A stochastic state-space model extends a deterministic state model with random disturbances and measurement uncertainty.

A common discrete-time linear model is

$$\mathbf x_k=A\mathbf x_{k-1}+B\mathbf u_k+\mathbf w_k,$$

$$\mathbf z_k=H\mathbf x_k+\mathbf v_k.$$

The process noise $\mathbf w_k$ represents disturbances or model error that affect how the hidden state evolves. The measurement noise $\mathbf v_k$ represents uncertainty in the observations.

Their covariance matrices are commonly written

$$Q=\operatorname{Cov}(\mathbf w_k),\qquad R=\operatorname{Cov}(\mathbf v_k).$$

The state $\mathbf x_k$ is therefore no longer known exactly even when the previous state estimate and input are known: uncertainty propagates through the dynamics and new uncertainty is added at each step.

The measurement need not reveal the state directly. The matrix $H$ maps the hidden state into the quantities observed by sensors.

This model separates three things: deterministic dynamics, uncertainty in those dynamics, and uncertainty in measurement. State estimators such as the Kalman filter use all three.