We establish the innovations conjecture for a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The approach uses only elementary stochastic analysis, together with a variant due to J.M.C. Clark of a theorem of Yamada and Watanabe on pathwise-uniqueness and strong solutions of stochastic differential equations.
"On the Innovations Conjecture of Nonlinear Filtering with Dependent Data." Electron. J. Probab. 13 2190 - 2216, 2008. https://doi.org/10.1214/EJP.v13-585