  In this paper we consider the problem of minimizing a quadratic functional
for a discrete-time linear stochastic system with multiplicative noise, on a
standard probability space, in infinite time horizon. We show that the
necessary and sufficient conditions for the existence of the optimal control
can be formulated as matrix inequalities in frequency domain. Furthermore, we
show that if the optimal control exists, then certain Lyapunov equations must
have a solution. The optimal control is obtained by solving a deterministic
linear-quadratic optimal control problem whose functional depends on the
solution to the Lyapunov equations. Moreover, we show that under certain
conditions, solvability of the Lyapunov equations is guaranteed. We also show
that, if the frequency inequalities are strict, then the solution is unique up
to equivalence.
