Kalman filter can accommodate time varying system matrices. Equations to run the filter are the same and it preserves its optimality under linear gaussian model.
My question is the following:
Can the evolution of time varying system matrices be stochastic? In some references I seem to read between the lines that they should evolve deterministically. Does it mean that the entire filter breaks or do we simply lose optimality by making them stochastic?
For reference, please peek at section 3.2 of the following paper:
http://www.ims.cuhk.edu.hk/~cis/2012.1/CIS%2012-1-05.pdf
A similar comment is in Harvey's book on Kalman Filter.
Best Answer
If your dynamic system is $$ x_t = A_t x_{t-1} + \eta_t $$ $$ y_t = B_t x_t + \varepsilon_t $$ Then when people say system matrices $A_t, B_t$ should be deterministic, this means that Kalman Filter gives you an estimate of state $x_t$ conditional on past and current values of parameters $$\mathbf E\left(x_t|\,y_t,\dots,y_1, \,A_t,\dots,A_1, \,B_t, \dots, B_1\right).$$
So when you do a filtering step to estimate this conditional expectation of state, you consider those matrices to be already known (observed) rather than unknown and random. Of course they can be realizations of some external random process (which is often the case) or be deterministic functions of time - this doesn't matter much.
What seems authors in above paper describe in 3.2 is an extension of KF when they assume $A_t, B_t$ to be random but they don't what to condition on their values when filtering. So they don't assume matrices to be known at the moment of filtering, but rather assume that they come from a distribution with known mean/variance.