The LQG problem for stochastic continuous-time systems subject to multirate sampling of both input and output variables is considered. By restating the problem as a discrete-time periodic LQG problem, a sufficient condition for the existence of an optimal stabilizing regulator is given in terms of the structural properties of the original system and the cost function. This improves on previous contributions, where optimal control schemes were proposed without addressing existence and/or stability issues. The possibility of incorporating an integral action within the optimal LQG regulator is also briefly discussed.