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What a fast-switching yield curve identifies

A first-order identifiability theorem, exact loading and initial-mixture rank boundaries, and an explicit finite-rate remainder.

Research note, 23 September 2026. This note develops the three-number example; the asymptotic statements concern a fixed finite maturity interval as the generator is sped up. This page is a summary; the full note has the proofs and the remaining results.

Model and convention

Let $Q_0$ be an irreducible generator on $n$ states, with stationary row vector $\pi$, and let the regime generator be $mQ_0$, $m\to\infty$. The short rate obeys $dr_t=\kappa(\theta_{y_t}-r_t)dt+\sqrt{s_{y_t}}\,dW_t$, with $\kappa>0$, $s_i\ge0$, and $y_t$ independent of $W$. Define

For a known initial rate $x$, write $P_i(T)=e^{-B(T)x}a_i(T)$ for a specified starting regime, $P_\pi(T)=e^{-B(T)x}\pi a(T)$ for a stationary start, and $P_\nu(T)$ for an initial regime distribution $\nu$. The exact finite-dimensional equation is

In particular, $g(0)=0$. Put $H=-Q_0^{\#}$. The symmetric Green–Kubo form is $\mathcal K_0^{\rm sym}(f,h) =\frac12\{\pi[f(Hh)]+\pi[h(Hf)]\}$ on centered vectors.

The coefficients below are computed with $Q_0$; their actual contribution at rate $mQ_0$ is divided by $m$.

The first-order information

For every fixed $L<\infty$, $\sup_{0\le T\le L}|E_\pi(T)|=O_L(m^{-2})$. With $\kappa$ and $x$ fixed, the four functions $I_1,\ldots,I_4$ are linearly independent on every open maturity interval. Hence an ideal first-order stationary curve uniquely determines these four coefficients.

It does not separately determine $\bar s$ and $\mathcal K_0(u,u)/m$ from $\beta_2$, nor the underlying chain. Two chains can share all four first-order coefficients and have different exact curves at order $m^{-2}$.

The $2\times2$ Green–Kubo matrix is positive semidefinite, so the observable restriction is $\beta_3^2\le4\beta_2\beta_4$. Strict inequality for the observable $\beta$'s alone cannot establish three regimes: $\bar s>0$ makes it strict even for two states.

Fixed features: rank counts regimes

Suppose the centered forcing separates as $g_i(t)-\pi g(t)=\sum_{r=1}^d\phi_r(t)f_r(i)$ with $\pi f_r=0$, and let $F=[f_1\ \cdots\ f_d]$ be the $n\times d$ feature matrix. Define the fixed-parameter Green–Kubo Gram matrix $G_{rs}=\mathcal K_0^{\rm sym}(f_r,f_s)$. Then, for every irreducible finite chain,

Thus an identified fixed-feature rank $r$ requires at least $r+1$ regimes. It does not identify the generator: many chains and feature embeddings produce the same $G$.

The same kernel appears at every finite $T>0$. For $Z_T=\int_0^T F^\top e_{Y_s}\,ds$, with $Y$ of generator $mQ_0$, and an arbitrary initial regime prior $\nu$, including a deterministic start,

An integrated rank is not a regime count

The rank statement applies to one fixed feature matrix. If $p$ contracts have a maturity-dependent loading matrix $L(t)\in\mathbb R^{p\times d}$, their integrated coefficient matrix is $D(T)=\int_0^T L(t)G L(t)^\top dt$. With $r=\operatorname{rank}G$ and any full-column factor $G=RR^\top$,

Integrating does not recover another property of the hidden chain; it measures the span swept out by the loaded fixed-feature range. The rank interpretation goes back to Kalman (1960).

The smallest counterexample uses only two regimes and one centered feature. Take scalar $G=[g]$ with $g>0$ and $L(t)=(1,t,t^2)^\top$: then $D(1)$ is $g$ times the positive-definite $3\times3$ Hilbert matrix. Hence $\operatorname{rank}D(1)=3$ even though $n-1=1$.

Known start and unobserved initial mixture

Let the initial regime have an unknown row distribution $\nu$. Uniformly on a fixed bounded maturity interval,

A specified starting state $i$ is the case of a point mass, with coefficients $(Hu)_i$ and $(Hv)_i$. The two added terms do not contribute two new independent shapes to a single curve: the six displayed functions span only $\{T,I_1,I_2,I_3,I_4\}$, a five-dimensional space.

The fixed-maturity outer expansion sees the unknown mixture only through $\nu H[u\ v]$. If the common stationary coefficients are known separately, the whole distribution $\nu$ is identified at this order exactly when $\operatorname{rank}[u\ v]=n-1$.

Scaled maturities: the Krylov criterion

For a general centered feature matrix $F\in\mathbb R^{n\times d}$ the exact mean is $\mathbb E_\nu Z_T=\frac1m\nu H(I-e^{mQ_0T})F$. For $\tau>0$ put $R(\tau)=H(I-e^{Q_0\tau})F$, the response at maturity $T=\tau/m$. The complete scaled-maturity mean curve identifies every initial distribution if and only if

This is the classical Kalman reachability condition on the centered state space, used here through its observability dual; the PBH form is in Hautus (1969). A repeated eigenspace is a genuine obstruction: varying maturity cannot create more than one direction from one vector inside that eigenspace.

By contrast, for fixed positive $T_j$ all response blocks converge to $HF$. Hence every singular value beyond $\operatorname{rank}F$ is exponentially small: exact finite-$m$ rank can survive, but the extra identification is exponentially ill-conditioned.

Which maturity tuples work

Let $r=n-1$ and let a centered scalar feature $F$ be cyclic, meaning $\operatorname{rank}[{\bf1}\ F\ Q_0F\ \cdots\ Q_0^{r-1}F]=n$. Suppose the restriction of $Q_0$ to the centered state space has only real eigenvalues, repeated eigenvalues and Jordan blocks being allowed. Then every ordered tuple $0<\tau_1<\cdots<\tau_r$ identifies the prior:

For an arbitrary real or complex spectrum, almost every ordered positive $r$-tuple identifies the initial prior. The real-spectrum theorem is stronger: it says the exceptional set is empty.

A complex spectrum marks a real boundary. For three states, a cyclic centered scalar feature has the all-pairs property if and only if the two nonzero eigenvalues are real. For a complex pair $-a\pm ib$, with $a,b>0$, the exact relation is

Hence every three-state complex-spectrum generator has an exceptional positive maturity pair.

Exact rank is not stable recovery

Suppose the nonzero eigenvalues are simple and real, $-a_1,\ldots,-a_r$ with $0<a_1<\cdots<a_r$, and the scalar feature is cyclic. Fix distinct offsets $c_1<\cdots<c_r$ and translate the entire scaled-maturity design by $M$, with $M+c_1>0$: $A_M=[{\bf1}\ R(M+c_1)\ \cdots\ R(M+c_r)]$.

Thus all $r$ distinct maturities retain exact rank for every finite $M$, while the weakest prior direction vanishes at the second-fastest decay rate.

With quote noise $Y=C_M^\top\nu+\sigma Z$, $Z\sim N(0,I_r)$, where $C_M=[R(M+c_1)\ \cdots\ R(M+c_r)]$, the weakest prior signal is $s_M=\Theta(e^{-a_{r-1}M})$ and this noise scale is sharp. Uniform consistent recovery along a translated long-end design is possible if $\sigma_M/s_M\to0$ and impossible if that ratio stays bounded away from zero along a subsequence.

Short windows are ill-conditioned polynomially instead. For continuous observation on $0\le\tau\le L$ with one scalar cyclic feature and scalar white-noise variance $\gamma>0$, the constrained Fisher eigenvalues, ordered decreasingly, obey

Here $\mathcal W_{L,\gamma}$ is the continuous observability Gramian, $U$ an orthonormal prior-tangent basis, $R_K$ the QR factor of the centered Krylov matrix and $C$ the Cholesky factor of a factorial Hilbert matrix. On long intervals, adding ever longer maturities keeps accumulating information only in the fixed-feature span.

The $T=O(m^{-1})$ boundary layer

The actual bond curve has a compatible, but one order smaller, initial layer because $g(0)=0$. Put $T=\tau/m$; then

In the important special case of constant $\theta$, $u=0$ and the order-$m^{-2}$ initial-prior signal vanishes. Switched variance then becomes the leading signal at order $m^{-3}$.

Headline numbers from the certificate

What the results do not claim

A rank lower bound on the number of regimes is valid for the fixed Green–Kubo Gram matrix after known loadings have been removed, not for a matrix obtained by integrating heterogeneous maturity loadings.

The observability, Krylov and Fisher results concern initial-distribution information, not the fixed-feature Green–Kubo rank and not the rank created by maturity-dependent contract loadings. The noise result is a local two-prior lower bound for the stated Gaussian observation model, not a claim about market microstructure or a change to noiseless identifiability.

The full note gives computable bounds on the $O(m^{-2})$ remainders. These are analytic upper bounds, not confidence intervals for noisy bond quotes; their rate threshold and constants are intentionally loose.

Scope: common fixed $\kappa$, fixed known $x$, finite $n$, and a fixed bounded maturity interval. The boundary theorem concerns $T=\tau/m$ with $\tau$ in a fixed compact interval; it is distinct from the fixed-maturity outer theorem. Stochastic reversion speed and simultaneous limits $T\to\infty$ require separate analyses.

The full note

The full note also covers the exact finite-horizon covariance formula, the block-Krylov Fisher hierarchy for several features, the defective (Jordan) and oscillatory long-end cases, correlated and deterministic quote errors, the complete three-state aliasing equation, small-maturity conditioning, determinant-optimal maturity designs, the explicit finite-rate bound, and the proofs.

Certificate: verify_identifiability.py.