A nonzero constant current cannot give an integrable, nonnegative density at both ends of the real line, so $j=0$. Integrating $p^{\prime }/p=-2cB/g^2$ and normalizing with the Gaussian integral givesThe stationary variance is $g^2/(2c)$. More explicitly the explicit Ornstein-Uhlenbeck solution has mean $\beta (0)e^{-ct}$ and variance $g^2(1-e^{-2ct})/(2c)$, confirming convergence to this density. Here $c>0$ and $g\ne 0$ are needed for the stated smooth stationary density; at $c=0$ there is no confinement, while $g=0$ gives a point mass at zero.

The correct state in a general Gaussian forward-rate field is the whole current curve. In time-to-maturity coordinates $g_t(x)=f(t,t+x)$, assume deterministic time-homogeneous volatility $\sigma (x)\in H$. Set $A(x)=⟨\sigma (x),{\int }_0^x\sigma (u)du⟩$. The Musiela forward-curve equation isUnder the appropriate maturity differentiability, or in its mild interpretation on a suitable function space, its solution isFor any future interval, the shifted current curve determines the deterministic part and future Brownian increments are independent of past information. This proves that the curve-valued process is time-homogeneous Markov. General observation-time-dependent deterministic volatility instead gives a time-inhomogeneous Markov curve. A low-dimensional state requires more: the maturity functions carrying the randomness must close under the shift evolution. Finitely many Brownian drivers by itself does not ensure that the short rate is Markov.

For a precise obstruction, take two independent stationary Ornstein-Uhlenbeck processes $Z^1,Z^2$ with distinct positive decay parameters $a_1,a_2$, and let $r_t=m+Z_t^1+Z_t^2$. Its centered covariance is $C(h)=c_1{e}^{-a_1|h|}+c_2{e}^{-a_2|h|}$ with $c_1,c_2>0$. A stationary scalar Gaussian Markov process must satisfy the covariance criterion for a stationary Gaussian Markov process:Indeed Gaussian conditioning gives $\mathbb{E}[r_{s+t}-m\mid r_s]=C(t)(r_s-m)/C(0)$; if the present is a Markov state, conditioning the future also on the earlier $r_0$ cannot change that mean. Multiplication by $r_0-m$ and taking expectations yields the equation. For the two-factor covariance the difference of its two sides isSo the short rate alone is not Markov, although the two-dimensional factor process is. Distinct exponentially decaying HJM volatility components realize such a multi-factor Gaussian curve model.

Stationarity is an additional condition, not a consequence of Gaussianity or time-homogeneity. A stationary Gaussian forward curve for the time-homogeneous model can be constructed whenis finite and the deterministic mean satisfies $m^{\prime }(x)+A(x)=0$. Initialize $g_0$ with mean $m$ and covariance $K$, independently of future noise. The mild solution has the same mean, since $m(x+t)+{\int }_0^{t}A(x+s)ds=m(x)$. Its covariance isThus its one-time Gaussian curve distribution is invariant, and the time-homogeneous transition law makes all time-shifted finite-dimensional distributions invariant as well. The initial field can explicitly be built from an independent noise on negative times using the same square-integrable kernels.

For example, ${\sigma }_j(x)={\eta }_j{e}^{-a_{j}x}$ with $a_j>0$ giveswhere $d{Z}_t^j=-a_j{Z}_t^{j}dt+{\eta }_{j}d{W}_t^j$ and the factors start in their stationary normal laws. This is a finite-dimensional Markov realization when the sum is finite. In contrast, a nonzero constant maturity volatility makes $K(x,x)$ infinite, excluding a finite-variance stationary curve of this form. A deterministic initial curve is generally nonstationary even when a stationary distribution exists. These distinctions separate Gaussianity, Markov state sufficiency, and stationarity.

The differential operator without its potential is the diffusion generator of the d-dimensional Ornstein-Uhlenbeck processIts explicit solution $X_s=e^{-\lambda s}x+{\int }_0^s{e}^{-\lambda (s-r)}d{B}_r$ defines its path law ${\mathbb{P}}^x$ on $C([0,t];{\mathbb{R}}^d)$. For the potential $V(x)={\lambda }^2|x{|}^2/2$ and initial function one, the Feynman-Kac formula gives the nonnegative path integralThe sign in the exponential is positive, because the PDE contains $+Vu$. One can first stop the diffusion in bounded balls, where the usual localized Feynman-Kac argument applies, and then examine finiteness of the resulting expectation. The critical quadratic potential for the Ornstein-Uhlenbeck generator below supplies that finiteness check.

For a discrete-time martingale relative to the filtration $({\mathcal{F}}_n)$, each $M_n$ is ${\mathcal{F}}_n$-measurable, $\mathbb{E}|M_n|<\infty$, and $\mathbb{E}[M_n\mid {\mathcal{F}}_{n-1}]=M_{n-1}$ for $n\ge 1$. The tower property then gives the corresponding identity conditional on any earlier time.

A bounded previsible process has ${\varphi }_i$ measurable at time $i-1$ and $|{\varphi }_i|\le C$. Its martingale transform $V$ is adapted and integrable, since each increment has absolute expectation at most $C(\mathbb{E}|M_i|+\mathbb{E}|M_{i-1}|)$. Moreover,Thus $V$ is a martingale, with $V_0=0$.

For the Itô integral, begin with a bounded adapted step integrand $H_s=H_i$ on $(t_i,t_{i+1}]$. Its integral is $I={\sum }_i{H}_i(W_{t_{i+1}}-W_{t_i})$. Each summand has zero conditional mean. For $i<j$, conditioning the cross product on ${\mathcal{F}}_{t_j}$ makes its expectation zero. The conditional second moment of a Brownian increment is $t_{i+1}-t_i$, henceApproximate the predictable integrand $g(s,W_s)$ in $L^2(dsd\mathbb{P})$ by such step processes. The same identity makes their integrals Cauchy in $L^2$ and defines the limit; both its first and second moments converge. This gives the requested mean-zero result and the Itô isometry:

For the Ornstein-Uhlenbeck process, the deterministic integrating factor has no quadratic covariation with $X$. The Itô formula therefore gives $d(e^{\kappa t}{X}_t)=\kappa \theta e^{\kappa t}dt+\sigma e^{\kappa t}d{W}_t$. Integrating and dividing by $e^{\kappa t}$ yieldsThe mean-zero integral and Itô isometry giveAt $\kappa =0$ these formulas are read by continuity: $X_t=x+\sigma W_t$, with mean $x$ and variance ${\sigma }^{2}t$.

For the CIR model, let $C_0=c_0\ge 0$ and use the usual positive parameters. Before a possible visit to zero, the derivatives of $\sqrt{c}$ are $1/(2\sqrt{c})$ and $-1/(4c^{3/2})$. Applying the Itô formula gives the Square root of a CIR diffusion:The PDF's claim under $2ab={\sigma }^2$ is false: that choice leaves the nonzero drift ${\sigma }^2/(8Z_t)-a{Z}_t/2$, which is not affine in $Z_t$. The actual interior cancellation condition is $4ab={\sigma }^2$. Even then the principal square root is nonnegative and cannot be an unrestricted Gaussian OU process.

To give the intended corrected construction, let a signed process solve $d{Y}_t=-a{Y}_{t}dt/2+(\sigma /2)d{B}_t$, with $Y_0=\sqrt{c_0}$. The Itô formula givesWriting ${\tilde{W}}_t={\int }_0^t\mathrm{sgn}(Y_s)d{B}_s$, with either sign chosen at zero, produces a Brownian motion; the diffusion term is $\sigma |Y_t|d{\tilde{W}}_t$. Thus $Y^2$ is a CIR model with $ab={\sigma }^2/4$, while $Z=|Y|$ is reflected at zero. The reflection contributes local time of a semimartingale, so the interior SDE alone is not a global unrestricted OU equation. Its second moment is $c_0{e}^{-at}+{\sigma }^2(1-e^{-at})/(4a)$, agreeing with the CIR mean for that corrected parameter choice.

The requested mean under the parameters actually printed is still available without the false OU assertion. Taking expectation in the original CIR equation gives $m^{\prime }=a(b-m)$; the stochastic integral has zero mean after the usual moment/localization justification. Consequently, for the printed choice as well as general admissible parameters,

An interest rate model must describe a whole term structure of interest rates, not just one future scalar rate. Let $P(t,T)$ be the time-$t$ price of a unit zero-coupon bond maturing at $T$. For a sufficiently regular maturity curve, define the instantaneous forward rate and short rate bySince $P(t,t)=1$, integration givesHere $B$ is the continuous-time bank account. Thus modelling the two-parameter field $f(t,T)$ determines both the bonds and the discounting account. A Gaussian random field means that every finite collection of its values is jointly Gaussian; merely making each individual forward rate normally distributed is insufficient.

A useful arbitrage-free construction works under a money-market risk-neutral measure $Q$. Take independent Brownian motions $W^j$, deterministic coefficient sequences ${\sigma }_j(t,T)$, and a deterministic initial forward curve $f(0,T)$. With the coefficients viewed as vectors in a separable Hilbert space $H$, writeFor infinitely many factors, the last term means the $L^2$ limit of ${\sum }_j{\sigma }_j(t,T)d{W}_t^j$, not an assumption that the infinite vector of Brownian motions is itself $H$-valued. Require ${\int }_0^t\Vert \sigma (s,T){\Vert }_H^{2}ds<\infty$ and the maturity/time regularity needed for stochastic integration and exchange of integrals. Deterministic drift and volatility make this a Gaussian forward-rate field.

The field covariance is determined byFor every finite set of maturities, $(c_t(T_i,T_j))$ must be positive semidefinite. A one-factor model has rank-one instantaneous covariance, while additional or infinitely many factors permit richer correlations along the yield curve. Equivalently one can drive the curve with a maturity-correlated Brownian field $Z_t(T)$ satisfying $d⟨Z(T),Z(U){⟩}_t=c(T,U)dt$. A Brownian sheet gives the exampleMultiplication by deterministic maturity-dependent amplitudes, or deterministic increasing changes of the two sheet coordinates, gives further Gaussian covariance structures. A covariance specification alone is not sufficient: the forward drift must also obey absence of arbitrage.

To derive that drift restriction, putDifferentiate the moving lower endpoint in $\log P(t,T)=-{\int }_t^{T}f(t,u)du$ to obtainThe Itô formula therefore givesDiscounted bond prices must be martingales under $Q$. Their drift vanishes precisely when ${\int }_t^T\alpha (t,u)du=\frac{1}{2}\Vert \Sigma (t,T){\Vert }_H^2$. Differentiating in maturity gives the Heath-Jarrow-Morton model restrictionFor scalar amplitudes $v(t,T)$ multiplying a field with covariance kernel $c(T,U)$, this becomes $\alpha (t,T)=v(t,T){\int }_t^{T}v(t,u)c(T,u)du$. The extra maturity correlation factor cannot in general be dropped.

The complete Gaussian specification is consequentlyIts mean and covariance follow immediately from deterministic integration and the Itô isometry. The initial yield curve is an input, so fitting initial bond prices requires choosing $f(0,T)$ rather than changing a few scalar parameters. With the displayed drift, the discounted bond obeysOn finite horizons, deterministic square-integrable bond volatility gives an expectation-one stochastic exponential, hence a true discounted bond price martingale. This supplies an arbitrage-free pricing model for admissible portfolios in any finite collection of these bonds. Under a physical measure, a market price of risk vector changes the drift: if $d{W}_t^Q=d{W}_t^P+{\lambda }_{t}dt$, then the physical forward drift is $\alpha (t,T)+⟨\sigma (t,T),{\lambda }_t{⟩}_H$. Historical mean changes must therefore not be inserted in place of the risk-neutral drift restriction. Gaussianity under both measures additionally requires suitable deterministic drift changes; state-dependent risk premiums need not preserve Gaussianity under the physical measure.

There is also a useful general consistency test for a proposed Gaussian field beyond this independent-increment construction. DefineIn the filtration generated by a Gaussian field, conditional distributions of such linear functionals are Gaussian. For $s\le t\le T$, the martingale requirement is therefore exactlyThis follows by taking the conditional exponential moment and equating it to $e^{-L_s^T}$. It constrains both the deterministic mean and the covariance-induced conditional mean. Matching only unconditional discounted bond expectations does not suffice. For the Brownian-field construction, the Heath-Jarrow-Morton model restriction realizes this consistency condition through local drift cancellation.

Gaussian structure makes risk-neutral pricing particularly tractable. As a finite-factor example, take the Vasicek model under $Q$:Solving this Ornstein-Uhlenbeck process gives, for $u\ge t$,Hence, with $\tau =T-t$ and $B(\tau )=(1-e^{-a\tau })/a$, the conditional Gaussian integral $I={\int }_t^T{r}_{u}du$ hasThe exponential moment identity for a normal variable now gives the zero-coupon bond price $P(t,T)={\mathbb{E}}_Q[e^{-I}\mid {\mathcal{F}}_t]=A(\tau )e^{-B(\tau )r_t}$, whereThis outlines the derivation of the familiar exponential-affine form without assuming a deterministic future discount rate. More general forward-field models need not reduce to a finite-dimensional Markov short-rate process.

For another application, a call at time $T$ on a unit maturity-$U$ bond, with $t<T<U$ and strike $K>0$, is priced under the T-forward measure, using $P(t,T)$ as numéraire. The bond-price ratio $F_s=P(s,U)/P(s,T)$ is a martingale under that measure and has deterministic volatility $-(\Sigma (s,U)-\Sigma (s,T))$. It is consequently lognormal, with integrated log-varianceIntegrating the lognormal payoff gives the Gaussian bond-option formulaAt $V=0$ use $(P(t,U)-KP(t,T){)}^{+}$. The same idea prices other suitable Gaussian-rate derivatives through forward measures and normal exponential moments.

The attractions are exact initial-curve fitting, explicit covariance control across maturities, and tractable bond and option prices. The limitations are also structural: Gaussian short rates and instantaneous forward rates generally can be negative; finite-factor specifications restrict maturity correlations; and deterministic volatilities cannot reproduce arbitrary stochastic changes in rate dispersion. An infinite-factor model allows richer covariance but typically leaves risks that cannot be spanned by finitely many traded bonds. Thus Gaussianity is a useful modelling choice, not by itself a guarantee of positivity, market completeness, or absence of arbitrage.

The strain is an Ornstein-Uhlenbeck process. Multiplying its equation by the integrating factor $e^{t/\tau }$ and using the zero initial condition givesThe second expression is a stochastic integral against Brownian motion. A deterministic linear functional of Gaussian white noise is Gaussian, and its expected value is zero. For $t\ge t^{\prime }$, the noise covariance givesFor $t^{\prime }\gg \tau$, the initial-condition term is negligible. The stationary autocorrelation is thereforeIts normalized exponential decay has correlation time $\tau$. The zero-start Ornstein-Uhlenbeck covariance also shows how stationarity is approached.