After looking various sources (1), I have found following equations for system description in kalman filter:
Measurement Equation as: $$ y_t=C_t x_t+r_t \tag{1} $$
and state vector equation as: $$x_{t+1} = A_t x_t + q_t \tag{2}$$
where $r_t \sim N(0, R_t)$ and $q_t \sim N(0, Q_T)$.
I have simple question. Why in equation 2 state vector equation is considered for $x_{t+1}$? Should we have $q_{t+1}$ on the right hand side? What would be the implication for the model if we have considered following equation for state vector equation: $$x_{t} = A_t x_{t-1} + q_t \tag{3}$$
Implication of equation 2 and 3:
If we consider equation 1 and 2, and given the filtration $F_{t-1}$, then in equation 1, only $r_t$ is random variable because $x_t$ is completely determined because $x_t = f(x_{t-1}, q_{t-1})$ but same can not be said about the $x_t$ in equation 1 if we consider equation 1 and 3. In equation 3, $x_t = f(x_{t-1}, q_t)$, it means that given the filtration $F_{t-1}$, $x_t$ is still random variable.
It means conditional expectation of $y_t$ considering (1) and (2) will be: \begin{align} E(y_t|F_{t-1}) =& C_t \, x_t + E(r_t|F_{t-1})\\ =&C_tx_t \tag{4} \end{align} Since, $x_t$ is completely determined given $F_{t-1}$.
If we considered equation (1) and equation (3), then conditional expectation of $y_t$ is :
\begin{align} E(y_t|F_{t-1}) =& C_t \, E(x_t|F_{t-1}) + E(r_t|F_{t-1})\\ =&C_t E(x_t|F_{t-1}) \tag{5} \end{align}
I think both the expression in equation 4 and 5 are different.
Thanks!