Motivation
DSY 2007 and the broader regime-switching ATSM literature give closed-form affine bond pricing recursions under regime switching. The Leather–Sagi setup needs exponential-quadratic factors because the CRE cap-rate observation involves products of state variables. This extension is currently handled by the project’s coupled Riccati recursion, but the literature does not yet have a clean general statement of when these recursions converge in the multivariate regime-switching case (Hansen–Scheinkman gives the operator framework but not the exponential-quadratic special case).
Hypothesis
The convergence of exponential-quadratic Riccati recursions under multivariate Markov switching is governed by the spectral radius of a regime-switching second-moment operator on the augmented state — analogous to the spectral-radius test for the affine case but on a higher-dimensional sequence space. Where this spectral radius is < 1 the recursion converges geometrically; where it is ≥ 1 the no-bubble condition fails and the asset is “infinitely valued” or undefined.
Approach sketch
- Formalize the exponential-quadratic regime-switching pricing recursion in the H^n / Costa-Fragoso-Marques operator-theoretic framework.
- Derive the second-moment operator T_2 on second-moment sequences and establish
convergence iff
r(T_2) < 1. - Connect to the project’s “absorbing spectral proxy” production gate (99.1 % accuracy) and to the exact CARE-existence theorems in CFT Appendix A.
- Apply to the Leather–Sagi setup as a worked example; verify the project’s current absorbing-NBC checks are consistent with the formal gate.
Expected outcome
A theorem (or at least a clean conjecture) characterizing convergence for the multivariate exponential-quadratic regime-switching pricing recursion, plus a finite-dimensional spectral test that improves on the current absorbing proxy.
Risks
- The H^n operator framework may not extend cleanly to second-moment sequences without additional regularity assumptions.
- The exact spectral test may be more expensive than the absorbing proxy without buying enough accuracy to justify replacing it in production.
- The project’s central claim about basin geometry could change if the formal gate disagrees with the absorbing proxy in regions of the parameter space.
Pilot results
(empty)
Lessons learned
(empty)