Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/eigenvector
The progress given so far appears to be correct. Finishing this problem just requires us to argue that $H$ must be chosen so that the equation
$$
(D' + H'(A - I)) x + (F + H'B)'(F + H'B) = \eta
$$
holds for all $x$. Thus, we must have $D' + H'(A - I) = 0$ and, consequently,
$H' = D'(I-A)^{-1}$. This implies that $\eta = (F' + H'B)'(F' + H'B)$.