A backward stochastic differential equation (BSDE) is an Ito stochastic differential equation (SDE) for which a random terminal condition on the state has been specified. The paper deals with estimation problems for partly observed stochastic processes described by linear SDEs with uncertain disturbances. The disturbances and unknown initial states are supposed to be constrained by the inequality including mathematical expectation of the integral quadratic cost. We consider our equations as BSDEs, and construct at given instant the random information set of all possible states which are compatible with the measurements and the constraints. The center of this set represents the best estimation of the process' state. The evolutionary equations for the random information set and for the best estimation are given. Some examples and applications are considered.