Continuity of the Filter with Unbounded Observation Coefficients

In this article, we study the continuity with respect to the trajectories of the observation process for the filter associated with nonlinear filtering problems when the coefficients depend on both the signal and the observation and the observation coefficient is unbounded. To achieve this task we define a formal unnormalized filter and prove by limiting arguments that it is related to the original filter through a generalized Bayes formula, and is locally Lipschitz continuous with respect to the uniform norm.