 Gaussian forward-rate field

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 $\alpha (t,T)=⟨\sigma (t,T),\Sigma (t,T)⟩$. Indeed, Itô formula gives bond drift $r_t-{\int }_t^T\alpha (t,u)du+\Vert \Sigma (t,T){\Vert }^2/2$; equating it to the short rate and differentiating in maturity yields this restriction. The field covariance isFinite-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.