Characterizing finite dimensional filters for the linear innovations of continuous time random processes

Motivated by recent work on nonlinear filtering, we examine the finite dimensional realizability issues associated with certain linear filtering problems. These problems depend only on the second order statistics of the underlying random process, with optimal linear filters being determined by linear integral equations of the Wiener-Hopf type. We are able to exploit linearity of the filter (i.e. of its linear input/output map) to obtain the conditions which are valid for a fairly general class of realizations, including linear and nonlinear ones. We show that a "semiseparable" covariance structure is necessary and sufficient for finite dimensionality, just as in the case when only linear realizability is considered.