A new performance measure for stochastic optimization in Hilbert space

A stochastic optimization theory predicated on the minimization of the memoryless part of an error covariance operator is formulated. The existence of an appropriate memoryless part transformator is verified and its properties delineated. The resultant memoryless part of the error covariance operator is then used as a performance measure in a stochastic filtering problem which is minimized in the partial ordering of the positive hermitian operators. This results in an explicit solution to the filtering problem which formally replicates the classical solution obtained via a mean squared error criterion but which bypasses the restrictive hypotheses required to guarantee the existence of a mean squared error.