  We study linear stochastic partial differential equations of parabolic type.
We consider a new boundary value problem where a Cauchy condition is replaced
by a prescribed average of the solution either over time and probabilistic
space for forward SPDEs and over time for backward SPDEs. Well-posedness,
existence, uniqueness, and a regularity of the solution for this new problem
are obtained. In particular, this can be considered as a possibility to recover
a solution of a forward SPDE in a setting where its values at the initial time
are unknown, and where the average of the solution over time and probability
space is observable, as well as the input processes.
