We study a nonlinear filtering problem in which the signal to be estimated is conditioned by the observations. The main results establish pathwise uniqueness for the unnormalized filter equation and uniqueness in law for the normalized and unnormalized filter equations.
"On Uniqueness of Solutions for the Stochastic Differential Equations of Nonlinear Filtering." Ann. Appl. Probab. 11 (1) 182 - 209, February 2001. https://doi.org/10.1214/aoap/998926990