Stochastic filtering involves estimating the state of a system that evolves over time and is subject to random disturbances, using observations that are themselves noisy and potentially incomplete. Stochastic evolution models are integral to this process as they mathematically describe how the system's state and the observations evolve. These models find applications in fields such as signal processing, finance, robotics, and control theory.

1. Overview of Stochastic Filtering

• Objective: The goal of stochastic filtering is to estimate an unobservable state $X(t)$ of a system using noisy observations $Y(t)$ . The true state evolves according to a stochastic differential equation (SDE), while the observations are also modeled by an SDE.
• Typical Problem: Given a system modeled by:

$dX(t) = f(X(t), t) dt + \sigma(X(t), t) dW_t^X,$

$dY(t) = h(X(t), t) dt + \gamma(X(t), t) dW_t^Y,$

where $W_t^X$ and $W_t^Y$ are independent Wiener processes, the task is to estimate $X(t)$ based on the information provided by $Y(t)$ .

2. Mathematical Framework

State Process (Signal)

• Evolution Equation: The state $X(t)$ follows a stochastic differential equation that captures the system's dynamics, typically represented as:

$dX(t) = a(X(t), t) dt + b(X(t), t) dW_t^X.$

Here, $a(X(t), t)$ is the drift term, and $b(X(t), t)$ is the diffusion coefficient.

Observation Process

• Measurement Equation: The observation $Y(t)$ is modeled as:

$dY(t) = c(X(t), t) dt + \sigma(Y(t), t) dW_t^Y,$

where $c(X(t), t)$ is the observation function, and $\sigma(Y(t), t)$ represents the noise affecting the measurement.

3. Examples of Stochastic Filtering Models

1. Kalman Filter (Linear Case)

• Model: Assumes linear dynamics with Gaussian noise. The state and observation equations are:

$dX(t) = AX(t) dt + Q dW_t^X,$

$dY(t) = CX(t) dt + R dW_t^Y,$

where $A$ and $C$ are matrices defining the system and observation models, and $Q$ and $R$ are noise covariance matrices.

• Properties: Provides an optimal estimator when both the process and observation noises are Gaussian and the system dynamics are linear.

2. Extended Kalman Filter (EKF)

• Use Case: Applies to non-linear systems by linearizing them around the current estimate.
• Approximation: The model is linearized using Taylor series expansion to update the state estimate and error covariance iteratively.

3. Particle Filter (Non-linear and Non-Gaussian)

• Approach: A Monte Carlo method that represents the probability distribution of the state using a set of weighted samples (particles).
• Update: The algorithm uses a sequential importance resampling approach to update the particle weights based on the likelihood of observations.
• Advantages: Can handle non-linear and non-Gaussian models, making it highly versatile.