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In this paper, we consider the class of infinite-dimensional discrete-time linear systems with multiplicative random disturbances (i.e. with the state multiplied by a random sequence), also known as stochastic bilinear systems. We formulate and solve the quadratic optimal-control problem for this class of systems subject to an arbitrary additive stochastic £ 2 input disturbance. Under assumptions that guarantee the existence of a solution to an algebraic Riccati-like operator equation (derived previously by the authors), we characterize a bounded linear operator that takes the additive stochastic (.2 input disturbance and the initial condition into the optimal control law. Such a result generalizes, to the infinite-dimensional bilinear stochastic case, some known results for the deterministic linear case.