Given a discrete-time linear time-varying system (LTV)

$$x(k+1) = A(k) x(k) + B(k) u(k)$$

where $A(k)$ and $B(k)$ are generated by a stationary random process. Is there an equivalent linear time-invariant (LTI) system which will calculate the expected trajectory of $x(k)$?

$$\mathbb E[x(k+1)] = z(k+1) = A_{\text{eq}} z(k) + B_{\text{eq}} u(k)$$

If so, how is it calculated?

If not, are there conditions where this can be calculated?