Filtering of diffusions controlled through their conditional measures

For the closed-loop nonlinear filtering problem with control in separated form (a functional of the conditional distribution measure), the Kallianpur-Striebel formula becomes a stochastic equation for the unnormalized conditional distribution, given the past of the observations. Existence, uniqueness and measurability of solutions to this equation are discussed. The results give a partially positive answer to the question of admissibility of separated control laws. However, "pathological" nonanticipative but noncausal solutions appear here, after the manner of Tsirelson's example.