Overview
A Markov jump linear system (MJLS) is a discrete-time linear system whose dynamics
matrices switch according to a finite-state Markov chain — i.e. x_{t+1} = A(θ_t) x_t + B(θ_t) u_t + w_t, with θ_t Markov. The MJLS framework provides the right theoretical
home for stability, optimal control, filtering, and H∞ analysis of regime-switching
linear-Gaussian models. Key technical objects are coupled Riccati difference / algebraic
equations (one Riccati per mode, coupled through the transition probabilities) and the
mean-square-stability (MSS) characterization via the spectral radius of an operator on
sequences of matrices. The canonical reference is the Costa–Fragoso–Todorov textbook;
risk-sensitive variants and impossibility results are due to Moon & Başar.
For the CRE asset-pricing project, MJLS theory is the mathematical backbone for: (a) checking determinacy / stability of the regime-switching macro RE solution, (b) understanding why the asset-pricing Riccati recursions converge (and when they don’t), and (c) providing the theoretical justification for the Rao-Blackwellized particle filter’s per-particle Kalman recursions.
Timeline
- 1990s — early MJLS results on coupled Riccati equations and MSS (Abou-Kandil et al.; Costa–Fragoso, Bertilsson; Krtolica et al.).
- 2005 — Costa, Fragoso & Marques: Discrete-Time Markov Jump Linear Systems (Springer textbook, often cited as CFT). The standard reference; consolidates stability, optimal LQR control, filtering, H∞ control, and design techniques.
- 2016 — Moon & Başar: risk-sensitive MJLS — proves that the clean risk-sensitive / H∞ duality from the jump-free LQG case does not extend to MJLS. No closed-form risk-sensitive optimal controller, no large-deviation or risk-neutral limits.
Seminal works
SOTA tracker
- MSS characterization: spectral radius of the
Toperator onH^nsequences; equivalent to existence of a coupled Lyapunov solution. Settled. - Optimal LQR control: coupled algebraic Riccati equation (CARE), maximal solution is the stabilizing one. Settled.
- Filtering with mode known vs mode unknown: mode-known case admits a coupled Kalman recursion (Markov jump filter); mode-unknown case has exponentially growing optimal filter complexity. The latter is the bottleneck that motivates particle filtering.
- H∞ control: state-feedback case has clean LMI / coupled-Riccati conditions; output-feedback is harder.
Open problems
- Risk-sensitive MJLS — no closed-form solutions; numerical approximation only (Moon & Başar 2016 negative result).
- Coupled Riccati existence under degenerate transition matrices (absorbing or near-absorbing regimes) — convergence is fragile, relevant to monetary policy regimes with very persistent states.
- Identifiability of mode-conditional parameters from observed mixed data is hard; related to the switching-state-estimation literature.
My position
The CRE project does not develop new MJLS theory but uses it as machinery. The most directly relevant chapters of the CFT textbook are:
- Ch.3 (Stability): justifies the spectral-radius determinacy check.
- Ch.4 (LQR / coupled Riccati): structural template for the asset-pricing Riccati recursion.
- Ch.5 (Filtering): the mode-unknown filtering result is exactly why the project needs a particle filter.
- Appendix A (CARE convergence): provides the convergence-of-Riccati existence theorems the asymptotic CRE pricing behavior depends on.
Research gaps
- Numerical convergence theory for coupled exponential-quadratic Riccati recursions under near-absorbing regimes is essentially absent; the project encounters this in the form of state-dependent T_bar requirements (Apartment is the slow asset).
- MJLS-aware particle filter design — the literature treats RBPF as a generic technique; there is little work on MJLS-specific variance-reduction or proposal distributions exploiting the mode-known Kalman structure.
Key people
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