This paper is concerned with existence and optimality properties of so-called guaranteed cost controllers for an uncertain system subject to structured uncertainty. The uncertainty in the system is assumed to have a stochastic character and to satisfy certain stochastic integral constraints. It is shown that a minimax optimal guaranteed cost state feedback controller for a stochastic system can be synthesized as a state feedback controller absolutely stabilizing this system. For each initial state of the system, this controller can be found by parametric optimization of solutions of a parameter-dependent generalized matrix Riccati equation arising in stochastic H-infinity theory.