Kalman Filters

The Kalman filter estimates the hidden state of a linear dynamical system from noisy observations, recursively.

State equation: $x_{t+1} = F x_t + w_t$, with $w_t \sim N(0, Q)$.

Observation: $y_t = H x_t + v_t$, with $v_t \sim N(0, R)$.

Each step has two phases. Predict: project the previous state estimate forward using $F$ and increase its uncertainty. Update: combine the prediction with the new observation, weighting each by inverse variance — the Kalman gain optimally blends old and new information.

In finance, Kalman filters power yield-curve modeling (extracting unobservable factors from observed bond prices), dynamic hedge ratio estimation, and any setting where you need to track a slow-moving latent variable from noisy observations. They're optimal for linear-Gaussian systems; for non-linear or non-Gaussian, extensions like the EKF, UKF, and particle filter take over.