Mean Square Stability via T-Operator Spectral Radius

Definition

A discrete-time MJLS x(k+1) = Γ_{θ(k)} x(k) is mean square stable (MSS) if for any initial condition (x_0, θ_0) there exist μ ∈ C^n and Q ∈ B(C^n)^+ (both independent of (x_0, θ_0)) such that

E[x(k)] → μ and E[x(k) x(k)*] → Q as k → ∞.

In the homogeneous case (no input) μ = 0 and Q = 0, so MSS reduces to E[x(k) x(k)*] → 0.

The operator-theoretic characterization (Theorem 3.9 in Costa–Fragoso–Marques) establishes the equivalence of six conditions, of which four are practically checkable:

1. The MJLS is MSS.
2. r_σ(T) < 1, where T : H^n → H^n is the second-moment operator T_j(V) = Σ_i p_{ij} Γ_i V_i Γ_i* (eq. 3.7).
3. r_σ(A_1) < 1, where A_1 ∈ B(C^{Nn²}) is the augmented matrix A_1 = (P’ ⊗ I_{n²}) · diag(Γ_i ⊗ Γ_i) (eq. 3.12d).
4. The coupled Lyapunov equation V − T(V) = S has a unique positive-definite solution V > 0 for every S > 0 in H^{n+}.

(Forms 5–6 are equivalent restatements via the operators L = T*, V, J.)

The “T-operator spectral radius < 1” condition is the cleanest: one number on the second-moment operator decides everything.

Intuition

Stability of a switched linear system is fundamentally about the long-run average of the per-mode contraction/expansion, weighted by how much time the chain spends in each mode (the stationary distribution π) AND by the correlations the chain induces on consecutive matrix products. The operator T encodes both weights at once: T_j packs the row of the transition matrix p_{ij} together with the per-mode second-moment maps Q_i ↦ Γ_i Q_i Γ_i*. Its spectral radius is exactly the asymptotic geometric growth rate of E[x(k) x(k)*], so r_σ(T) < 1 is the natural drop-in replacement for the linear case |λ_max(Γ)| < 1.

The MSS criterion is famously not the same as per-mode stability:

• A system whose every Γ_i is stable can fail MSS (Example 3.17 in CFM): the Markov chain spends just enough time alternating between modes that the second moments grow.
• A system with all Γ_i unstable can be MSS (Example 3.18): the chain switches fast enough to average out the per-mode blowup.

So the right invariant is r_σ(T), not max_i r_σ(Γ_i).

Formal notation

Let H^n be the Hilbert space of N-tuples of n × n complex matrices, with inner product ⟨V; S⟩ = Σ_i tr(V_i* S_i). Define five linear operators on H^n:

E_i(V) = Σ_j p_{ij} V_j (eq. 3.6) T_j(V) = Σ_i p_{ij} Γ_i V_i Γ_i* (eq. 3.7) L_i(V) = Γ_i* (Σ_j p_{ij} V_j) Γ_i (eq. 3.8) V_j(V) = Σ_i p_{ij} Γ_j V_i Γ_j* (eq. 3.9) J_i(V) = Σ_j p_{ij} Γ_j* V_j Γ_j (eq. 3.10)

with T = L* and V = J* under the Hilbert inner product (Proposition 3.2). By Remark 3.3, r_σ(T) = r_σ(L) = r_σ(V) = r_σ(J) = r_σ(A_1).

The augmented matrix realization (eq. 3.12d): A_1 = (P’ ⊗ I_{n²}) · diag(Γ_i ⊗ Γ_i) ∈ B(C^{Nn²})

is the same operator T expressed as an Nn² × Nn² matrix; r_σ(A_1) is what one actually computes numerically.

The coupled Lyapunov equation for testing MSS (Corollary 3.21) is

V − T(V) = S, V, S ∈ H^{n+}, S > 0.

Existence of a positive-definite V is equivalent to r_σ(T) < 1 and hence to MSS.

Variants

• Stochastic stability (SS): Σ_k E[‖x(k)‖²] < ∞. For finite-state MJLS this is equivalent to MSS (Theorem 3.9 item 6). For countably infinite chains the equivalence breaks down (CFM Remark 3.12, ref [67]).
• Almost sure asymptotic stability (a.s.a.s.): x(k) → 0 with probability 1. Implied by MSS for finite-state MJLS (CFM §3.6 gives sufficient conditions via per-mode norms and limit probabilities).
• Mean square stabilizability: there exists a feedback F = (F_1, …, F_N) such that the closed-loop MJLS with Γ_i = A_i + B_i F_i is MSS, i.e. r_σ(T) < 1 with these Γ_i.
• Mean square detectability: dual to stabilizability via Proposition A.3 in Appendix A. Required for CARE existence.
• Non-homogeneous MSS: time-varying matrices and transitions; equivalent to asymptotic wide-sense-stationary stability under second-order stationary inputs (Section 3.4).

Comparison

Compared to linear stability (single matrix Γ):

• Linear case: |λ_max(Γ)| < 1 ⇔ Γ is Schur ⇔ Lyapunov V − Γ*VΓ = S has V > 0 solution. Three formulations, all equivalent.
• MJLS case: r_σ(T) < 1 ⇔ r_σ(A_1) < 1 ⇔ coupled Lyapunov V − T(V) = S has V > 0 solution. Three formulations, all equivalent — and r_σ(A_1) is the analog of |λ_max(Γ)|.

Compared to per-mode stability (max_i |λ_max(Γ_i)|):

• This is the wrong test. It is neither necessary nor sufficient for MSS. CFM’s Examples 3.17 and 3.18 show both directions fail.

When to use

• Verifying that a regime-switching state-space model is well-posed before estimating it.
• As a no-bubble / feasibility constraint in asset pricing (the “absorbing-NBC spectral proxy” used in the parent CRE project is an application of exactly this idea: r_σ of an absorbing-state augmented operator < 1 ⇒ no bubbles).
• As the existence pre-condition for the mean-square-stabilizing CARE solution in Chapter 4 / Appendix A.

Known limitations

• Computing r_σ(A_1) needs an Nn² × Nn² eigendecomposition; for large N or large n the augmented matrix gets unwieldy (though typical MJLS have small N).
• MSS is a second-moment condition; it does not control higher moments or pathwise blow-up except under additional assumptions (CFM §3.6).
• The equivalence MSS ⇔ stochastic stability does not extend to countably infinite chains.

Open problems

• Sharp convergence rates for E[x(k) x(k)*] → 0 as a function of the spectral gap 1 − r_σ(T).
• Robust MSS under uncertain transition probabilities — the LMI route is one answer (CFM §5.5) but tight conditions are open.
• Extension to MJLS over Polish-state Markov chains where the operator T is defined on an infinite-dimensional space.

Key papers

• costa-fragoso-marques-mjls-textbook — Theorem 3.9 (the equivalence), Examples 3.16–3.18 (counterexamples to per-mode reasoning), Section 3.6 (almost-sure convergence).
• understanding-markov-switching-rational-expectations-models — FWZ (2009) applies MSS to MSRE determinacy: the spectral radius of M_1(Phi_{i,j}) determines whether sunspot solutions exist.
• determinacy-classification-markov-switching-rational-expectations — Cho (2020) extends MSS characterization to the MOD solution’s two-spectral-condition classification.

My understanding

This concept is the load-bearing wall of the entire MJLS theory in CFM. Once you have the operator T and the spectral radius condition r_σ(T) < 1, you get MSS, you get the existence of stabilizing CARE solutions in Appendix A, you get the stationary filter convergence in Theorem 5.12, and you get the dual duality between control and filtering CARE. For the CRE asset-pricing project, this is the formal foundation under the project’s “absorbing NBC” spectral proxy: the project’s no-bubble check is exactly a r_σ(T) < 1 test on an absorbing-state augmented operator, and the project’s claim that “old get_lim_η is deprecated” is the empirical corollary that approximate or per-mode tests miss the MJLS subtleties.