Partially Observed Linear SDEs

• Partially observed linear SDEs are systems where latent states evolve according to linear stochastic differential equations while observations are noisy or incomplete.
• The framework employs Kalman–Bucy filtering, quasi-maximum likelihood, and adaptive methods to estimate hidden states and unknown parameters accurately.
• Applications span quantitative finance, signal processing, biology, and control theory, supported by numerical solvers and structured approximations for computational efficiency.

A partially observed linear stochastic differential equation (SDE) system describes the evolution of a high-dimensional latent state governed by linear SDEs, with access only to noisy or incomplete (typically discrete or low-dimensional) observations. Such systems are foundational in state-space modeling, stochastic filtering, time series analysis, and stochastic control. The canonical case is a latent Ornstein–Uhlenbeck process whose trajectory drives observed data through an additional linear SDE, while parameters of the process and observation structure are unknown and must be inferred from partial—frequently discrete in time—samples. This class encompasses numerous models in quantitative finance, signal processing, biology, and control theory, and underpins the classical Kalman–Bucy filtering framework and its modern generalizations.

1. Mathematical Formulation

A prototypical partially observed linear SDE comprises a latent process $X_t \in \mathbb{R}^{d_1}$, typically an ergodic Ornstein–Uhlenbeck process, and an observed process $Y_t \in \mathbb{R}^{d_2}$ driven linearly by $X_t$ and independent observation noise: $$\begin{align*} dX_t &= -a(\theta_2)\, X_t\, dt + b(\theta_2)\, dW^1_t, \ dY_t &= c(\theta_2)\, X_t\, dt + \sigma(\theta_1)\, dW^2_t, \end{align*}$$ where $W^1$, $W^2$ are mutually independent Wiener processes, and the parameter $\theta = (\theta_1, \theta_2)$ is unknown. The observed data typically consists of the discrete-time samples $\{Y_{t_i}\}_{i=0}^n$, while $X_t$ is never directly observed (Kurisaki, 2022).

Key features of these models include:

• Ergodicity and Stationarity: Ergodicity is ensured by uniform positivity of the spectrum of $a(\theta_2)$. The model is stationary for fixed coefficients under suitable initial conditions.
• Linearity and Gaussianity: Both the signal and the observation models are linear and driven by Brownian noise, yielding all conditional distributions as (multivariate) Gaussian.

2. Filtering and State Estimation

For known parameters, optimal estimation of the hidden state sequence from noisy observations is realized by the Kalman–Bucy filter. The filtering equations for $m(t) = \mathbb{E}[X_t | \mathcal{F}_t^Y]$ and its error variance $P(t)$ are given by

$$\begin{align*} dm(t) &= -[a + \Gamma(t)]\, m(t)\, dt + \Gamma(t) \sigma^{-2} dY_t,\ dP(t)/dt &= -2a\, P(t) - \sigma^{-2} c^2\, P(t)^2 + b^2, \end{align*}$$

with $\Gamma(t) = \sigma^{-2} c^2 P(t)$ and Riccati structure for $P(t)$ (Kutoyants, 2023).

Filter design under unknown parameters requires an adaptive approach. If a sequence of parameter estimates $\theta_t^*$ is available online, the adaptive filter recursively embeds $\theta_t^*$ in the gain computations, yielding an estimator that retains asymptotic efficiency as $t \to \infty$, in the sense of achieving the minimal limiting Bayesian risk (Kutoyants, 2023).

3. Parameter Estimation Methodologies

3.1 Quasi-Maximum Likelihood

In the canonical framework, parameter estimation proceeds by quasi-maximum likelihood estimation (QMLE), exploiting the linear-Gaussian structure of the conditional increments of the discretely observed process $Y$:

• Stage 1: Estimate the observation noise scale $\theta_1$ by maximizing the closed-form likelihood of the observed increments, treating the drift due to the latent process as unknown (Kurisaki, 2022).
• Stage 2: Estimate the signal parameters $\theta_2$ using a (one-step) Kalman predictor $\hat{m}_{i-1}$, where the filter gain is set to the stationary solution of the algebraic Riccati equation.

Both stages yield consistent and asymptotically normal estimators. In particular, for the one-dimensional case,

$$\hat{\theta}_1^n = \sqrt{\tfrac{1}{t_n} \sum (\Delta_i Y)^2}$$

is closed-form (Kurisaki, 2022).

3.2 Adaptive and Efficient Procedures

Adaptive estimation can be constructed via the method of moments for initial parameter identification, followed by recursive one-step Le Cam corrections on incoming data. This approach enables embedding the updated parameter estimates directly into the Kalman–Bucy filter gain, with joint analysis establishing asymptotic minimax efficiency for the filtering error (Kutoyants, 2023).

3.3 Bayesian and Monte Carlo Inference

Bayesian inference for such models, especially when extended to nonlinear or mechanistically specified SDE settings, can be made tractable via Gaussian process approximations and linear noise approximations (LNA). The LNA yields a piecewise-linear Gaussian state-space model for filtering and likelihood computations, enabling efficient gradient-based Markov Chain Monte Carlo samplers such as MALA (Xu et al., 2024). For arbitrary (possibly nonlinear) partially observed SDEs, sequential Monte Carlo (SMC) smoothers with unbiased transition density estimation (e.g., via Poisson estimators) achieve online posterior inference, though for linear models the transition density is exact and fully tractable (Gloaguen et al., 2017).

3.4 Structural Identifiability

Structural identifiability analysis is addressed by considering the deterministic moment equations induced by the SDE model,

$\frac{d}{dt} m_\alpha(t) = \cdots,$

and eliminating unobserved moments to derive input–output differential relations involving only observed quantities. The identifiable combinations of parameters correspond to the (non-leading) coefficients of the resulting monic highest-derivative polynomials. Properties such as ergodicity, model dimension, and initial condition significantly affect identifiability (Browning et al., 25 Mar 2025).

4. Control and Game-Theoretic Extensions

Partially observed linear SDEs are the foundation of stochastic linear-quadratic (LQ) control and dynamic games under partial information. The introduction of control in either the drift or diffusion terms, and observation equations with unbounded drift, leads to nontrivial complications:

• Separation and Innovation Principle: The classical separation theorem holds: optimal controls are computed by replacing state variables with their filters.
• Stochastic Maximum Principle: For models with control in diffusion or unbounded observation drift, direct application of Girsanov’s theorem may fail. Solutions rely on restricting control to bounded sets, applying Girsanov’s theorem conditional on observation paths, and then relaxing the restriction via limit arguments to recover near-optimal controls (Sun et al., 13 Jun 2025).
• Forward-Backward SDEs: In Stackelberg games and leader-follower models, optimal controls for both leader and followers involve filtered states and are characterized by coupled systems of forward-backward SDEs and Riccati equations (Zheng et al., 2020, Li et al., 2024).

These settings require handling non-classical filtrations, circular dependence between control and observation, and the solution of high-dimensional matrix Riccati equations for feedback synthesis.

5. Computational Implementations

Implementation is dominated by the need for stable numerical solvers for:

• The algebraic Riccati equation that determines the stationary Kalman gain (numerically implemented via matrix factorization or eigenvalue methods).
• Recursive filter and predictor updates, typically via autoregressive or Euler–Maruyama schemes.
• Optimization or sampling routines for parameter inference (e.g., gradient-based maximization, grid search, or MCMC).

Particle-based smoothers such as the PaRIS algorithm for general SDEs reduce to O(N) complexity per time-step in the linear case, utilizing the explicit form of the transition density and avoiding discretization bias (Gloaguen et al., 2017). For large-scale or high-dimensional systems, sparsity or low-rank approximations to the Riccati solution are proposed (Kurisaki, 2022). Empirical studies confirm asymptotic theory and support these routines with model-specific simulation evidence (Xu et al., 2024).

6. Limitations and Extensions

Current methodologies crucially depend on linearity and Gaussianity—Kalman-type structure—both for ease of likelihood computation and filter implementation. Discretization bias in the observational scheme is non-negligible and must be controlled ($n h^2 \to 0$ as sampling interval $h \to 0$) (Kurisaki, 2022). In very high dimensions, direct Riccati inversion becomes numerically ill-conditioned, motivating structured approximations.

Extensions of this theory include:

• Nonlinear Filtering: Embedding quasi-likelihood approaches in extended or unscented Kalman filters to allow for weakly nonlinear dynamics.
• Jumps and Lévy noise: Replacing Brownian noise with jump-driven (e.g., Lévy or compound Poisson) processes, necessitating new innovation and estimation procedures.
• Nonparametric Drift Learning: Incorporation of Gaussian process regression for drift estimation from partially observed data, leveraging expectation-maximization with bridging for hidden-state imputation (Batz et al., 2017).
• Structural Identifiability under Alternative Initializations: Exploiting non-stationary or perturbed initial conditions to increase identifiable parameter combinations (Browning et al., 25 Mar 2025).

7. Applications and Practical Significance

Partially observed linear SDEs are central in the design and analysis of stochastic filters, signal processing pipelines, adaptive control systems, and quantitative models in systems biology. Their theoretical structure underlies:

• Adaptive filter design: On-line efficient smoothing in signal extraction (Kutoyants, 2023),
• Multi-agent formation control and Stackelberg games: Decentralized optimal strategies in distributed stochastic environments (Li et al., 2024),
• Stochastic parameter identification: Robust inference for biological reaction networks with partial and noisy measurements (Xu et al., 2024),
• Nonparametric system learning: Gaussian process-based discovery of drift and diffusion structures from irregular and sparse samples (Batz et al., 2017).

The general methodology, facilitation of tractable inference and control in high-dimensional, noisy, and partially observed systems, and extension to structurally identifiable estimation, confirm the foundational significance of partially observed linear SDEs in mathematical modeling and applied stochastic analysis.

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References:

(Kurisaki, 2022, Kutoyants, 2023, Gloaguen et al., 2017, Xu et al., 2024, Zheng et al., 2020, Sun et al., 13 Jun 2025, Batz et al., 2017, Li et al., 2024, Browning et al., 25 Mar 2025)