= Gaussian forward-rate field
{c}
{title2=$df(t,T)=\alpha(t,T)dt+\langle\sigma(t,T),dW_t\rangle$}
With deterministic Hilbert-space-valued volatility, a deterministic initial curve and the <Heath-Jarrow-Morton model> drift restriction, the <instantaneous forward rate> is a <Gaussian random field> indexed by observation time and maturity. Put $\Sigma(t,T)=\int_t^T\sigma(t,u)du$. Absence of <arbitrage> requires $\alpha(t,T)=\langle\sigma(t,T),\Sigma(t,T)\rangle$. Indeed, <Itô formula> gives bond drift $r_t-\int_t^T\alpha(t,u)du+\|\Sigma(t,T)\|^2/2$; equating it to the <short rate> and differentiating in maturity yields this restriction. The field covariance is
$$
\operatorname{Cov}(f(t,T),f(s,U))=\int_0^{\min(t,s)}\langle\sigma(v,T),\sigma(v,U)\rangle dv.
$$
Finite-dimensional volatility spaces give finite-factor models; an infinite-dimensional space permits much richer maturity correlations. Gaussian rates can be negative. Deterministic volatility gives explicit <zero-coupon bond> and bond-option formulas.
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