This paper deals with the computation of upper bounds for the state covariance matrix of discrete-time linear systems subject to stochastic excitation and additive time-varying uncertainty in the system dynamic matrix. Such upper bounds are obtained as the stabilizing solutions of suitable Hinfinity-type Riccati equations. A necessary and sufficient condition for the existence of such solutions is given in terms of the Hinfinity-norm of a suitable transfer function. As for the computation of the optimal bound, it is demonstrated that the bounds are a convex function of a scalar parameter, so that efficient numerical schemes can be worked out.