  The explicit solution of the discrete time filtering problems with
exponential criteria for a general Gaussian signal is obtained through an
approach based on a conditional Cameron-Martin type formula. This key formula
is derived for conditional expectations of exponentials of some quadratic forms
of Gaussian sequences. The formula involves conditional expectations and
conditional covariances in some auxiliary optimal risk-neutral filtering
problem which is used in the proof. Closed form recursions of Volterra type for
these ingredients are provided. Particular cases for which the results can be
further elaborated are investigated.
