Pathwise Zakai Equation

• The pathwise Zakai equation is a deterministic filtering method that uniquely determines the unnormalized conditional law in partially observed stochastic systems.
• It refines traditional SPDE formulations by treating observation paths as fixed, enabling robust analysis in diffusion, jump, and rough-path models.
• This approach supports pathwise uniqueness and underpins advanced numerical methods and variational inference for high-dimensional filtering problems.

The pathwise Zakai equation is a central object in stochastic filtering for partially observed stochastic dynamical systems, providing a linear stochastic (or deterministic/pathwise) evolution for the unnormalized conditional law of the hidden signal, conditioned on the observation path. Unlike the traditional (probabilistic) Zakai stochastic partial differential equation (SPDE), the pathwise formulation regards observed data as fixed (or as a realized rough path), yielding deterministic or rough-path equations that admit robust, sample-pathwise analysis. The notion of pathwise uniqueness for Zakai equations is fundamental: it ensures that for a given realization of observed noise, the unnormalized filter is uniquely determined by the observed trajectory. This concept has been systematically developed for classical diffusion models, systems with jump (Lévy) noise, McKean-Vlasov (distribution-dependent) stochastic systems, and more recently, for rough-path and variational inference settings. Below, main mathematical claims and frameworks are detailed.

1. Mathematical Formulation of the Pathwise Zakai Equation

The classical nonlinear filtering problem seeks the conditional law of a Markov signal process $X$ given observations $Y$. The Zakai equation describes the evolution of the (unnormalized) conditional measure-valued process, commonly written (after Girsanov change of measure) as a linear SPDE: $$\pi_t(F) = \pi_0(F) + \int_0^t \pi_s(\mathcal{L}_s F) \, ds + \int_0^t \pi_s\big(F\,h(s,\cdot)\big)\, dY_s + \int_{0}^{t} \int_{U} \pi_{s-}\big(F(\cdot + f(\cdot,u)) - F(\cdot)\big)\, \tilde{N}(ds,du)$$ where $F$ is a suitable test function, $\mathcal{L}_s$ is the generator of $X$, $h$ is derived from the observation coefficients, and $\tilde N$ denotes compensated Poisson random measures for models with jump noise (Qiao, 2017).

The pathwise Zakai equation refines this framework by conditioning on the realized observation path $y$. For nondegenerate diffusions with observation model $dY_t = h(t,X_t)dt + k(t)dB_t$,

$$\partial_t q_t^y(x) = -\nabla_x \cdot \big[ \beta^y(t,x) q_t^y(x) \big] + \frac{1}{2} \sum_{i,j} \partial_{x_i x_j} [a_{ij}(t,x) q_t^y(x)] + G(t,x;y) q_t^y(x),$$

where $\beta^y$, $a_{ij}$, $G$ are explicit functions of the signal and observation coefficients and $y$ (Yang, 29 Jan 2026). This is a deterministic PDE for $q_t^y(x)$, parameterized by the observation path.

In rough-path setups, the evolution of the filter is expressed via rough integrals, e.g.,

$$\mu_t^Y(\varphi) = \mu_0^Y(\varphi) + \int_0^t \mu_r^Y( \mathcal{A}_r^Y \varphi)\, dr + \int_0^t \mu_r^Y ( \Gamma_r^Y \varphi)\, dY_r,$$

where $Y$ is interpreted as a deterministic rough path and the integral is defined in the sense of rough-path theory (Bugini et al., 15 Sep 2025).

For distribution-dependent (McKean-Vlasov) systems, the Zakai equation becomes an infinite-dimensional SPDE on the product space $(x, \mu)$, with measure-derivative operators (Liu et al., 2020).

2. Existence, Uniqueness and Pathwise Uniqueness

Pathwise uniqueness for Zakai equations asserts that if two solutions, defined on the same probability space and driven by the same noise, agree at the initial time and are subjected to the same realization of the observation, then they are indistinguishable: $$\text{If } \mu^1_0 = \mu^2_0, \text{ then } \mu^1_t = \mu^2_t \quad \text{a.s. for all } t.$$ This is established for a wide range of models, including continuous-state diffusions, systems with Lévy jumps, and McKean-Vlasov equations (Qiao, 2017, Ceci et al., 2012, Liu et al., 2020, Qiao, 2019).

Typical technical conditions for uniqueness include:

• Lipschitz continuity and linear growth for signal and observation coefficients in $x$ and possibly the measure argument,
• Uniform non-degeneracy for observation noise,
• Integrability and boundedness for jump coefficients and compensators.

The principal proof methods exploit quadratic-difference (energy) estimates, martingale problems, and duality with Fokker-Planck equations or backward Kolmogorov equations (Qiao, 2017, Ceci et al., 2012). In McKean-Vlasov and space-distribution dependent settings, equivalence with uniqueness for associated filtered martingale problems provides the main route to uniqueness (Liu et al., 2020).

3. Generalizations: Jump Noise, Correlated Lévy Systems, and Distribution Dependence

For models incorporating Lévy (jump) noise, the Zakai equation includes integral terms with respect to compensated Poisson random measures. Pathwise uniqueness has been established for filtering problems with correlated Brownian and Lévy noises under suitable integrability and invertibility hypotheses for the jump structures (Qiao, 2017, Qiao, 2019). Importantly, uniqueness holds as soon as the jump coefficients and intensities admit sufficient regularity (boundedness, Lipschitz continuity), and the signal still admits strong existence and uniqueness.

In distribution-dependent (McKean-Vlasov) settings, both signal and observation coefficients may depend on the law of the signal, making the Zakai equation a (nonlinear) SPDE on the Wasserstein space. Pathwise uniqueness is deduced from the uniqueness of the associated filtered martingale problem, relying on regularity and Lipschitz-type bounds for the coefficients in both state and measure arguments (Liu et al., 2020).

Table: Model Classes and Uniqueness Results

Model Class	Zakai Equation Type	Pathwise Uniqueness Reference
Diffusions with jumps (Lévy)	Measure-valued SPDE	(Qiao, 2017, Qiao, 2019)
Correlated jump-diffusions	Measure-valued SPDE	(Ceci et al., 2012)
McKean-Vlasov (distribution-dependent)	Hilbert-space valued SPDE	(Liu et al., 2020)
Rough-path setup	Rough-path driven integral eq.	(Bugini et al., 15 Sep 2025)

4. Pathwise Formulations: Rough Paths, Deterministic PDEs, and Small-Time Approximations

Modern research extends the Zakai equation into rough-path and deterministic formulations, allowing explicit pathwise dependence on sample paths of the observation, suitable for robustness and numerical implementation.

• Rough-path Zakai equations: By representing $Y$ as a $\alpha$-Hölder rough path, measure-valued robust integral equations can be solved pathwise for each deterministic $Y$, and only at the end is randomization over $Y$ restored. With suitable dimension-independent regularity ($\operatorname{Lip}^{1,\gamma}$ for drift and observation coefficients, $\gamma>1/\alpha$), uniqueness and well-posedness hold (Bugini et al., 15 Sep 2025).
• Deterministic pathwise PDEs: For fixed observations, the Zakai equation reduces to a deterministic parabolic PDE with observation path appearing as a "parameter" or drift term. This pathwise formulation is particularly effective for small-time approximations, where the solution over short intervals can be closely approximated by a deterministic Kolmogorov equation, yielding explicit error bounds (Lanconelli et al., 2021).
• Finite-interval robust PDEs: The robust Duncan-Mortensen-Zakai equation applies a robustification transformation to yield a deterministic PDE with frozen observation paths. Regularity theory in weighted Sobolev spaces and novel quantized tensor-train solvers guarantee convergence and high-dimensional tractability (Meng et al., 23 Sep 2025).

5. Connections to Control, Variational Inference, and Learning

The pathwise Zakai PDE admits a reinterpretation as a dynamic programming (Hamilton-Jacobi-Bellman) equation for a stochastic optimal control problem, with the observation path acting as a control input: $$\partial_t V = \tfrac{1}{2} \operatorname{tr}(a \nabla_x^2 V) + H(x,t,\nabla_x V)$$ with explicit feedback policy $u^*(t,x)$ recovering the posterior path-measure as the law of the optimally controlled diffusion (Yang, 29 Jan 2026). This insight allows pathwise learning of the posterior by parameterizing feedback controls informed by observation paths. Variational inference principles (via pathwise ELBOs and Girsanov-type KL objectives) enable efficient amortized learning of state and control SDEs, with practical neural SDE solvers supplanting direct SPDE integration for high-dimensional systems (Yang, 29 Jan 2026).

6. Numerical Methods and Applications

The pathwise structure of the Zakai equation underpins advanced numerical approaches:

• Galerkin-type and quantization methods for linear SPDEs;
• Sparse tensor-train approximations exploiting polyadic structure in drift and observation coefficients for high dimensions, with provable convergence and regularity guarantees in weighted Sobolev frameworks (Meng et al., 23 Sep 2025);
• Particle approximations remain standard for normalized filters, but pathwise methods afford guaranteed accuracy and efficiency, particularly for multi-modal and non-Gaussian posteriors;
• Pathwise approximation yields $O(T)$ small time error rates for deterministic PDE surrogates of the Zakai equation (Lanconelli et al., 2021).

7. Equivalence to the Kushner-Stratonovich Equation and Martingale Problem Techniques

Pathwise uniqueness for the Zakai equation is strongly related to uniqueness for the corresponding Kushner-Stratonovich (normalized filter) equation. Via the Kallianpur-Striebel formula, normalized and unnormalized filtering equations are equivalent. The filtered martingale problem approach provides a unifying technical framework: pathwise uniqueness and strong uniqueness for the nonlinear filtering problem are equivalent under minimal integrability and continuity assumptions (Ceci et al., 2012, Liu et al., 2020). This framework enables extension to models with pure-jump signals, correlated jumps, and state-dependent intensity processes.

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In summary, the pathwise Zakai equation provides a mathematically robust and uniquely solvable formulation of nonlinear filtering for systems with a broad range of noise and dependency structures (including jumps, rough paths, and measure-dependence). Its pathwise uniqueness is a cornerstone for both rigorous theory and reliable numerical inference in high- and infinite-dimensional stochastic filtering problems (Qiao, 2017, Ceci et al., 2012, Liu et al., 2020, Bugini et al., 15 Sep 2025, Meng et al., 23 Sep 2025, Yang, 29 Jan 2026, Lanconelli et al., 2021, Qiao, 2019).