Gaussian forward-rate field ( $df (t, T) = α (t, T) d t + ⟨ σ (t, T), d W_{t} ⟩$ )

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

Σ (t, T) = \int_{t}^{T} σ (t, u) d u

. Absence of arbitrage requires

α (t, T) = ⟨ σ (t, T), Σ (t, T)⟩

. Indeed, Itô formula gives bond drift

r_{t} - \int_{t}^{T} α (t, u) d u + ∥Σ (t, T) ∥^{2} /2

; equating it to the short rate and differentiating in maturity yields this restriction. The field covariance is

Cov (f (t, T), f (s, U)) = \int_{0}^{m i n (t, s)} ⟨ σ (v, T), σ (v, U)⟩ d v .

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.

Integrated Gaussian forward-rate process ( $Y (s, T) = \int_{0}^{T} X (s \land u, u) d u$ )

Cov (X (s, u), X (r, w)) = c (s \land r, u, w)

, integrate the short-rate past and the current forward curve together using the displayed mean-square integral. Its variance is

v (s, T) = \int_{0}^{T} \int_{0}^{T} c (s \land u \land w, u, w) d u d w .

For fixed

T

, its increments are independent of the full earlier forward-rate filtration: their covariance with

X (z, w)

z \leq r \leq s

is the integral of

c (s \land u \land z, u, w) - c (r \land u \land z, u, w) = 0

c (0, u, w) = 0

, this centered process starts at zero.

μ_{s, T}

c (s \land r, T, U)

, with deterministic initial curve and suitable integrability and maturity regularity. Its discounted zero-coupon bond prices are martingales exactly when the displayed mean identity holds. Put

A (s, T) = \int_{0}^{s} μ_{u, u} d u + \int_{s}^{T} μ_{s, u} d u .

The discounted bond price is

exp [- A (s, T) - Y (s, T)]

, where

Y

is the integrated Gaussian forward-rate process. Its conditional Gaussian exponential expectation gives the equivalent identity

A (s, T) - A (0, T) = v (s, T) /2

. Differentiating this in maturity gives the displayed restriction. Conversely, integrating that restriction on the diagonal and along the current curve recovers the half-variance identity, by symmetry of the covariance over the maturity square. The Gaussian exponential martingale with deterministic variance then verifies the full conditional martingale property. This formulation does not require a time derivative of

c

For time-to-maturity coordinates

g_{t} (x) = f (t, t + x)

and time-homogeneous deterministic Heath-Jarrow-Morton model volatility,

A (x) = ⟨ σ (x), \int_{0}^{x} σ (u) d u ⟩

. The displayed evolution contains the maturity shift. Its mild solution is

g_{t} (x) = g_{0} (x + t) + \int_{0}^{t} A (x + s) d s + \int_{0}^{t} ⟨ σ (x + t - s), d W_{s} ⟩

. The whole curve is a time-homogeneous Markov process under suitable function-space well-posedness conditions; the current short rate alone need not be a sufficient state.

If the displayed covariance is finite and the mean obeys

m^{'} (x) + A (x) = 0

, initialize the Musiela forward-curve equation with the corresponding Gaussian random field, independently of future noise. The resulting curve is stationary in time. Indeed shifting the initial covariance contributes the integral over