Nonlinear Stochastic Filtering with Poisson Measures

• Nonlinear stochastic filtering with Poisson random measures is defined by recursive equations that estimate hidden signals affected by both continuous and jump noise.
• The methodology employs Zakai and Kushner–Stratonovich formulations, leading to infinite-dimensional SPDEs that manage complex noise structures.
• Applications include ensemble filtering and feedback synthesis in optimal control and stochastic games, enabling real-time Bayesian estimation under jump-diffusion uncertainty.

Non-linear stochastic filtering with Poisson random measures addresses the real-time estimation of a hidden (signal) process whose evolution and partial observations are affected by both continuous (Brownian) and discontinuous (jump, specifically Poisson) stochastic perturbations. The theory, foundational to modern partially observed systems with jumps, underpins a range of feedback control and stochastic game formulations, especially those involving mean-variance objectives, risk, and regime-switching phenomena. At its core, the approach provides recursive equations—often infinite-dimensional, measure-valued stochastic partial differential equations (SPDEs)—that propagate the conditional distribution or its moments for the unobserved signal conditioned on the observed noisy and jump-driven data.

1. Mathematical Formalism and Model Classes

Fix a complete filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ supporting two independent standard Brownian motions and two independent Poisson random measures with respective compensators. The hidden state (signal) process $x(t) \in \mathbb{R}^n$ evolves according to a multidimensional Itô SDE with both diffusive and jump terms: $$\begin{aligned} dx(t) &= a(t, x(t))\,dt + \sigma_1(t, x(t))\,dW_1(t) + \sigma_2(t, x(t))\,dW_2(t) \ &\quad+ \int_{E_1} \varpi_1(t, x(t^-), e)\,\tilde N_1(de, dt) + \int_{E_2} \varpi_2(t, x(t^-), e)\,\tilde N_2(de, dt) \end{aligned}$$ The observation $Y(t)$ may be driven by a subset of these noise sources, e.g.,

$$dY(t) = H(t, x(t))\,dt + K(t)\,dW_1(t) + \int_{E_1} c(t, e)\,\tilde N_1(de, dt)$$

Assumptions include joint measurability and adaptedness of all coefficients, square-integrability to ensure well-posedness, and invertibility/boundedness of gain coefficients such as $K(t)$, $c(t,e)$ (Lin et al., 24 Jan 2026, Zheng et al., 2021).

Such frameworks encompass settings where the measurement process is an inhomogeneous Cox process with state-dependent intensity, or where observation noise and signal noise may be correlated or exhibit structured dependence (e.g., via Lévy copulas) (Fernando et al., 2015).

2. Stochastic Filtering Equations: Zakai and Kushner–Stratonovich Forms

The central objects are measure-valued processes that recursively encode the conditional law of the hidden state. Two main formulations appear:

• Zakai Equation (Unnormalized Filter):

$$\begin{aligned} d\eta_t(f) &= \eta_t(\mathcal{L} f)\,dt + \eta_{t^-}(f H^\top) K^{-1}\,dY^c(t) + \int_{E_1} \eta_{t^-}(f c(\cdot,e))\,\tilde N_1(de,dt) \end{aligned}$$

where $\mathcal{L}$ is the generator for the signal process.

• Kushner–Stratonovich Equation (Normalized Filter):

$$\begin{aligned} d\hat f(t) &= \widehat{\mathcal{L} f}\,dt + \bigl\{\widehat{f H^\top} - \hat f \, \hat H^\top\bigr\} K^{-1}\,dY^c(t) \ &\quad+ \int_{E_1} \bigl\{\widehat{f \kappa} - \hat f\, \widehat{\kappa}\bigr\}\,\tilde N_1(de,dt) \end{aligned}$$

Here, $\eta_t(f)$ is the unnormalized conditional expectation of $f(x(t))$, $\hat f(t)$ is the normalized conditional expectation, and $\kappa$ is the normalized jump-gain. These equations rigorously define the time evolution of the conditional distribution in the presence of both diffusive and jump (Poisson) observations and are substantiated under regularity and integrability of the coefficients (Lin et al., 24 Jan 2026, Zheng et al., 2021).

For vector-valued states or nonlinear test functions, these filters become infinite-dimensional SPDEs; for linear-Gaussian cases, they reduce to jump-diffusion Kalman-Bucy-type recursions.

3. Existence, Uniqueness, and Regularity Conditions

Well-posedness of the Zakai equation in the nonlinear, jump-driven context requires several analytic assumptions:

• Regularity: Drift $a$, diffusions $\sigma_i$, and jump coefficients are required to be Lipschitz or $C^{1,1}_b$; the jump measure needs appropriate Blumenthal–Getoor indices.
• Integrability: The process and observation model coefficients must satisfy square-integrability and boundedness, guaranteeing solutions in $L^\beta(\Omega; C([0,T]))$ for $\beta \geq 2$.
• Noise Structure: Invertibility and boundedness of gain coefficients (e.g., observation noise matrices) are crucial to ensure the innovation process is a well-defined Brownian motion or martingale.
• Dependence: In the presence of correlated or coupled noise—the case for Lévy copula dependence—further conditions on the scaling property and second derivatives of the copula need to be satisfied for the joint Lévy intensity and conditional laws to be well-defined. See Theorem 3.1 and corresponding remarks in (Fernando et al., 2015, Zheng et al., 2021).

Under these conditions, the Zakai equation admits a unique, measure-valued, mild solution, and the normalized Kushner–Stratonovich filter is well-defined via Itô quotient calculus (Fernando et al., 2015, Zheng et al., 2021).

4. Orthogonal Decomposition and the Separation Principle

Non-linear filtering with Poisson random measures is central to achieving observable state-feedback control laws, particularly in stochastic Stackelberg games and mean-variance optimal control. In these settings, a circular dependence arises: optimal controls depend on filtered estimates, which in turn depend on the control policy. The orthogonal decomposition method addresses this by decomposing any partially observed process into a $\mathcal{Y}_t$-measurable estimate and an orthogonal innovation: $$X^{u_1, u_2}(t) = \hat{X}^{u_1, u_2}(t) + \tilde{X}^{u_1, u_2}(t), \quad \hat{X}(t)=\mathbb{E}[X(t)|\mathcal{Y}^1_t]$$ Here, $\hat{X}$ evolves as a fully observed SDE driven by the innovation process and compensated Poisson measure (see Lemma 3.1 of (Lin et al., 24 Jan 2026)). The orthogonal component does not affect the cost and decouples, allowing the partially observed control problem to be reformulated as a sequence of fully observed (filter-driven) stochastic LQ problems. For leader–follower structures, this technique extends to FBSDEs with jump terms and yields state-feedback Stackelberg equilibria via Riccati and backward SDEs (Lin et al., 24 Jan 2026).

5. Numerical Schemes and Ensemble Filtering

For practical filtering in high dimensions or nonlinear regimes, ensemble-based recursive algorithms have been developed. Notably, the Ensemble Kushner–Stratonovich–Poisson Filter (EKSPF) approximates the time-discretized filtering equation for systems with jump (Poisson) observations (Venugopal et al., 2014). The algorithm consists of:

• Prediction: Each particle is propagated according to the process SDE.
• Update: The conditional means are adjusted via an additive, gain-based correction computed from ensemble sample covariances. For Poisson-type observations, this gain corrects each particle toward the observed jump increments, mitigating weight-collapse endemic to importance-sampling particle filters.
• Stability: Additive updates and gain regularization, when needed, ensure stability even for non-Gaussian and nonlinear settings. Monte Carlo methods based on this structure demonstrate efficacy in applications ranging from target tracking with Poissonized range–bearing data to structural health monitoring and adaptive control of nonlinear oscillators (Venugopal et al., 2014).

6. Applications in Optimal Control and Stochastic Games

Non-linear filtering over Poisson random measures enables feedback synthesis in mean–variance Stackelberg differential games with partial observation under jump-diffusion uncertainty (Lin et al., 24 Jan 2026). Both the follower’s and leader’s control problems are embedded into auxiliary fully-observed stochastic LQ or FBSDE problems with Poisson jumps. The state filters obtained via the Zakai/Kushner–Stratonovich equations directly enter the design of optimal policies and feedback gains, which in turn are computed via Riccati (and backward) equations. This technique realizes the separation principle in partially observed jump-diffusion control problems, underpinning observable state-feedback equilibria in games and optimal control (Lin et al., 24 Jan 2026, Zheng et al., 2021).

7. Generalizations and Extensions

The basic nonlinear filtering theory extends to models with:

• Correlated Jump and Diffusion Noise: Filter derivations incorporating Lévy copulas and cross-covariations between signal and observation jumps (Fernando et al., 2015).
• Nonconvex Control Domains and FBSDEs: Solutions for global maximum principles in partially observed FBSDEs with jumps (Zheng et al., 2021).
• Computation of Path-dependent Functionals: Probabilities of threshold exceedances, necessary for risk quantification, can be directly evaluated from the (unnormalized) Zakai filter by integrating against indicator functions (Fernando et al., 2015).

The explicit theoretical structure of Zakai/Kushner–Stratonovich equations, unique solution results, and practical numerical filtering algorithms collectively provide the foundation for real-time Bayesian estimation in systems with discrete-event sensors, abrupt transitions, and measurement bursts typical of modern engineering, finance, and robotics.

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Key references: (Lin et al., 24 Jan 2026, Venugopal et al., 2014, Fernando et al., 2015, Zheng et al., 2021).