Stochastic Filtering

Stochastic filtering is the problem of estimating the state of a dynamical system from noisy observations. It is fundamental to signal processing, control theory, and econometrics.

Problem setup

State: $X_t$ evolves as $dX_t = b(X_t) dt + \sigma dW_t$ (hidden process).
Observation: $Y_t = \int_0^t h(X_s) ds + V_t$ (noisy measurement).
Goal: Compute $\hat{X}_t = \mathbb{E}[X_t \mid \mathcal{F}_t^Y]$ , the conditional expectation given observations up to time $t$ .

Kalman filter

Theorem8.1Kalman filter

For linear Gaussian systems:

$dX_t = A X_t dt + dW_t, \quad dY_t = C X_t dt + dV_t,$

the optimal filter is:

$d\hat{X}_t = A \hat{X}_t dt + K_t (dY_t - C \hat{X}_t dt),$

where $K_t$ is the Kalman gain, computed from the Riccati equation. The filter is linear and optimal (minimum mean-square error).

ExampleGPS navigation

A GPS receiver observes noisy position measurements. The Kalman filter combines these observations with a model of vehicle dynamics (e.g., constant velocity) to produce smoothed position estimates.

Nonlinear filtering

For nonlinear systems, the Zakai equation or Kushner-Stratonovich equation governs the conditional density. Practical implementations use particle filters (sequential Monte Carlo) or extended Kalman filters (linearization).

Summary

Stochastic filtering uses martingale theory and stochastic calculus to optimally estimate hidden states from noisy observations, with applications to navigation, robotics, finance, and signal processing.