Introduction to Local Level Model and Kalman Filter

Steady State Kalman Filter
Kalman filter converges to a positive value, say Pt → ¯ P. We would
then have
Ft → ¯P + σ 2 ε, Kt → ¯P/(¯P + σ 2 ε).
The state prediction variance updating leads to
¯P = ¯P
1 −
which reduces to the quadratic
¯P
¯P + σ 2 ε
x 2 − xq − q = 0,
+ σ 2 η,
where x = ¯ P/σ 2 ε and q = σ 2 η/σ 2 ε, with solution
¯P = σ 2
ε q + q2 + 4q /2.