Nonlinear filtering with continuous time perfect observations and noninformative quadratic variation

We consider the problem of estimating the state of a diffusion process, based on continuous time observations in singular noise. As long as the observations are regular values of the observation function, we derive an equation for the density (w.r.t. the canonical Lebesgue measure on the corresponding level set) of the conditional probability distribution of the state, given the past observations. The proof is based on the idea of decomposition of solutions of SDE, as introduced by Kunita (1981).