Kalman Filter Optimal State Estimation and Riccati Recursion

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding State Space Models / Optimal Linear Filtering.

Your Task
Consider the linear discrete-time state-space model: $x_k = F x_{k-1} + w_k, \quad w_k \sim \mathcal{N}(0, Q)$ $y_k = H x_k + v_k, \quad v_k \sim \mathcal{N}(0, R)$ Derive the Kalman Filter measurement update equations via orthogonal projection (BLUE): $K_k = P_{k|k-1} H^T (H P_{k|k-1} H^T + R)^{-1}$ $\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (y_k - H \hat{x}_{k|k-1})$ $P_{k|k} = (I - K_k H) P_{k|k-1}$

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.