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In this paper we consider the problem of minimizing a quadratic functionalfor a discrete-time linear stochastic system with multiplicative noise, on astandard probability space, in infinite time horizon. We show that thenecessary and sufficient conditions for the existence of the optimal controlcan be formulated as matrix inequalities in frequency domain. Furthermore, weshow that if the optimal control exists, then certain Lyapunov equations musthave a solution. The optimal control is obtained by solving a deterministiclinear-quadratic optimal control problem whose functional depends on thesolution to the Lyapunov equations. Moreover, we show that under certainconditions, solvability of the Lyapunov equations is guaranteed. We also showthat, if the frequency inequalities are strict, then the solution is unique upto equivalence.

Frequency Theorem for discrete time stochastic system with multiplicative noise