= Solution
A <Gaussian forward-rate field> models the whole <instantaneous forward rate> curve. Let $H$ be a real separable <Hilbert space>, let $W$ be a <cylindrical Brownian motion>, and let deterministic $\sigma(t,T)\in H$ satisfy the maturity-integrability and square-integrability conditions needed below. For a deterministic initial curve, specify under $Q$
$$
df(t,T)=\alpha(t,T)dt+\langle\sigma(t,T),dW_t\rangle,\qquad 0\leq t\leq T.
$$
Using an orthonormal basis, the noise is $\sum_j\sigma_j(t,T)dW_t^j$; an infinite-dimensional cylindrical noise need not exist as an $H$-valued random vector. Its covariation kernel across maturities is the positive semidefinite kernel $C_t(T,U)=\langle\sigma(t,T),\sigma(t,U)\rangle$. Thus arbitrary suitable maturity-correlation kernels can be represented by Hilbert-space factors, while finitely many factors give finite-rank kernels.
No <arbitrage> requires a drift restriction. Define the <zero-coupon bond> price and integrated volatility by
$$
P(t,T)=\exp\!\left(-\int_t^T f(t,u)du\right),\qquad\Sigma(t,T)=\int_t^T\sigma(t,u)du,\qquad r_t=f(t,t).
$$
Differentiation of the moving lower endpoint and the <Itô formula> give
$$
\frac{dP(t,T)}{P(t,T)}=\left[r_t-\int_t^T\alpha(t,u)du+\frac12\|\Sigma(t,T)\|^2\right]dt-\langle\Sigma(t,T),dW_t\rangle.
$$
The discounted bond must have zero drift. Differentiating $\int_t^T\alpha(t,u)du=\tfrac12\|\Sigma(t,T)\|^2$ with respect to maturity yields the <Heath-Jarrow-Morton model> restriction
$$
\boxed{\alpha(t,T)=\langle\sigma(t,T),\Sigma(t,T)\rangle=\int_t^T C_t(T,u)du.}
$$
This is why forward-rate drift and covariance cannot be selected independently in an arbitrage-free random-field model. With deterministic volatilities square-integrable on the finite horizon, the discounted bond stochastic exponential is a true <martingale>, not only a <local martingale>.
The solution is explicitly Gaussian:
$$
f(t,T)=f(0,T)+\int_0^t\alpha(s,T)ds+\int_0^t\langle\sigma(s,T),dW_s\rangle,
$$
with mean given by the first two terms and covariance
$$
\boxed{\operatorname{Cov}(f(t,T),f(s,U))=\int_0^{\min(t,s)}C_v(T,U)dv.}
$$
The entire <Gaussian random field> can therefore be calibrated through its initial curve and maturity covariance structure. Bond prices are exponential functionals of these Gaussian rates; conditional bond ratios have lognormal laws under appropriate <forward measures>. Negative rates are possible. A large or infinite volatility space provides rich maturity correlations, but Gaussianity alone supplies neither a low-dimensional Markov state nor positivity of rates.
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