A two-criteria dynamic stochastic decision problem in separable Hilberi space with two different information patterns is discussed. The cost functional is assumed to be quadratic. Under the perfect measurement information pattern, three types of disturbance—control-dependent noise, state-dependent noise, and purely additive noise—are taken into account. Under the one-step delay observation sharing pattern and the assumption of underlying gaussian statistics, some existing theorems in finite dimensional space are extended. The explicit solutions are derived for both cases. The results in this paper can be used to study the partial differential games when the system is specialized to discrete-time distributed parameter system. Furthermore, some results are new even in a finite-dimensional space setting.