Filtering of a reflected Brownian motion with respect to its local time

We consider a filtering problem when the state process is a reflected Brownian motion X t and the observation process is its local time Λ s , for s ≤ t . For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λ t . The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azéma martingale.