Zakai Equation: Linear SPDE for Filtering

• Zakai equation is a linear stochastic partial differential equation that defines the unnormalized conditional density in nonlinear filtering.
• It enables rigorous analysis and numerical approximations using methods such as Galerkin projections, kernel collocation, and Monte Carlo simulations.
• Extensions addressing jumps, fractional derivatives, and nonlocal dynamics make it essential for modern stochastic filtering research.

The Zakai equation is a class of linear stochastic partial differential equations (SPDEs) that play a central role in nonlinear filtering theory. It governs the evolution of the unnormalized conditional density (or measure) of a stochastic signal process given noisy, possibly discontinuous observations. Unlike the nonlinear Kushner–Stratonovich (KS) equation, the Zakai equation is linear in the unnormalized filter and is thus more amenable to rigorous analysis and numerical approximation. The formulation and rigorous analysis of the Zakai equation under general, especially discontinuous (jump) settings, as well as advanced numerical and approximation schemes—including those based on Galerkin projections, kernel methods, Monte Carlo, and Wong–Zakai approximations—form a major research direction at the intersection of SPDEs, stochastic analysis, and computational filtering.

1. Origins and General Formulation

The Zakai equation was originally introduced as an SPDE governing the unnormalized conditional distribution of a signal process $X_t$ given an observation process, typically of the form

$$du(t, x) = L_x u(t, x)\,dt + h(x) u(t, x)\,dY_t, \quad u(0, x) = u_0(x),$$

where $L_x$ is the (formal adjoint of the) generator of the signal process and $Y_t$ is the observation process. For systems with jumps, the equation involves additional terms accounting for the compensated jump measure: $$dv(t, x) = v(t, x)[p(x, y) - 1]\,\Theta(dt, dy) + \ldots,$$ with the structure depending on the specifics of the jump mechanisms and observation processes (Mikulevicius et al., 2010, Ceci et al., 2012). The process $v(t, x)$ encodes the unnormalized filtering density; normalization recovers the standard filter via the Kallianpur–Striebel formula.

In the case of McKean–Vlasov or mean-field type models, the Zakai equation becomes "space-distribution dependent": $$\langle \widetilde{\mu}_t, F \rangle = \langle \widetilde{\mu}_0, F \rangle + \int_0^t \langle \widetilde{\mu}_s, L_s F \rangle ds + \int_0^t \langle \widetilde{\mu}_s, F h^v(s, \cdot, \cdot; Y_s) \rangle dV_s,$$ where $L_s$ involves both pointwise and law (distribution) dependencies (Liu et al., 2020).

For time-changed or "fractional" models, fractional derivatives replace standard derivatives, reflecting memory effects: $$d\varphi(t, x) = D_t^{1-\beta} [A^*\varphi(t, x)]\,dt + \sum_k h_k(x)\varphi(t, x)\,dZ_t^{(k)}.$$ Here $D_t^{1-\beta}$ is the Riemann–Liouville fractional derivative (Umarov et al., 2013).

2. Existence, Uniqueness, and Analytical Properties

The analytic well-posedness of the Zakai equation under general conditions is highly non-trivial, especially when the signal and/or observation processes exhibit jumps or nonlocal dynamics:

• Integro-differential Zakai equations: For systems with jumps, the Zakai equation takes the form of a stochastic parabolic integro-differential equation. Existence and uniqueness in Hölder spaces of "fractional smoothness" were established, involving advanced Fourier analysis and Littlewood–Paley decompositions. Solutions are constructed as

$u(t, x) = R^x f(t, x) + R^x g(t, x),$

where these are given by Fourier inversion integrals controlled via a priori estimates (Mikulevicius et al., 2010).

• Jump-diffusion and distribution-dependent settings: For correlated jump-diffusions with both diffusive and jump noise, strong (pathwise) uniqueness of measure-valued solutions is achieved using martingale problem formulations. The equivalence between uniqueness for the KS and Zakai equation is shown, establishing robust well-posedness of the filtering equation (Ceci et al., 2012, Liu et al., 2020).
• Fractional/generalized Zakai equations: For time-changed state or observation processes, the Riemann–Liouville fractional derivative appears; existence (pathwise or in law) is shown under weaker conditions, generalizing the classical Markovian case to non-Markovian (memory) settings (Umarov et al., 2013).
• Nonlocal and homogenized equations: When non-Gaussian Lévy noise is present, the Zakai equation involves nonlocal (or fractional Laplacian) generators. Upon effective reduction (homogenization), the complex nonlocal SPDE converges (in law) to a simpler, effective SPDE, which retains or averages out the nonlocal effects depending on the integrability of the jump kernel (Lin et al., 2018).

3. Numerical and Approximation Methodologies

a. Galerkin Approximations

The Galerkin method projects the SPDE onto finite-dimensional subspaces $\mathcal{H}_n$, yielding deterministic or stochastic ODE systems for expansion coefficients. In the context of the Zakai equation: $$dq_t^{(n)} = (A^*)^{(n)}q_t^{(n)}\,dt + (B)^{(n)}q_t^{(n)}\,dZ_t + (C)^{(n)}q_t^{(n)}\,dY_t,$$ with convergence in mean-square proven under density of the basis (Frey et al., 2013). Hermite polynomial bases are especially effective due to analytical tractability and orthogonality.

b. Kernel-based Collocation Methods

Collocation methods employ positive-definite kernels (e.g., Wendland kernels) for spatial interpolation at a quasi-uniform grid of points. The time-stepping procedure updates the filter density via kernel-based interpolation at each spatial point: $$u^h(t_i, x) = u^h(t_{i-1}, x) + \sum_k L_k I(u^h(t_{i-1}, \cdot))(x) \Delta W_k(t_i).$$ Under proper conditions on the mesh and kernel, stability and error bounds $$\mathcal{O}(\sqrt{\Delta t} + (\Delta x)^{\tau - 3/2})$$ are obtained (Nakano, 2017).

c. Monte Carlo and Feynman–Kac Transformations

A modern direction involves transforming the Zakai SPDE into a random PDE with coefficients that depend functionally on observation (sample) paths: $$X_t(x) = u_{Z}(t, x) \exp(\langle h(x), Z_t \rangle),$$ where $u_Z$ solves a deterministic linear parabolic PDE for each fixed observation sample path. Using the Feynman–Kac formula, one computes $u_Z$ as an expectation over SDE trajectories: $$u_Z(t, x) = \mathbb{E}[\varphi(R^{Z, t, x}_t) \exp(\int_0^t B_Z(t-s, R^{Z, t, x}_s) ds)],$$ where $R^{Z, t, x}$ solves a drifted SDE (Beck et al., 2022). This method, coupled with Monte Carlo sampling over observation paths and SDE trajectories, achieves high efficiency and scalability—demonstrated up to 25 dimensions with practical run times.

d. Wong–Zakai and Support Theorems

Wong–Zakai approximations replace the rough driving noise (e.g., Brownian paths) with smooth approximations (typically polygonal or piecewise-linear), leading to pathwise (deterministic) equations augmented by suitable correction terms (Itô–Stratonovich corrections). Rigorous $L^p$ convergence rates for SPDEs, support theorems, and structure-preserving geometric integrators have been proven in this context (Aida et al., 2013, Yastrzhembskiy, 2018, Nakayama et al., 2019, Liu et al., 2017, Liu et al., 2019, Lanconelli et al., 2021, Cui et al., 25 Mar 2025).

4. Advanced Topics: Jumps, Fractional Models, and Nonlocality

• Non-Gaussian and jump-driven Zakai equations: The Zakai equation under general Lévy noise settings (integrable or non-integrable jump kernels) involves nonlocal operators, such as:

$(Bu)(x) = \int_{\mathbb{R}} c(z)[u(x+z) - u(x)]dz.$

Effective reduction via homogenization yields, in the integrable case, a local (diffusion) limit; in the non-integrable (α-stable) case, a fractional Laplacian persists in the effective SPDE (Lin et al., 2018).

• Fractional Zakai equations: For time-changed processes, the emergence of Riemann–Liouville fractional derivatives makes the Zakai equation intrinsically non-Markovian, encoding long-range dependence and delay effects:

$$d\varphi(t, x) = D_t^{1-\beta}[A^* \varphi(t, x)]\,dt + \sum_{k} h_k(x)\varphi(t, x)\,dZ_t^{(k)}.$$

This broadens the classical theory to encompass systems with anomalous diffusion and nonlocal memory (Umarov et al., 2013).

• Support, uniqueness, and Fokker–Planck correspondence: Strong (pathwise) uniqueness of the Zakai equation—and the established superposition principle connecting Zakai and Fokker–Planck equations—ensures well-posedness and provides important analytical tools for mean-field and interacting particle systems (Liu et al., 2020, Yastrzhembskiy, 2018).
• Low regularity and resonance-based integrators: For stochastic dispersive equations (notably, the nonlinear Schrödinger equation with white noise dispersion), resonance-based schemes and Wong–Zakai regularization enable provable strong convergence even for low-regularity ($L^2$) initial data, using phase "twisting" and exact computation of resonant integrals (Cui et al., 25 Mar 2025).

5. Practical Implications and Applications

• Filtering in jump-diffusion environments: The robust analytic theory for integro-differential and nonlocal Zakai equations directly supports nonlinear filtering when jump processes and discontinuous observations are present—scenarios typical in quantitative finance and high-frequency networked systems (Mikulevicius et al., 2010, Lin et al., 2018).
• Efficient high-dimensional computation: Transformation-based Monte Carlo and kernel-collocation methods break the curse of dimensionality, enabling practical filtering in large-scale systems. Numerically, these approaches outperform traditional grid-based or particle filter methods, as verified by case studies with high-dimensional Ornstein–Uhlenbeck signals and mixed observation models (Frey et al., 2013, Nakano, 2017, Beck et al., 2022).
• Well-posedness and algorithmic stability: Pathwise uniqueness results and support theorems provide theoretical guarantees for the correctness and stability of numerical schemes. Wong–Zakai and Galerkin approximations, combined with explicit error bounds, offer a rigorous basis for the development of filtering algorithms (Aida et al., 2013, Yastrzhembskiy, 2018, Liu et al., 2017, Nakayama et al., 2019).
• Fractional and mean-field systems: Extensions to fractional Zakai equations and distribution-dependent nonlinear filtering enable modeling of systems with long-range memory and interacting agents—from anomalous diffusive phenomena in natural sciences to mean-field control in engineered networks (Umarov et al., 2013, Liu et al., 2020).
• Numerical schemes for low-regularity data: The resonance-based Wong–Zakai integrators enable accurate numerical solution of stochastic dispersive systems at low Sobolev regularity, which is unattainable by conventional methods (Cui et al., 25 Mar 2025).

6. Connections, Limitations, and Research Directions

The range of analytical and numerical methodologies for Zakai equations is now broad, encompassing functional analytic, probabilistic, and computational perspectives. The adaptations for jumps, fractional derivatives, and nonlocality align the theory with modern stochastic dynamics observed in a host of applications. Kernel- and Monte Carlo-based methods provide practical tools for high-dimensional and non-Markovian filtering.

Remaining challenges include:

• Weakening regularity and boundedness assumptions for transformation methods in real-world signal/observation models (Beck et al., 2022).
• Deriving sharp error bounds and stability results in the presence of non-Lipschitz or distribution-dependent coefficients (Liu et al., 2020).
• Extending resonance-based integrators and Wong–Zakai-type reductions to more general SPDE filters in nonlinear and networked settings (Cui et al., 25 Mar 2025).
• Investigating homogenization and multiscale effects in nonlocal agents and anomalous diffusion (Lin et al., 2018).

The Zakai equation remains a central analytical and numerical object for stochastic filtering, with ongoing advances positioning it as a key interface between probability, analysis, numerical methods, and data assimilation.