Koopman's Kalman Filter Derivation

Question

i am currently doing my undergraduate thesis which uses State Space Model and Kalman Filter as my main model. The State Space Model and Kalman Filter is based on the one used in "Time Series Analysis by State Space Methods 2nd edition (2012)" by J. Durbin and S. J. Koopman. The state and observation equation is shown as follow

Observation Eq : $y_t=Z_tα_t+ε_t, ε_t$ ~ $N(0,H_t)$

State Eq: $α_{t+1}=T_tα_t+R_tη_t, η_t$ ~ $N(0,Q_t)$

There is a part of a derivation that shows :

$v_t=y_t-Z_ta_t=Z_tα_t+ε_t-Z_ta_t$

$Cov(α_t,v_t)=E[α_tv_t'|Y_{t-1}]=E[α_t(Z_tα_t+ε_t-Z_ta_t)'|Y_{t-1}]=E[α_t(α_t-a_t)'Z'_t|Y_{t-1}]$

Why did the $ε_t$ is directly ignored? i understand that the mean of $ε_t$ is 0, but doesn't it got multiplicated by $α_t$ first?, wouldn't there be $E[α_tε_t|Y_{t-1}]$ ? Any help would be meaningful. Thank you in advance!!

Answer 1

That term still goes to zero when you evaluate the expected value. The measurement noise is independent of the state, so we can write $E[\alpha_t \varepsilon_t] = E[\alpha_t] \cdot E[\varepsilon_t] = E[\alpha_t] \cdot 0 = 0$