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Green–Kubo geometry of joint default

Positive CIR intensities, an identifiable instantaneous matrix, and a maturity-integrated rank trap.

Research note, 23 September 2026; extended 30 September 2026. The constant-hazard example gives the starting point; this note changes the loadings while retaining nonnegative intensities. This page is a summary; the full note has the proofs.

The model

Conditionally independent Cox defaults have intensities $X_j(t)$, $j=1,\ldots,d$. A common irreducible $n$-state chain $y_t$ with stationary law $\pi$ switches their CIR mean levels:

Assume $\kappa_j>0$, $\sigma_j>0$, $X_j(0)=x_j>0$, independent Brownian motions and independent unit-exponential default triggers. Taking $2\kappa_j\min_i\theta_{j,i}>\sigma_j^2$ keeps each intensity strictly positive over every finite time interval, even as regimes switch.

For a subset $A$ of names the joint survival price is $S_A(T)=\mathbb E\exp(-\int_0^T\sum_{j\in A}X_j(t)\,dt)$. The CIR Riccati loading $B_j$ and the regime-dependent factor obey

This is an exact finite-dimensional reduction for any initial regime prior $\nu$. The loadings $B_j$ vary with maturity and can differ across names, although $Q_0$ is shared.

The first-order matrix

Let $h_j=-\kappa_j(\theta_j-\bar\theta_j{\bf1})$, where $\bar\theta_j=\pi\theta_j$, and define the fixed Green–Kubo matrix and the loading integrals

Here $\mathcal K^{\rm sym}_0$ uses $Q_0$ and is the half-asymptotic covariance of the two centered additive functionals. On each fixed interval $0\le T\le L$, the stationary-start fast-switching expansion gives

The $O_L(m^{-2})$ error is uniform in $T$ for fixed model parameters. This is an asymptotic statement, not an exact low-rank statement at finite $m$.

An arbitrary initial regime prior

Hiddenness does not imply a stationary start. Let $\nu$ be any initial prior for the regime, put $H=-Q_0^\#$ and $f_A(t)=\sum_{j\in A}B_j(t)h_j$. Then

The term is genuine start-regime memory and vanishes for $\nu=\pi$, since $\pi H=0$. It cancels in the pairwise ratio:

Thus the first-order pairwise log-survival ratio and the identification of $K_{jk}$ are robust to an unknown common prior, even though every marginal survival curve changes at order $m^{-1}$. Its first-order sign is the sign of $K_{jk}$. Off-diagonal values can be negative while the entire $K$ matrix remains positive semidefinite.

Default order detects cycle direction

Unordered joint survival sees only the symmetric Green–Kubo matrix because hazard multiplication operators commute. The identity of the first defaulter is different: it is an absorbing-channel label and retains time orientation.

Let two names have positive regime-dependent hazards $h_1,h_2$, put $h=h_1+h_2$, and start the regime in stationarity. The exact probability that name 1 defaults first is $p_1=\pi^\top(\operatorname{diag}h-mQ_0)^{-1}h_1$. With the unscaled oriented matrix

reversing the cycle changes $p_1$ at first order:

Thus only two event types are required; at least three hidden states are needed for a finite irreducible chain to violate detailed balance.

With $d$ competing causes, one unweighted race does not generally identify the whole oriented matrix. Let $w_a>0$ scale cause $a$ and write $A=\tfrac12(K^{(0)}-K^{(0)\top})$. One unweighted race sees only $A\mathbf1$, a $(d-1)$-dimensional contrast, whereas $q=d-1$ linearly independent positive designs $w$ identify every antisymmetric $d\times d$ matrix, and this number is minimal.

Which rank bound survives?

For fixed centered feature vectors $h_1,\ldots,h_d$, the matrix $K\succeq0$ has rank at most $n-1$. The integrated matrix $D(T)=K\circ J(T)$ is also positive semidefinite, because it is a time integral of positive semidefinite matrices. Its rank need not be at most $n-1$.

Time-dependent diagonal rescalings rotate a low-dimensional range as $t$ varies. In particular, a rank test applied directly to $D(T)$ is not a lower bound on the number of regimes. This distinction is absent for constant hazards: all $B_j(t)=1$ and $D(T)=TK$.

There is nevertheless an exact replacement for the false regime-count bound. Let $r=\operatorname{rank}K$ and $s=\operatorname{rank}J(T)$, where $s$ is the dimension of the span of the loading functions $B_1,\ldots,B_d$ in $L^2[0,T]$. Then

Since $r\le n-1$, the integrated rank $R=\operatorname{rank}D(T)$ and the known loading-span dimension $s$ imply

This is the strongest possible lower bound from these ranks alone. With one common loading shape, $s=1$ and the familiar rule $n\ge R+1$ survives. With $d$ independent loading shapes, a full-rank $d\times d$ integrated matrix says only $n\ge2$.

Two regimes can create arbitrarily large integrated rank

For any number $d$ of names and any $T>0$, a two-regime model can have $\operatorname{rank}K=1$ but $\operatorname{rank}D(T)=d$. Therefore no bound depending only on the number of regimes survives after maturity integration.

The exact rank is known. Partition names into clusters $C_c$ having identical full pairs $(\kappa_i,\sigma_i)$. For arbitrary positive $\kappa_i$, nonnegative $\sigma_i$, $K\succeq0$, and every $T>0$,

In particular, if every full pair is distinct, positive diagonal entries of $K$ are necessary and sufficient for full integrated rank, whatever the fixed rank of $K$. If every loading is identical, there is one cluster and integration preserves $\operatorname{rank}K$.

Full algebraic rank does not mean uniform information across maturities, and does not make nearby loading parameters stably identifiable. For common volatility and distinct mean-reversion rates, the eigenvalues of the loading Gramian $J(T)$ near zero maturity have orders $T^3,T^5,\ldots$ and $\operatorname{cond}_2J(T)=\Theta(T^{-2(d-1)})$. For pairwise distinct full pairs $(\kappa_j,\sigma_j)$, at long maturity only one eigenvalue grows, $\lambda_1(T)=\|b\|^2T+O(1)$ with $b_j=\lim_{t\to\infty}B_j(t)$, and the rest stay bounded.

When the mean-reversion rates coalesce as $\kappa_j=\kappa_0+\epsilon x_j$ with a common $\sigma$, $\operatorname{cond}_2J_\epsilon=\Theta(\epsilon^{-2(d-1)})$. Thus two regimes still give full integrated rank for every $\epsilon>0$, while the weakest identified direction vanishes as $\epsilon^{2(d-1)}$.

Headline numbers

The two-name benchmark has a symmetric two-state generator; $K$ has rank one, yet $D(5)$ has rank two. The eight-name case uses distinct $\kappa_j$ between $0.2$ and $5$ and strict Feller mean levels; every integrated matrix is full rank.

What the curves identify, and what is not claimed

Suppose $x_j,\kappa_j,\sigma_j$ are known, every name shares the same stationary-start chain, and ideal first-order marginal and pairwise survival curves are available over a maturity interval. These curves identify $\bar\theta_j$ and the whole rate-scaled matrix $K$ at first order, although they do not identify $Q_0$, $\theta_{j,i}$, or $m$ separately.

After removing the known loadings, the correct regime-count restriction is $\operatorname{rank}K\le n-1$, or equivalently $\operatorname{rank}[\operatorname{diag}(B(t))^{-1} D'(t)\operatorname{diag}(B(t))^{-1}]\le n-1$ for $t>0$. The corresponding bound on $\operatorname{rank}D(T)$ is false.

The long-maturity spectrum is a deterministic loading-Gram theorem at fixed CIR parameters. It does not make the earlier $O_L(m^{-2})$ pricing remainder uniform as $T\to\infty$ and does not interchange the switching-speed and maturity limits. The coalescence result is a fixed-$d$, fixed-$T$ statement; it is not uniform in the number of names or maturity.

Scope: fixed CIR $\kappa_j,\sigma_j$, switching mean levels, independent Brownian motions and Cox triggers, arbitrary initial regime prior independent of the specified $x_j$, and first-order fast-switching accuracy. Switching $\kappa$ or $\sigma$ changes the Riccati reduction and requires its own identification argument.

The full note also covers the proofs, the multi-name race expansion and the conditioning of its designs, the explicit short-maturity, long-maturity and coalescence constants, partial collisions of loading shapes, and the two-name benchmark in detail. See the full note and the certificate.