Nonlinear filtering with perfect discrete time observations

We consider the problem of estimating the state of a diffusion process, based on discrete time observations in singular noise. We reduce the problem to a static problem, and we show that the solution is provided by the area or co-area formula of geometric measure theory, provided the observed value is a regular value of the observation function. In order to address the case of singular values, we propose another approach, based on small-noise perturbation and asymptotics of Laplace integrals.