Title: Variance Decay Property for Filter Stability
Domain: Math

Question: How can one rigorously establish the stability of nonlinear filters in continuous-time hidden Markov models with white noise observations, going beyond classical ergodicity and observability arguments, to prove asymptotic forgetting of initial conditions in the χ²-divergence sense? Specifically, under what precise conditions does stability hold, how does variance decay serve as the central mechanism, and how can advanced tools such as minimum variance duality transformations, Poincaré-type inequalities for nonlinear filters, and stochastic observability analysis be used to derive the results?

Background: The stability of nonlinear filters is a central problem in stochastic systems and signal processing. A nonlinear filter estimates the hidden state of a system based on noisy observations. In continuous-time hidden Markov models (HMMs) with white noise observations, the filter evolves as a conditional distribution over states. The main difficulty lies in whether the filter can forget its initial condition: if started with an incorrect prior distribution, does the filter eventually converge to the one started with the correct prior as time grows? | | In the linear Gaussian case (Kalman filter), stability is well understood. The Riccati equation governing the error covariance shows that under mild observability and controllability conditions, the filter error variance decays and stability follows. | | For nonlinear filters, however, the situation is far more subtle. Two mechanisms are known to support stability: | 1. Ergodicity of the hidden Markov process, which ensures the hidden state mixes over time and thus smooths out dependence on initial conditions. | 2. Observability of the noisy measurements, which allows the filter to correct for incorrect priors because the observations are informative enough to distinguish states. | | Classical results treated these two cases separately. Ergodicity arguments alone fail when the signal is not mixing but the observations are informative, while observability arguments alone fail when the observations are weak but the hidden process mixes strongly. This created a long-standing gap: a unified and quantitative framework for stability of nonlinear filters was missing.

Definitions: 1. Nonlinear Filter | For f \in C_b(S), the nonlinear filter is the conditional expectation | \pi_t(f) := \mathbb{E}[f(X_t) \mid \mathcal{Z}_t], | where X_t is the hidden process, and \mathcal{Z}_t is the observation σ-algebra. It evolves according to the Kushner–Stratonovich equation under Novikov’s condition. | | | | 2. Filter Stability in χ²-Divergence | | Given two priors \mu, \nu with \mu \ll \nu, the filters are stable in χ²-divergence if | \mathbb{E}_\mu\big[\chi^2(\pi_T^\mu \,\|\, \pi_T^\nu)\big] \xrightarrow[T\to\infty]{} 0. | This is the central criterion, since χ²-stability implies stability in KL divergence, total variation, and L^2. | | | | 3. Backward Map (Definition by the authors) | | For likelihood ratio \gamma_T(x) = \tfrac{d\pi_T^\mu}{d\pi_T^\nu}(x), the backward map is defined as | y_0(x) := \mathbb{E}_\nu[\gamma_T(X_T)\mid X_0=x]. | This maps the random terminal likelihood ratio back to a deterministic function at initial time, and its variance controls stability. | | | | 4. Variance Decay Property (Author’s innovation) | | The variance decay property asserts | \mathrm{Var}\nu(y_0(X_0)) \leq \mathrm{Var}\nu(\gamma_T(X_T)). | This inequality shows that variance contracts along the backward map. Exponential decay of χ²-divergence is obtained when combined with the Poincaré inequality. | | | | 5. Minimum Variance Duality | | A dual formulation of the filtering problem: the nonlinear filter can be analyzed through a backward stochastic differential equation (BSDE) optimal control problem, where minimizing variance corresponds to optimality of the control system. | | | | 6. Poincaré Inequality for Nonlinear Filters (Author’s innovation) | | A new functional inequality tailored for nonlinear filters: | \mathbb{E}\rho[\Gamma f] \geq c\rho \,\mathrm{Var}\rho(f), | where \Gamma is the carré du champ operator, \rho is a distribution, and c\rho > 0 is the Poincaré constant. This inequality links filter stability directly to ergodicity, observability, and noise non-degeneracy. | | | | 7. BSDE Optimal Control Formulation | | The backward map Y_t satisfies the BSDE: | -dY_t(x) = \Big[(A Y_t)(x) + h(x)^\top(U_t + V_t(x))\Big]dt - V_t(x)^\top dZ_t, | \quad Y_T(x) = \gamma_T(x). | This connects filtering stability with stochastic control optimization. | | | | 8. Stability Metrics and Implications | • KL divergence: D(\mu\|\nu). | • Total variation: \|\mu-\nu\|_{\mathrm{TV}}. | • L^2-stability: mean-square convergence. | • Relations: χ²-stability ⇒ KL ⇒ TV ⇒ L^2. | |

Methodology: ### 1. Preliminaries | | * Unnormalized filter $\sigma^\rho_t$ vs. normalized filter $\pi^\rho_t$. | * Conditional covariance $V^\rho_t(f,g)$ as variance measure under the filter. | * Carré du Champ operator $\Gamma$ captures local variance from the generator $\mathcal{A}$. | * Itô–Wentzell formula enables dynamics of processes $Y_t(X_t)$. | | ### 2. Conditional Variance SDE | | * Target: obtain SDE for $V^\rho_t(f,Y_t)$. | * Path 1 (Optimal Control): use Hamiltonian duality, momentum process $P_t$, and Itô’s rule. | * Path 2 (Filter Dynamics): apply Kushner–Stratonovich to covariance definition. | * Both approaches yield the same SDE, confirming variance decay dynamics. | | ### 3. Likelihood Ratio Connection | | * Show BSDE solution $Y_t$ acts as likelihood ratio for the filter. | * Uniqueness of Itô representation forces optimal control $ U_t^{(\text{opt})}=0$. | * Hence $Y_t(x)=\mathbb{E}^\rho[F(X_T)\mid \mathcal{Z}_t, X_t=x]$, proving likelihood ratio property. | | ### 4. Poincaré Constant | | * Define $\beta^\rho_\tau$ as minimal energy to sustain unit variance. | * Existence follows from weak compactness (Banach–Alaoglu). | * Continuity established via norm equivalence across measures. | * Provides quantitative bound linking energy functional and conditional variance. | | ### 5. Markov Property and Inequality | | * Optimal control BSDE inherits Markov structure. | * Energy functional conditional expectation reduces to a Poincaré inequality. | * Leads to key inequality: expected future variance is exponentially bounded by present variance. | | ### 6. Variance Decay Theorem | | * Case 1: Uniformly integrable likelihood ratio → variance decays in expectation. | * Case 2: Vanishing Poincaré constants → variance tends to zero almost surely. | * Core message: filter variance always decays under these structural conditions. | | ### 7. Positivity of Poincaré Constant | | * For ergodic, observable, or detectable HMMs, assume $c_\rho=0$ leads to contradiction. | * Therefore $c_\rho>0$, guaranteeing non-trivial variance decay. | | ### 8. Filter Stability Theorem | | * With $c_\rho>0$, conditional variance $V^\nu_T(\gamma_T)\to 0$. | * This implies convergence of $\pi^\mu_T$ to $\pi^\nu_T$ in total variation norm. | | ### 9. Key Assumptions | | * Equivalent initial measures ensure likelihood ratio well-defined. | * Finite-state space simplifies Γ operator and TV distance. | * Lipschitz coefficients guarantee unique solvability and boundedness. | | ### 10. Linear Case Sanity Check | | * For Gaussian linear systems, framework reduces to classical Riccati equation. | * Poincaré constant coincides with observability criterion. | * Confirms general theory recovers Kalman filter stability. | |

Answer: We address the problem of proving stability of nonlinear filters in continuous-time hidden Markov models (HMMs) with white noise observations. Stability means asymptotic forgetting of the initial condition, measured here in the χ²-divergence sense. The proof combines minimum variance duality, a newly defined Poincaré inequality (PI) for nonlinear filters, and the variance decay property. | | Step 1. Mathematical Setup | • Hidden process: X_t is a Feller-Markov process on state space S with infinitesimal generator A. | • Observation process: | dZ_t = h(X_t)\,dt + dW_t, | where h: S \to \mathbb{R}^m is the observation function and W_t is an m-dimensional Brownian motion, independent of X_t. | • Filter definition: | For f \in C_b(S), | \pi_t(f) := \mathbb{E}[f(X_t) \mid \mathcal{Z}_t], | where \mathcal{Z}_t = \sigma\{Z_s: 0 \leq s \leq t\}. | The process \pi_t is called the nonlinear filter. | • Stability question: Suppose the filter is initialized with two different priors \mu,\nu \in \mathcal{P}(S) with \mu \ll \nu. Denote the two filters \pi_t^\mu and \pi_t^\nu. Stability means: | \mathbb{E}_\mu\big[\chi^2(\pi_T^\mu \,\|\, \pi_T^\nu)\big] \xrightarrow[T \to \infty]{} 0. | | Step 2. Likelihood Ratio and Backward Map | | Define the likelihood ratio at time T: | \gamma_T(x) := \frac{d\pi_T^\mu}{d\pi_T^\nu}(x). | This random variable is \mathcal{Z}_T-measurable. | | Introduce the backward map: | y_0(x) := \mathbb{E}_\nu[\gamma_T(X_T)\mid X_0=x]. | • y_0 is deterministic and satisfies \nu(y_0)=1. | • Crucially, it connects the terminal likelihood ratio \gamma_T back to the initial state. | | Variance bound: | \left(\mathbb{E}\mu\big[\chi^2(\pi_T^\mu \| \pi_T^\nu)\big]\right)^2 | \;\leq\; \mathrm{Var}\nu(y_0(X_0)) \cdot \chi^2(\mu \| \nu). | | Thus, to prove stability, it suffices to show | \mathrm{Var}_\nu(y_0(X_0)) \xrightarrow[T\to\infty]{} 0. | | Step 3. Variance Decay Property | | From the definition of the backward map and Jensen’s inequality, | \mathrm{Var}\nu(y_0(X_0)) \;\leq\; \mathrm{Var}\nu(\gamma_T(X_T)). | | This is the variance decay property: the variance at the initial time (backward-projected) is always smaller than or equal to the variance at the terminal time. | | This property shows that variance contracts along the backward map. To obtain exponential decay, we need a structural inequality that quantifies this contraction — the Poincaré inequality. | | Step 4. Embedding into BSDE Optimal Control | | The variance decay is formalized by embedding the backward map into a backward stochastic differential equation (BSDE): | -dY_t(x) = \Big[(A Y_t)(x) + h(x)^\top(U_t + V_t(x))\Big] dt - V_t(x)^\top dZ_t, | \quad Y_T(x) = \gamma_T(x). | | The associated cost functional is | J(U) = \mathrm{Var}\nu(Y_0(X_0)) + \mathbb{E}\nu\left[\int_0^T \ell(Y_t,V_t,U_t;X_t) dt\right], | with running cost \ell(y,v,u;x) = \Gamma y(x) + \|u+v(x)\|^2, where \Gamma is the carré du champ operator of A. | | This control problem reveals that variance decay is equivalent to optimality of the controlled BSDE system. | | Step 5. Poincaré Inequality for Nonlinear Filters | | For a Markov process with invariant distribution \bar\mu, the classical Poincaré inequality is: | \bar\mu(\Gamma f) \geq c \,\mathrm{Var}_{\bar\mu}(f), | where c>0 is the Poincaré constant. | | In the nonlinear filter setting, an analogous inequality is defined for the dual BSDE system: | \mathbb{E}\rho\Big[\int_0^\tau \pi_t(\Gamma Y_t) + \|V_t\|^2 + |V\rho(h,Y_t)|^2 \, dt \Big] \;\geq\; \beta_\rho^\tau \,\mathrm{Var}\rho(Y_0(X_0)). | This inequality ensures that variance contracts at rate \beta\rho^\tau, giving a filter-specific Poincaré constant c_\rho. | | Step 6. Exponential Decay of χ²-Divergence | | From the variance decay inequality and the PI, one derives: | \mathbb{E}\mu[\chi^2(\pi_T^\mu \| \pi_T^\nu)] \;\leq\; \frac{1}{a}\, e^{-c(T-\tau)} \,\chi^2(\mu\|\nu), | where a = \mathrm{ess\,inf}{x\in S} \gamma_0(x). | | Thus, χ²-divergence decays exponentially, proving stability. | | By known inequalities: | \chi^2 \text{-stability} \;\;\implies\;\; \text{KL-stability, total variation stability, and \(L^2\)-stability}. | | Step 7. Connection to Model Properties | | The positivity of the Poincaré constant is tied to structural properties of the HMM: | 1. Ergodicity of the hidden process: ensures variance contracts due to mixing in the state dynamics. | 2. Observability of the observation model: ensures distinct states produce distinct observation statistics. | 3. Non-degenerate noise: ensures information in the observations is not lost. | | In finite-state HMMs: | • Ergodicity (irreducibility + aperiodicity of the generator), | • Observability (observation matrix full rank), or | • Detectability (all unobservable modes are stable) | → each implies c_\rho > 0, hence stability. | | Step 8. Broader Interpretation | • The variance decay property generalizes the Kalman filter’s covariance decay to nonlinear filtering. | • The new PI unifies previously separate ergodic and observable cases. | • The framework yields not only qualitative stability but also explicit convergence rates, important for algorithm design (e.g., particle filters, reinforcement learning in partially observed systems). | | Final Conclusion | | By reformulating nonlinear filtering through minimum variance duality and BSDE control, introducing the variance decay property, and defining a Poincaré inequality for nonlinear filters, we obtain a complete framework for stability. | | Result: Under conditions of ergodicity of the hidden Markov process, observability under white-noise measurements, and non-degenerate noise distributions, the χ²-divergence between filters with different priors decays exponentially. Therefore, nonlinear filters are stable in χ²-divergence, and consequently also in KL divergence, total variation, and L^2. | | This provides rigorous conditions and proofs ensuring asymptotic forgetting of initial conditions and long-term reliability of nonlinear filtering

Checklist: 1. Defines unnormalized filter $\sigma^\rho_t$ and normalized filter $\pi^\rho_t = \sigma^\rho_t(f)/\sigma^\rho_t(1)$, introduces conditional covariance $V^\rho_t(f,g)$ and variance $V^\rho_t(f)$, and states the carré du champ operator $\Gamma$ with its role in local variance. | 2. Lists key mathematical tools (Itô–Wentzell formula, Banach–Alaoglu theorem, Dominated Convergence Theorem) and clarifies their relevance to subsequent proofs. | 3. Derives the conditional variance SDE in two ways: (i) Hamiltonian/optimal control via momentum process $P_t$ with Itô’s formula and cross-term cancellation, (ii) Kushner–Stratonovich equation specialized to $g=Y_t$ with innovation process substitution, confirming both yield the same SDE. | 4. Proves the likelihood ratio property by assuming $\pi^\rho_\tau(F)=c$, invoking measure equivalence $\tilde P^\rho\sim P^\rho$, applying uniqueness of Itô representation to show $U_t^{(\text{opt})}=0$, and deriving conditional expectation form of $Y_t(x)$, concluding $Y_t$ is a likelihood ratio with $\pi^\nu_t(Y_t)=1$. | 5. Establishes the Poincaré constant’s existence and continuity: verifies norm identity, boundedness of infimizing sequence, weak convergence (Banach–Alaoglu), convexity for minimality, compactness of $\mathcal{L}_0$ for strong convergence, and norm equivalence for upper/lower bounds leading to continuity. | 6. States the Markov BSDE with terminal condition $Y_T(x)=\gamma_T(x)$, proves consistency and independence from $\mathcal{Z}_t$, rewrites energy functional with $\mathbb{E}^{\pi^\nu_t}$, and derives inequality $\mathbb{E}^\nu[V^\nu_{t+\tau}(Y_{t+\tau}) \mid \mathcal{Z}_t] \ge e^{\tau c_{\pi^\nu_t}} V^\nu_t(Y_t)$. | 7. Validates variance decay under two scenarios: (i) uniformly integrable likelihood ratios via $\epsilon$–$K$ decomposition and DCT, (ii) vanishing Poincaré constants implying Dirac convergence of the filter and a.s. variance collapse, with both showing $\mathbb{E}^\nu[e^{-\tau C_N}V^\nu_T(\gamma_T)]\to0$. | 8. Confirms positivity of the Poincaré constant by contradiction ($c_\rho=0$ ⇒ $\beta^\rho_\tau=0$), showing impossibility under (i) ergodic processes, (ii) observable HMMs, and (iii) detectable HMMs, each yielding contradiction to $\mathrm{var}^\rho(Y_0)=1$. | 9. Proves filter stability for detectable HMMs in three steps: (i) boundedness of $\gamma_T$ ensures uniform integrability of $V^\nu_T(\gamma_T)$, (ii) variance decay with $C_N\to\infty$ plus Vitali theorem and contradiction argument shows a.s. convergence, (iii) total variation norm linked to $\mathbb{E}^\nu[|\gamma_T(X_T)-1|\mid\mathcal{Z}_T]$ bounded by $\sqrt{V^\nu_T(\gamma_T)}$, yielding $\|\pi^\mu_T-\pi^\nu_T\|_{TV}\to0$. | 10. Ensures consistency of notation ($\rho,\mu,\nu,\pi^\rho_t,\sigma^\rho_t,V^\rho_t,\Gamma,\beta^\rho_\tau$) and assumptions (equivalent initial measures, finite state space, Lipschitz coefficients), and validates generality by recovering Kalman filter stability as a special case. | |