The Zakai equation is a fundamental equation in the theory of stochastic filtering, which deals with estimating the state of a stochastic process based on noisy observations. It plays an essential role in the mathematical formulation of optimal filtering problems, especially when dealing with nonlinear and non-Gaussian processes.

1. Stochastic Filtering Problem Overview

The goal of stochastic filtering is to estimate the state $X_t$ of a system at time $t$ , given noisy observations $Y_t$. The state process $X_t$ is typically modeled by a stochastic differential equation (SDE):

$dX_t = b(X_t, t) \, dt + \sigma(X_t, t) \, dW_t,$

where $W_t$ is a standard Brownian motion, $b$ is the drift term, and $\sigma$ is the diffusion coefficient.

The observation process $Y_t$ is often given by:

$dY_t = h(X_t, t) \, dt + dV_t,$

where $V_t$ is another Brownian motion independent of $W_t$ , and $h$ represents the observation function.

2. The Zakai Equation

The Zakai equation provides a way to describe the unnormalized conditional probability density of the state $X_t$ given observations up to time $t$ . It is a linear stochastic partial differential equation (SPDE) for the unnormalized conditional density $\pi_t$ .

The Zakai equation is expressed as:

$d\pi_t(x) = \mathcal{L}^* \pi_t(x) \, dt + \pi_t(x) h(x, t) \cdot (dY_t - \bar{h}_t \, dt),$

where:

• $\pi_t(x)$ is the unnormalized conditional density of $X_t$ .
• $\mathcal{L}^*$ is the adjoint of the generator $\mathcal{L}$ of the state process $X_t$ .
• $\bar{h}_t = \int h(x, t) \pi_t(x) \, dx$ is the expectation of $h(x, t)$ under the current unnormalized density $\pi_t$ .

3. Significance of the Zakai Equation

• Simplification for Numerical Methods: The Zakai equation is easier to handle numerically than the Kushner-Stratonovich equation, which is a nonlinear filtering equation for the normalized conditional density.
• Linear Form: The linearity of the Zakai equation allows for certain mathematical tools and techniques (e.g., SPDE methods) to be used for analysis and approximation.
• Connection to Conditional Density: The normalized conditional density $\rho_t(x)$ , representing the true posterior distribution of $X_t$ given $Y_t$ , can be obtained from $\pi_t(x)$ by normalizing:

$\rho_t(x) = \frac{\pi_t(x)}{\int \pi_t(x) \, dx}.$

4. Derivation and Intuition

The Zakai equation is derived using a change of measure technique on the probability space, leading to a martingale representation of the filtering problem. The derivation involves:

• Girsanov’s Theorem: Used to change the measure under which the observed process $Y_t$ is a standard Brownian motion.
• Stochastic Calculus: It incorporates Itô's calculus for handling the stochastic integrals that arise in the derivation.

5. Applications

• Signal Processing: The Zakai equation helps in filtering problems where the goal is to recover signals that are corrupted by noise.