Coupling Divergence Under Regime Switching: A Methodology for Structural Systemic Risk in Heterogeneous Subsystems

Systemic risk in heterogeneous multi-subsystem settings has been addressed by composite stress indices, spectral entropy of correlation matrices, and regime-switching copula models; none directly measures structural divergence between regime-conditional coupling matrices under an explicit hidden-regime model.

We embed whitened subsystem indicators in a two-regime Gaussian-copula hidden Markov process and define the coupling divergence as the matrix relative entropy between regime-conditional correlation matrices. We establish non-negativity, reduction to scalar Kullback-Leibler divergence between sorted eigenvalue distributions under commutativity, orthogonal invariance, and vanishing under the no-regime-switching null.

On stylized simulation, the framework separates regime-switching from single-regime null cases at an operating window T ∈ [250, 1000]; it isolates eigenbasis-rotation signals invisible to any sorted-eigenvalue method, with 99.9% of the divergence in the rotation regime residing in the non-commutative component; it tolerates Gaussian-copula misspecification under heavy-tailed processes with a quantifiable upward bias; and expectation-maximization convergence behavior serves as an auxiliary null-identification diagnostic.

The framework composes existing primitives into a regime-to-regime structural divergence and isolates a compositional mode of regime change beyond scalar methods. Results are internal-validity claims on synthetic data; external validation on real multi-subsystem data is an open question.